Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games

Manuscript submitted to AIMS Journal doi:10.3934/xx.xxxxxxx MINIMAX SOLUTIONS OF P A TH-DEPENDENT HAMIL TON–JA COBI EQUA TIONS UNDER WEAKENED ASSUMPTIONS WITH APPLICA TION TO DIFFERENTIAL GAMES Gomoyuno v Mikhail  1 , 2 1 Krasovskii Institute of Mathematics and Mechanics Ural Branc h of Russian Academ y of Sciences, Russia 2 Ural F ederal University , Russia (Comm unicated b y Handling Editor) Abstract. W e study minimax (generalized) solutions of a Cauch y problem for a (ﬁrst-order) path-dep endent Hamilton–Jacobi equation with co-in v arian t deriv ativ es under a righ t-end b oundary condition. Under assumptions on the Hamiltonian that are more general than those previously considered in the lit- erature and allow, in particular, a measurable dep endence on the ﬁrst (time) v ariable, we establish existence, uniqueness, stability , and consistency results for minimax solutions. As an application, we consider a zero-sum diﬀerential game for a time-delay system and prov e that this game has a v alue under assumptions more general than the kno wn ones but rather natural being con- sistent with the Carath´ eodory conditions. 1. In tro duction and ov erview of the main results. 1.1. P ath-dep enden t Hamilton–Jacobi equation. Let num bers n ∈ N , T > 0, h ≥ 0 b e ﬁxed. Let C ([ − h, T ] , R n ) b e the Banach space of all contin uous functions x : [ − h, T ] → R n with the standard norm ∥ x ( · ) ∥ ∞ : = max τ ∈ [ − h,T ] ∥ x ( τ ) ∥ , where ∥ · ∥ denotes the Euclidean norm in R n . Giv en ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), w e deﬁne a function x ( · ∧ t ) ∈ C ([ − h, T ] , R n ) by x ( τ ∧ t ) : = x ( τ ) if τ ∈ [ − h, t ] and b y x ( τ ∧ t ) : = x ( t ) if τ ∈ ( t, T ]. W e assume that the segmen t [0 , T ] is equipp ed with the Leb esgue measure µ and let L 1 ([0 , T ] , R ) b e the linear space of all measurable functions c : [0 , T ] → R suc h that ∥ c ( · ) ∥ 1 : = R T 0 | c ( τ ) | d τ < + ∞ . W e consider a Cauchy pr oblem for a p ath-dep endent Hamilton–Jac obi e quation ∂ t φ ( t, x ( · )) + H  t, x ( · ) , ∇ φ ( t, x ( · ))  = 0 , (1) where ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), under a right-end b oundary c ondition φ ( T , x ( · )) = σ ( x ( · )) , (2) where x ( · ) ∈ C ([ − h, T ] , R n ). In this problem, φ : [0 , T ] × C ([ − h, T ] , R n ) → R is an unknown functional , ∂ t φ ( t, x ( · )) ∈ R and ∇ φ ( t, x ( · )) ∈ R n are so-called c o-invariant 2020 Mathematics Subje ct Classiﬁc ation. Primary: 35F25, 49L12; Secondary: 49L25, 49N70. Key wor ds and phr ases. P ath-dep endent Hamilton–Jacobi equation, co-inv ariant deriv atives, minimax solution, well-posedness, semiclassical solution, zero-sum diﬀerential game, time-delay system, game v alue. This work is supp orted by RSF grant 25-11-00269, https://rscf.ru/en/pro ject/25-11-00269/ . 1 2 MIKHAIL GOMOYUNO V derivatives of this functional at a p oin t ( t, x ( · )) (see Deﬁnition 2.1 in Section 2 ), the Hamiltonian H : [0 , T ] × C ([ − h, T ] , R n ) × R n → R as w ell as the b oundary functional σ : C ([ − h, T ] , R n ) → R are giv en. The goal of the paper is to dev elop a theory of generalized (in the minimax sense) solutions of the Cauch y problem ( 1 ), ( 2 ) under the following standing assumption on H and σ . Assumption 1.1. The following conditions hold. (A.1) F or any ﬁxed x ( · ) ∈ C ([ − h, T ] , R n ), s ∈ R n , the function t 7→ H ( t, x ( · ) , s ), [0 , T ] → R , is measurable. (A.2) There exists a set E (A . 2) ⊂ [0 , T ] with µ ( E (A . 2) ) = T and such that the mapping below is contin uous for every ﬁxed t ∈ E (A . 2) : ( x ( · ) , s ) 7→ H ( t, x ( · ) , s ) , C ([ − h, T ] , R n ) × R n → R . (3) (A.3) F or ev ery compact set D ⊂ C ([ − h, T ] , R n ), there exists a non-negativ e function λ H ( · ) : = λ H ( · ; D ) ∈ L 1 ([0 , T ] , R ) satisfying the following prop ert y: for any x 1 ( · ), x 2 ( · ) ∈ D , s ∈ R n , there exists a set E (A . 3) : = E (A . 3) ( D , x 1 ( · ) , x 2 ( · ) , s ) ⊂ [0 , T ] with µ ( E (A . 3) ) = T and suc h that, for all t ∈ E (A . 3) , | H ( t, x 1 ( · ) , s ) − H ( t, x 2 ( · ) , s ) | ≤ λ H ( t )(1 + ∥ s ∥ ) ∥ x 1 ( · ∧ t ) − x 2 ( · ∧ t ) ∥ ∞ . (4) (A.4) There exist a non-negativ e function c H ( · ) ∈ L 1 ([0 , T ] , R ) and a set E (A . 4) ⊂ [0 , T ] with µ ( E (A . 4) ) = T such that | H ( t, x ( · ) , s 1 ) − H ( t, x ( · ) , s 2 ) | ≤ c H ( t )(1 + ∥ x ( · ∧ t ) ∥ ∞ ) ∥ s 1 − s 2 ∥ (5) for all t ∈ E (A . 4) , x ( · ) ∈ C ([ − h, T ] , R n ), s 1 , s 2 ∈ R n . (A.5) F or ev ery compact set D ⊂ C ([ − h, T ] , R n ), there exists a non-negativ e function m H ( · ) : = m H ( · ; D ) ∈ L 1 ([0 , T ] , R ) satisfying the following prop ert y: for ev ery x ( · ) ∈ D , there exists a set E (A . 5) : = E (A . 5) ( D , x ( · )) ⊂ [0 , T ] with µ ( E (A . 5) ) = T and suc h that, for all t ∈ E (A . 5) , | H ( t, x ( · ) , 0) | ≤ m H ( t ) . (6) (A.6) The functional σ is contin uous. Note that (see Corollary 2.10 in Section 2 ) conditions (A.1)–(A.5) imply the follo wing property of the Hamiltonian H : for every compact set D ⊂ C ([ − h, T ] , R n ), there exists a set E H : = E H ( D ) ⊂ [0 , T ] with µ ( E H ) = T and such that, for all t ∈ E H , x ( · ) ∈ D , s ∈ R n , H ( t, x ( · ) , s ) = H ( t, x ( · ∧ t ) , s ) . (7) This pr op erty of non-anticip ation plays an imp ortan t role and justiﬁes the require- men t that a solution φ of the Cauch y problem ( 1 ), ( 2 ) should b e non-anticip ative , whic h means that, for all ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), φ ( t, x ( · )) = φ ( t, x ( · ∧ t )) . Recall that Cauc hy problems of t yp e ( 1 ), ( 2 ) arise, for example, as Bellman and Isaacs equations in (deterministic) optimal control problems and diﬀerential games for functional diﬀerential equations of retarded type. Similarly to the classical (i.e., non-path-dep enden t) case, the Cauch y problem ( 1 ), ( 2 ) rarely has a solution in terpreted in the classical sense (even under far more restrictive assumptions than Assumption 1.1 ), and, thus, appropriate generalized solutions should be considered. P A TH-DEPENDENT HJ EQUA TIONS 3 With regard to the Cauc hy problem ( 1 ), ( 2 ), the minimax appr o ach to the notion of a generalized solution (see, e.g., [ 35 , 36 , 37 ]) was historically the ﬁrst to b e de- v elop ed. The reader is referred to [ 29 , 18 , 17 ] for a discussion of the corresp onding results obtained under v arious groups of assumptions on H and σ and also to a re- cen t pap er [ 1 ]. Later, a n um b er of pap ers appeared devoted to the developmen t of the visc osity solution te chnique [ 14 , 12 ] (see also, e.g., [ 13 ]). W e limit ourselv es here to referring to [ 26 , 15 , 34 , 16 , 4 , 40 , 10 , 33 , 11 , 22 , 41 , 19 ] (see also a discussion in [ 17 , Section 3.8]). It should be emphasized that it is the theory of minimax solutions that has b een constructed in its most complete form and under the most general assumptions b y now. T o some extent, this is due to the fact that the minimax ap- proac h partially av oids the diﬃculties caused by the fact that the Hamilton–Jacobi equation ( 1 ) is considered o v er the inﬁnite dimensional path space C ([ − h, T ] , R n ), in which the prop ert y of lo cal compactness is absent and the norm has rather po or diﬀeren tiability prop erties. Since, in this pap er, we make a next step and further w eaken assumptions on H and σ to Assumption 1.1 , it seems natural to take this approac h as a basis. The p ossibilit y of dev eloping the viscosity approac h under Assumption 1.1 is an interesting question, but it is b ey ond the scop e of this pap er. 1.2. Discussion of the standing assumption. Let us discuss Assumption 1.1 in a more detailed wa y . In the case where a Hamiltonian H is c ontinuous , the most general assumption under which the theory of minimax solutions was developed is the following (see, e.g., [ 18 ] and also [ 17 , Section 3.1]). Assumption 1.2. The following conditions hold. (B.1) The mapping H : [0 , T ] × C ([ − h, T ] , R n ) × R n → R is con tinuous. (B.2) F or every compact set D ⊂ C ([ − h, T ] , R n ), there exists a num b er λ H : = λ H ( D ) > 0 such that, for all t ∈ [0 , T ], x 1 ( · ), x 2 ( · ) ∈ D , s ∈ R n , | H ( t, x 1 ( · ) , s ) − H ( t, x 2 ( · ) , s ) | ≤ λ H (1 + ∥ s ∥ ) ∥ x 1 ( · ∧ t ) − x 2 ( · ∧ t ) ∥ ∞ . (B.3) There exists a num b er c H > 0 such that | H ( t, x ( · ) , s 1 ) − H ( t, x ( · ) , s 2 ) | ≤ c H (1 + ∥ x ( · ∧ t ) ∥ ∞ ) ∥ s 1 − s 2 ∥ for all t ∈ [0 , T ], x ( · ) ∈ C ([ − h, T ] , R n ), s 1 , s 2 ∈ R n . (B.4) The functional σ : C ([ − h, T ] , R n ) → R is con tinuous. It is clear that Assumption 1.2 imply Assumption 1.1 . The case of a time-me asur able Hamiltonian H (i.e., measurable with respect to the ﬁrst v ariable t and con tinuous with resp ect to other v ariables x ( · ) and s ) w as considered in the recen t paper [ 1 ], which served as a main motiv ation for the presen t study . In that pap er, a path-dep enden t Hamilton–Jacobi equation of a more general type was considered, in which the Hamiltonian H may depend on φ ( t, x ( · )) also. Nevertheless, the sp eciﬁcation of Assumptions 4.1 and 4.2 from [ 1 ] in relation to the Cauc hy problem ( 1 ), ( 2 ) shows that, in essence, the diﬀerence is the follo wing: instead of integrable functions λ H ( · ), c H ( · ), and m H ( · ) in, resp ectiv ely , conditions (A.3)–(A.5), num b ers λ H , c H , and m H are inv olved, which is clearly more restrictiv e. In addition, note that the (lo cal) boundedness condition (A.5) is clearly weak er than the corresp onding condition (vi) of Assumption 4.2 from [ 1 ], which has the form of a (global) sublinear growth condition (in the spirit of condition (A.4)). It is worth emphasizing that suc h a weak ening of the assumptions seems rather natural and imp ortan t since it is consistent with the Car ath ´ eo dory 4 MIKHAIL GOMOYUNO V c onditions in the theory of ordinary and functional diﬀerential equations and allows us to cov er more general classes of optimal control problems and diﬀerential games in applications. On the other hand, this w eakening requires some eﬀorts and is therefore not purely formal. Th us, to the b est of our knowledge, there are no results in the literature devoted to the developmen t of the theory of generalized solutions of the Cauc hy problem ( 1 ), ( 2 ) under rather general and natural Assumption 1.1 . The presen t pap er is in tended to ﬁll this gap. In conclusion of the discussion, it is also necessary to note that, in the the- ory of minimax solutions of (non-path-dep enden t) Hamilton–Jacobi equations with ﬁrst-order partial deriv atives, a group of conditions closest to Assumption 1.1 was considered in [ 38 , Chapter 12] (in this connection, see also, e.g., [ 39 ]). Ho wev er, there w as an additional assumption of p ositive homo geneity of a Hamiltonian H with resp ect to the third v ariable s . Among the related papers on the theory of viscosit y solutions of such equations, we mention [ 21 , 2 , 24 ] (see also [ 8 ]). 1.3. Deﬁnition of a minimax solution and consistency . In the pap er, w e pro- p ose notions of upp er , lower , and minimax solutions of the Cauch y problem ( 1 ), ( 2 ) under Assumption 1.1 . These deﬁnitions are in natural agreement with the deﬁ- nitions given earlier in the case of Assumption 1.2 (see, e.g., [ 18 ] and [ 17 , Section 3.3]). F rom a substan tiv e p oint of view, the deﬁnition of an upp er (resp ectiv ely , lo wer) solution is based on a prop erty of weak inv ariance of the epigraph (resp ec- tiv ely , hypograph) of the solution with respect to so-called c haracteristic diﬀerential inclusions. The ﬁrst question we address is the c onsistency pr op erty of minimax solutions. Giv en that the Hamiltonian H is only measurable with resp ect to the ﬁrst v ariable t , we follow [ 38 , Chapter 11] and prop ose a notion of a semiclassic al solution of the Cauc hy problem ( 1 ), ( 2 ). T o the b est of our knowledge, such a notion has not b een previously introduced in the literature for the considered case of path-dependent Hamilton–Jacobi equations. W e prov e that a semiclassical solution (if it exists) must b e a minimax solution (see Theorem 3.4 ). On the other hand, we establish that, if a minimax solution φ is co-inv arian tly diﬀeren tiable at a point ( t, x ( · )) b elonging to a certain special set, then it satisﬁes the Hamilton–Jacobi equation ( 1 ) at this p oin t (see Theorems 3.6 and 3.8 ; a similar fact in the non-path-dep enden t case is [ 38 , Theorem 12.5, ii)]). Note that the arguments use an auxiliary result on the L eb esgue p oints (see, e.g., [ 30 , p. 255] and [ 7 , p. 562]) of the Hamiltonian H with resp ect to the ﬁrst v ariable t (see Lemma 2.8 ), whic h, in particular, allo ws us to justify equalit y ( 7 ). 1.4. Existence, uniqueness, and stability results for minimax solutions. The next result that w e prov e is a c omp arison the or em for upper and lo wer solutions (see Theorem 4.1 and also Theorem 4.2 ). In general, the pro of follows the scheme of the proof of [ 37 , Theorem 8.1] and is close to the pro of of [ 17 , Lemma 1], where the case of Assumption 1.2 w as considered. W e only need to mo dify the Lyapunov– Kr asovskii functional accordingly and use appropriate prop erties of the solution sets of functional diﬀerential inclusions of retarded type. The comparison theorem immediately implies the uniqueness of a minimax solution (see Theorem 4.4 ). In addition, we note that the proof of the comparison theorem diﬀers from that of [ 1 , Corollary 5.4], where the approach prop osed in [ 4 ] w as developed. The pro of also diﬀers from that of [ 38 , Lemma 12.12], where, due to the additional assumption of P A TH-DEPENDENT HJ EQUA TIONS 5 p ositiv e homogeneity of H , a diﬀerent notion of a minimax solution was considered, and the corresp onding constructions from earlier works [ 35 , 36 ] w ere dev elop ed. F urther, we establish a stability r esult for the minimax solution stating that the minimax solution depends contin uously on v ariations of the Hamiltonian H and the b oundary functional σ (see Theorem 5.1 ). The pro of relies on the comparison the- orem and again go es back to the pro ofs of the corresp onding results in the classical case (see, e.g., [ 36 , Section 4.4]) and in the path-dep endent case under Assumption 1.2 (see, e.g., [ 29 , Theorem 9.1] and also [ 25 , Theorem 6.1]). A distinctiv e feature of the presented stability result is that the closeness of the Hamiltonians with resp ect to the ﬁrst v ariable t is understoo d not in the uniform norm, but in the inte gr al norm ∥ · ∥ 1 (in this connection, see also, e.g., [ 21 , Proposition 7.1]). This feature is imp ortan t in the context of Assumption 1.1 , since it allows us to appr oximate the original time-measurable Hamiltonian H b y contin uous Hamiltonians. Finally , w e pro ve an existenc e the or em for the minimax solution. T o this end, w e p erform the Steklov tr ansformation of the Hamiltonian H with resp ect to the ﬁrst v ariable t (see, e.g., [ 31 , p. 212]) and obtain a sequence of Hamiltonians H k , k ∈ N , each of whic h satisﬁes conditions (B.1)–(B.3) from Assumption 1.2 . F urther, for every k ∈ N , we take a minimax solution φ k of the Cauc hy problem ( 1 ), ( 2 ) with the Hamiltonian H k (existence and uniqueness of φ k are established in [ 18 , Theorem 1]). Then, after v erifying that the sequence H k , k ∈ N , conv erges to H in the appropriate sense, w e apply the stability theorem and derive the existence of a minimax solution φ of the original Cauc h y problem ( 1 ), ( 2 ). Summarizing, in this pap er, we introduce a notion of a minimax solution of the Cauch y problem ( 1 ), ( 2 ) under Assumption 1.1 and prov e that this solution is wel l-p ose d and c onsistent with a notion of a solution in the classical sense. 1.5. Application to diﬀeren tial games. The main results obtained in the pap er are then applied to prov e the existe nce of a v alue of a zero-sum diﬀeren tial game for a time-delay system under weak ened assumptions. Namely , we consider a zer o-sum diﬀer ential game describ ed by an initial data ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), the dynamic e quation ˙ y ( τ ) = f ( τ , y ( · ) , u ( τ ) , v ( τ )) , (8) where τ ∈ [ t, T ], the initial c ondition y ( τ ) = x ( τ ) for all τ ∈ [ − h, t ], and the c ost functional J ( t, x ( · ) , u ( · ) , v ( · )) : = σ ( y ( · )) − Z T t χ ( τ , y ( · ) , u ( τ ) , v ( τ )) d τ . (9) Here, τ is time , y ( τ ) ∈ R n is the curren t state of the system, ˙ y ( τ ) : = d y ( τ ) / d τ , y ( · ) ∈ C ([ − h, T ] , R n ) is the system motion , u ( τ ) ∈ P and v ( τ ) ∈ Q are the current c ontr ol actions of the ﬁrst and se c ond players , resp ectively , P ⊂ R n P and Q ⊂ R n Q are compact sets, n P , n Q ∈ N . The goal of the ﬁrst pla y er is to minimize the v alue of the cost functional ( 9 ) via u ( · ) ∈ U [ t, T ], while the goal of the second play er is to maximize this v alue via v ( · ) ∈ V [ t, T ]. F or (rather standard) deﬁnitions of the sets of pla yers’ admissible c on- tr ols U [ t, T ] and V [ t, T ], the motions y ( · ) : = y ( · ; t, x ( · ) , u ( · ) , v ( · )) of system ( 8 ), the lower ρ − ( t, x ( · )) and upp er ρ + ( t, x ( · )) game values in the classes of non-anticip ative str ate gies , and the game value ρ ( t, x ( · )), see Section 7 . 6 MIKHAIL GOMOYUNO V W e study the diﬀerential game ( 8 ), ( 9 ) under the following standing assumption on the mapping ( f , χ ) : [0 , T ] × C ([ − h, T ] , R n ) × P × Q → R n × R and the functional σ : C ([ − h, T ] , R n ) → R . Assumption 1.3. The following conditions hold. (C.1) F or any ﬁxed y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , and v ∈ Q , the mapping τ 7→ ( f ( τ , y ( · ) , u, v ) , χ ( τ , y ( · ) , u, v )), [0 , T ] → R n × R , is measurable. (C.2) There exists a set E (C . 2) ⊂ [0 , T ] with µ ( E (C . 2) ) = T and such that the mapping below is contin uous for every ﬁxed τ ∈ E (C . 2) : ( y ( · ) , u, v ) 7→ ( f ( τ , y ( · ) , u, v ) , χ ( τ , y ( · ) , u, v )) , C ([ − h, T ] , R n ) × P × Q → R n × R . (10) (C.3) F or every compact set D ⊂ C ([ − h, T ] , R n ), there exist a non-negativ e function λ f ,χ ( · ) : = λ f ,χ ( · ; D ) ∈ L 1 ([0 , T ] , R ) and a set E (C . 3) : = E (C . 3) ( D ) ⊂ [0 , T ] with µ ( E (C . 3) ) = T such that, for all τ ∈ E (C . 3) , y 1 ( · ), y 2 ( · ) ∈ D , u ∈ P , v ∈ Q , ∥ f ( τ , y 1 ( · ) , u, v ) − f ( τ , y 2 ( · ) , u, v ) ∥ + | χ ( τ , y 1 ( · ) , u, v ) − χ ( τ , y 2 ( · ) , u, v ) | ≤ λ f ,χ ( τ ) ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ . (11) (C.4) There exist a non-negative function c f ( · ) ∈ L 1 ([0 , T ] , R ) and a set E (C . 4) ⊂ [0 , T ] with µ ( E (C . 4) ) = T such that ∥ f ( τ , y ( · ) , u, v ) ∥ ≤ c f ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) (12) for all τ ∈ E (C . 4) , y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . (C.5) F or every compact set D ⊂ C ([ − h, T ] , R n ), there exist a non-negativ e function m χ ( · ) : = m χ ( · ; D ) ∈ L 1 ([0 , T ] , R ) and a set E (C . 5) : = E (C . 5) ( D ) ⊂ [0 , T ] with µ ( E (C . 5) ) = T such that, for all τ ∈ E (C . 5) , y ( · ) ∈ D , u ∈ P , v ∈ Q , | χ ( τ , y ( · ) , u, v ) | ≤ m χ ( τ ) . (13) (C.6) The functional σ is contin uous. Note that condition (C.3) imply the following prop ert y of the mapping ( f , χ ): for ev ery compact set D ⊂ C ([ − h, T ] , R n ), there exists a set E f ,χ : = E f ,χ ( D ) ⊂ [0 , T ] with µ ( E f ,χ ) = T and suc h that, for all τ ∈ E f ,χ , y ( · ) ∈ D , u ∈ P , v ∈ Q , f ( τ , y ( · ) , u, v ) = f ( τ , y ( · ∧ τ ) , u, v ) , χ ( τ , y ( · ) , u, v ) = χ ( τ , y ( · ∧ τ ) , u, v ) . (14) In particular, this pr op erty of non-anticip ation allows us to assert that the dynamic equation ( 8 ) is a functional diﬀer ential e quation of r etar de d typ e and call the dy- namical system under consideration a time-delay system . By the mapping ( f , χ ), deﬁne the lower H − : [0 , T ] × C ([ − h, T ] , R n ) × R n → R and upp er H + : [0 , T ] × C ([ − h, T ] , R n ) × R n → R Hamiltonians b y H − ( τ , y ( · ) , s ) : = max v ∈ Q min u ∈ P  ⟨ s, f ( τ , y ( · ) , u, v ) ⟩ − χ ( τ , y ( · ) , u, v )  , H + ( τ , y ( · ) , s ) : = min u ∈ P max v ∈ Q  ⟨ s, f ( τ , y ( · ) , u, v ) ⟩ − χ ( τ , y ( · ) , u, v )  (15) if τ ∈ E (C . 2) and b y H − ( τ , y ( · ) , s ) : = 0 , H + ( τ , y ( · ) , s ) : = 0 (16) otherwise. Here, y ( · ) ∈ C ([ − h, T ] , R n ), s ∈ R n , and E (C . 2) is the set from condition (C.2). Observe that H − and H + satisfy conditions (A.1)–(A.5). P A TH-DEPENDENT HJ EQUA TIONS 7 The central result that we pro ve in this part of the pap er is that the lower value functional ρ − : [0 , T ] × C ([ − h, T ] , R n ) → R (respectively , upp er value func- tional ρ + : [0 , T ] × C ([ − h, T ] , R n ) → R ) coincides with the minimax solution φ − (resp ectiv ely , minimax solution φ + ) of the Cauch y problem ( 1 ), ( 2 ) with H = H − (resp ectiv ely , with H = H + ) and the boundary functional σ from the cost func- tional ( 9 ) (see Theorem 7.2 ). In particular, this fact conﬁrms the me aningfulness of the notion of a minimax solution considered in the pap er. Recall that, according to [ 3 , Theorem 7.1] (in this connection, see also, e.g., [ 27 , 28 ]), the stated result is v alid under the following assumption. Assumption 1.4. The following conditions hold. (D.1) The mapping ( f , χ ) : [0 , T ] × C ([ − h, T ] , R n ) × P × Q → R n × R is con- tin uous. (D.2) F or every compact set D ⊂ C ([ − h, T ] , R n ), there exists a n umber λ f ,χ : = λ f ,χ ( D ) > 0 such that, for all τ ∈ [0 , T ], y 1 ( · ), y 2 ( · ) ∈ D , u ∈ P , v ∈ Q , ∥ f ( τ , y 1 ( · ) , u, v ) − f ( τ , y 2 ( · ) , u, v ) ∥ + | χ ( τ , y 1 ( · ) , u, v ) − χ ( τ , y 2 ( · ) , u, v ) | ≤ λ f ,χ ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ . (D.3) There exists a num b er c f > 0 such that ∥ f ( τ , y ( · ) , u, v ) ∥ ≤ c f (1 + ∥ y ( · ∧ τ ) ∥ ∞ ) for all τ ∈ [0 , T ], y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , and v ∈ Q . (D.4) The functional σ : C ([ − h, T ] , R n ) → R is con tinuous. It is clear that Assumption 1.4 implies Assumption 1.3 . In addition, conditions (C.1)–(C.5) seem rather natural since they are consistent with the Carath´ eo dory conditions. In order to prov e the result, w e construct an appr oximating se quenc e of mappings ( f k , χ k ), k ∈ N , each of which satisﬁes conditions (D.1)–(D.3), and compute the limits of the sequences of low er and upp er v alue functionals of the approximating diﬀeren tial games ( 8 ), ( 9 ) with ( f , χ ) = ( f k , χ k ), k ∈ N , as w ell as the limits of the sequences of the minimax solutions of the Cauch y problems ( 1 ), ( 2 ) with the Hamiltonians H = H − k and H = H + k , k ∈ N , corres ponding to these games. In the latter case, we use the established stability result for minimax solutions. The construction of the appropriate approximation ( f k , χ k ) of the mapping ( f , χ ) is based, in particular, on the Sc orza Dr agoni the or em (see, e.g., [ 23 , Theorem 2]) and the McShane–Whitney extension the or em (see, e.g., [ 9 , Theorem 4.1.1]). As a corollary , and o wing to the uniqueness result for minimax solutions, w e then deriv e that the diﬀerential game ( 8 ), ( 9 ) has a value under Assumption 1.3 and the follo wing assumption. Assumption 1.5. F or every compact set D ⊂ C ([ − h, T ] , R n ) and ev ery y ( · ) ∈ D , there exists a set E : = E ( D , y ( · )) ⊂ [0 , T ] with µ ( E ) = T and suc h that H − ( τ , y ( · ) , s ) = H + ( τ , y ( · ) , s ) for all τ ∈ E , s ∈ R n . Moreo ver, we obtain that the value functional ρ : [0 , T ] × C ([ − h, T ] , R n ) → R of the diﬀeren tial game ( 8 ), ( 9 ) coincides with the minimax solution φ of the Cauch y problem ( 1 ), ( 2 ) with H = H − (or, equiv alently , with H = H + ) (see Theorem 7.6 ). 8 MIKHAIL GOMOYUNO V 1.6. Organization of the pap er. In Section 2 , w e in tro duce the notions of co- in v ariant diﬀeren tiability and co-inv arian t deriv atives and provide deﬁnitions of up- p er, lo wer, and minimax solutions of the Cauch y problem ( 1 ), ( 2 ). F urthermore, we presen t a result sho wing that, under Assumption 1.1 , conditions (A.3) and (A.5) can b e somewhat strengthened (see Corollary 2.9 ) and justify equalit y ( 7 ). In Section 3 , we prop ose the notion of a semiclassical solution of the Cauch y problem ( 1 ), ( 2 ) and in vestigate the consistency properties of minimax solutions. In Sections 4 – 6 , w e form ulate and prov e the comparison, uniqueness, stability , and existence theorems for minimax solutions. In Section 7 , we giv e the results concerning the diﬀerential game ( 8 ), ( 9 ). 2. Deﬁnition of a minimax solution. W e b egin b y in tro ducing the notion of co- in v ariant diﬀeren tiability ( ci -diﬀerentiabilit y , for short) and co-inv ariant deriv atives ( ci -deriv atives, for short), whic h are inv olv ed in the Hamilton–Jacobi equation under consideration (see ( 1 )). F or every p oin t ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), denote by AC ( t, x ( · )) the set of all functions y ( · ) ∈ C ([ − h, T ] , R n ) such that y ( · ∧ t ) = x ( · ∧ t ) and the restriction y | [ t,T ] ( · ) of the function y ( · ) to the interv al [ t, T ] is absolutely contin uous. Deﬁnition 2.1. A functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R is called ci -diﬀer en- tiable at a p oin t ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) if there exist ∂ t φ ( t, x ( · )) ∈ R and ∇ φ ( t, x ( · )) ∈ R n with the following prop ert y: for every function y ( · ) ∈ AC ( t, x ( · )), there exists a function o : (0 , + ∞ ) → R such that φ ( τ , y ( · )) − φ ( t, x ( · )) = ∂ t φ ( t, x ( · ))( τ − t ) + ⟨∇ φ ( t, x ( · )) , y ( τ ) − x ( t ) ⟩ + o ( τ − t ) (17) for all τ ∈ ( t, T ] and lim τ → t + o ( τ − t ) τ − t + R τ t ∥ ˙ y ( ξ ) ∥ d ξ = 0 . (18) In this case, ∂ t φ ( t, x ( · )) and ∇ φ ( t, x ( · )) are called ci -derivatives of φ at ( t, x ( · )). Note that the ci -deriv ativ es ∂ t φ ( t, x ( · )) and ∇ φ ( t, x ( · )) are determined uniquely . F or a discussion of the notion of ci -diﬀerentiabilit y , see, e.g., [ 17 , Section 3]. Remark 2.2. In the literature, a common deﬁnition of ci -diﬀerentiabilit y of a functional φ at a p oin t ( t, x ( · )) uses a narro w er class of “right extensions” y ( · ) of the point ( t, x ( · )). Speciﬁcally , it is required that equalit y ( 17 ) holds only for functions y ( · ) ∈ AC ( t, x ( · )) such that y | [ t,T ] ( · ) is Lipschitz contin uous. In this case, relation ( 18 ) is naturally replaced b y lim τ → t + o ( τ − t ) / ( τ − t ) = 0. Ho wev er, in the presen t pap er, we cannot restrict ourselves to the class of Lipsc hitz con tinuous “righ t extensions”, whic h leads to a slight mo diﬁcation of the deﬁnition. T o giv e deﬁnitions of upp er, lo wer, and minimax solutions of the Cauc hy problem ( 1 ), ( 2 ), we need to carry out auxiliary constructions. Giv en a p oin t ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) and a non-negative function c ( · ) ∈ L 1 ([0 , T ] , R ), consider the set Y ( t, x ( · ); c ( · )) : =  y ( · ) ∈ AC ( t, x ( · )) : ∥ ˙ y ( τ ) ∥ ≤ c ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) for a.e. τ ∈ [ t, T ]  . (19) Note that Y ( t, x ( · ); c ( · ))  = ∅ since x ( · ∧ t ) ∈ Y ( t, x ( · ); c ( · )). In addition, we hav e the follo wing result, which is used several times in the pap er. P A TH-DEPENDENT HJ EQUA TIONS 9 Prop osition 2.3. L et p oints ( t, x ( · )) , ( t k , x k ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) , k ∈ N , and non-ne gative functions c ( · ) , c k ( · ) ∈ L 1 ([0 , T ] , R ) , k ∈ N b e such that lim k →∞  | t k − t | + ∥ x k ( · ) − x ( · ) ∥ ∞ + ∥ c k ( · ) − c ( · ) ∥ 1  = 0 . (20) Then, for every se quenc e y k ( · ) ∈ Y ( t k , x k ( · ); c k ( · )) , k ∈ N , ther e exist a subse quenc e y k i ( · ) , i ∈ N , and a function y ( · ) ∈ Y ( t, x ( · ); c ( · )) such that ∥ y k i ( · ) − y ( · ) ∥ ∞ → 0 as i → ∞ . Pr o of. F or every k ∈ N , we hav e ∥ y k ( τ ) ∥ ≤ ∥ x k ( t k ) ∥ + Z τ t k ∥ ˙ y k ( ξ ) ∥ d ξ ≤ ∥ x k ( t k ) ∥ + Z τ t k c k ( ξ )(1 + ∥ y k ( · ∧ ξ ) ∥ ∞ ) d ξ for all τ ∈ [ t k , T ], whic h yields 1 + ∥ y k ( · ∧ τ ) ∥ ∞ ≤ 1 + ∥ x k ( · ∧ t k ) ∥ ∞ + Z τ t k c k ( ξ )(1 + ∥ y k ( · ∧ ξ ) ∥ ∞ ) d ξ for all τ ∈ [ t k , T ]. Therefore, applying the Gron w all inequalit y , we derive 1 + ∥ y k ( · ∧ τ ) ∥ ∞ ≤ (1 + ∥ x k ( · ∧ t k ) ∥ ∞ ) exp  Z τ t k c k ( ξ ) d ξ  for all τ ∈ [ t k , T ], k ∈ N . Then, thanks to ( 20 ), the sequence y k ( · ), k ∈ N , is uniformly bounded, i.e., there exists R > 0 such that ∥ y k ( · ) ∥ ∞ ≤ R for all k ∈ N . F urther, consider the auxiliary functions w ( τ ) : = (1 + R ) Z τ 0 c ( ξ ) d ξ , w k ( τ ) : = (1 + R ) Z τ 0 c k ( ξ ) d ξ , where τ ∈ [0 , T ], k ∈ N . By ( 20 ), max τ ∈ [0 ,T ] | w k ( τ ) − w ( τ ) | → 0 as k → ∞ , and, therefore, the sequence w k ( · ), k ∈ N , is uniformly equicontin uous due to the Arzel` a–Ascoli theorem. F or any k ∈ N , τ 1 , τ 2 ∈ [ t k , T ] with τ 2 > τ 1 , w e deriv e ∥ y k ( τ 2 ) − y k ( τ 1 ) ∥ ≤ Z τ 2 τ 1 ∥ ˙ y k ( ξ ) ∥ d ξ ≤ Z τ 2 τ 1 c k ( ξ )(1 + ∥ y k ( · ∧ ξ ) ∥ ∞ ) d ξ ≤ (1 + R ) Z τ 2 τ 1 c k ( ξ ) d ξ = (1 + R )( w k ( τ 2 ) − w k ( τ 1 )) . (21) Hence, the uniform equicontin uity of the sequences x k ( · ), w k ( · ), k ∈ N , implies the uniform equicon tin uity of the sequence y k ( · ), k ∈ N . As a result, according to the Arzel` a–Ascoli theorem, there exists a subsequence y k i ( · ), i ∈ N , and a function y ( · ) ∈ C ([ − h, T ] , R n ) suc h that ∥ y k i ( · ) − y ( · ) ∥ ∞ → 0 as i → ∞ , and it remains to verify that y ( · ) ∈ Y ( t, x ( · ); c ( · )). Note that 0 = lim i →∞ ∥ y ( · ∧ t ) − y k i ( · ∧ t k i ) ∥ ∞ = lim i →∞ ∥ y ( · ∧ t ) − x k i ( · ∧ t k i ) ∥ ∞ = ∥ y ( · ∧ t ) − x ( · ∧ t ) ∥ ∞ , 10 MIKHAIL GOMOYUNO V whic h means that y ( · ∧ t ) = x ( · ∧ t ). F urther, based on ( 20 ) and ( 21 ), we obtain ∥ y ( τ 2 ) − y ( τ 1 ) ∥ ≤ Z τ 2 τ 1 c ( ξ )(1 + ∥ y ( · ∧ ξ ) ∥ ∞ ) d ξ for all τ 1 , τ 2 ∈ [ t, T ] with τ 2 > τ 1 . Therefore, y | [ t,T ] ( · ) is absolutely con tin uous and ∥ ˙ y ( τ ) ∥ ≤ c ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) for every τ ∈ ( t, T ) suc h that the deriv ative ˙ y ( τ ) exists and τ is a Leb esgue p oint of the function c ( · ), i.e., for a.e. τ ∈ [ t, T ]. Consequen tly , y ( · ) ∈ Y ( t, x ( · ); c ( · )), and the pro of is complete. In particular, Proposition 2.3 implies that the set Y ( t, x ( · ); c ( · )) is compact in C ([ − h, T ] , R n ) for every ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) and every non-negativ e function c ( · ) ∈ L 1 ([0 , T ] , R ). Deﬁnition 2.4. A functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R is called an upp er so- lution of the Cauc hy problem ( 1 ), ( 2 ) if it is non-anticipativ e, low er semicontin uous, satisﬁes the b oundary condition φ ( T , x ( · )) ≥ σ ( x ( · )) (22) for all x ( · ) ∈ C ([ − h, T ] , R n ), and p ossesses the following prop erty . (U) F or any ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), s ∈ R n , τ ∈ ( t, T ], there exists a function y ( · ) ∈ Y ( t, x ( · ); c H ( · )) suc h that φ ( τ , y ( · )) − Z τ t  ⟨ s, ˙ y ( ξ ) ⟩ − H ( ξ , y ( · ) , s )  d ξ ≤ φ ( t, x ( · )) . (23) Deﬁnition 2.5. A functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R is called a lower solu- tion of the Cauc h y problem ( 1 ), ( 2 ) if it is non-anticipativ e, upp er semicontin uous, satisﬁes the b oundary condition φ ( T , x ( · )) ≤ σ ( x ( · )) (24) for all x ( · ) ∈ C ([ − h, T ] , R n ), and p ossesses the following prop erty . (L) F or any ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), s ∈ R n , τ ∈ ( t, T ], there exists a function y ( · ) ∈ Y ( t, x ( · ); c H ( · )) suc h that φ ( τ , y ( · )) − Z τ t  ⟨ s, ˙ y ( ξ ) ⟩ − H ( ξ , y ( · ) , s )  d ξ ≥ φ ( t, x ( · )) . (25) In Deﬁnitions 2.4 and 2.5 , the set Y ( t, x ( · ); c H ( · )) is deﬁned according to ( 19 ) with c ( · ) = c H ( · ), where c H ( · ) is the function from condition (A.4) of Assumption 1.1 . Note also that, thanks to conditions (A.1), (A.2), (A.4), and (A.5), for every x ( · ) ∈ C ([ − h, T ] , R n ) and ev ery con tinuous function s : [0 , T ] → R n , the function t 7→ H ( t, x ( · ) , s ( t )), [0 , T ] → R , b elongs to L 1 ([0 , T ] , R ). In particular, this implies that the integrals in ( 23 ) and ( 25 ) are well-deﬁned. Deﬁnition 2.6. A functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R is called a minimax solution of the Cauch y problem ( 1 ), ( 2 ) if it is an upper solution as well as a low er solution of this problem. In particular, a minimax solution φ is non-an ticipative, contin uous, and satisﬁes the boundary condition ( 2 ) for all x ( · ) ∈ C ([ − h, T ] , R n ). Remark 2.7. Let Hamiltonians H 1 and H 2 satisfy conditions (A.1)–(A.5) and a b oundary functional σ satisfy condition (A.6). In addition, let condition (A.4) for H 1 and for H 2 b e fulﬁlled with c H 1 ( · ) = c H 2 ( · ) = c ( · ) for some non-negativ e function c ( · ) ∈ L 1 ([0 , T ] , R ). Supp ose that, for every x ( · ) ∈ C ([ − h, T ] , R n ), there exists a P A TH-DEPENDENT HJ EQUA TIONS 11 set E : = E ( x ( · )) ⊂ [0 , T ] with µ ( E ) = T and suc h that H 1 ( t, x ( · ) , s ) = H 2 ( t, x ( · ) , s ) for all t ∈ E , s ∈ R n . Then, it follows directly from the ab o ve deﬁnitions that a functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R is an upper (respectively , low er, minimax) solution of the Cauch y problem ( 1 ), ( 2 ) with H = H 1 if and only if it is an upp er (resp ectiv ely , lo wer, minimax) solution of the Cauc hy problem ( 1 ), ( 2 ) with H = H 2 . In this regard, we note that conditions (A.2) and (A.4) can be somewhat strengthened when dealing with upp er, low er, or minimax solutions. Namely , w e can assume that mapping ( 3 ) is con tin uous for all t ∈ [0 , T ] and that inequalit y ( 5 ) holds for all t ∈ [0 , T ], x ( · ) ∈ C ([ − h, T ] , R n ), s 1 , s 2 ∈ R n . Concluding this section, we giv e an auxiliary result on the Leb esgue points of the Hamiltonian H with respect to the ﬁrst v ariable t . The pro of follows the same lines as the pro of of [ 38 , Lemma 12.2]. Lemma 2.8. Supp ose that c onditions (A.1) – (A.5) of Assumption 1.1 hold. Then, for every c omp act set D ⊂ C ([ − h, T ] , R n ) , ther e exists a set E 1 H : = E 1 H ( D ) ⊂ [0 , T ) with µ ( E 1 H ) = T and such that, for any t ∈ E 1 H , x ( · ) ∈ D , s ∈ R n , lim τ → t + 1 τ − t Z τ t H ( ξ , x ( · ) , s ) d ξ = H ( t, x ( · ) , s ) . (26) Pr o of. Consider the set E (A . 2) and the functions λ H ( · ) : = λ H ( · ; D ) and c H ( · ) from conditions (A.2)–(A.4). Denote by E 1 the set of all t ∈ [0 , T ) such that t is a Leb esgue p oin t of b oth functions λ H ( · ) and c H ( · ). T ak e an at most countable set D ∗ ⊂ D that is dense in D and let E 2 b e the set of all t ∈ [0 , T ) such that t is a Leb esgue p oint of each of the functions τ 7→ H ( τ , x ( · ) , s ), [0 , T ] → R , where x ( · ) ∈ D ∗ , s ∈ Q n . Put E 1 H : = E (A . 2) ∩ E 1 ∩ E 2 and observ e that µ ( E 1 H ) = T . Fix t ∈ E 1 H , x ( · ) ∈ D , s ∈ R n and let ε > 0. Since t ∈ E 1 , there exists M > 0 suc h that, for every τ ∈ ( t, T ], 1 τ − t Z τ t c H ( ξ ) d ξ ≤ M , 1 τ − t Z τ t λ H ( ξ ) d ξ ≤ M . (27) Due to the inclusion t ∈ E (A . 2) , there exists δ 1 > 0 such that | H ( t, y ( · ) , r ) − H ( t, x ( · ) , s ) | ≤ ε/ 3 (28) for all y ( · ) ∈ D and r ∈ R n with ∥ y ( · ) − x ( · ) ∥ ∞ ≤ δ 1 and ∥ r − s ∥ ≤ δ 1 . Deﬁne δ 2 : = min { δ 1 , ε/ (6 M (1 + ∥ s ∥ )) , ε/ (6 M (2 + ∥ x ( · ) ∥ ∞ )) , 1 } . T ake y ( · ) ∈ D ∗ and r ∈ Q n satisfying the inequalities ∥ y ( · ) − x ( · ) ∥ ∞ ≤ δ 2 and ∥ r − s ∥ ≤ δ 2 . Owing to the inclusion t ∈ E 2 , there exists δ 3 ∈ (0 , T − t ] such that, for ev ery τ ∈ ( t, t + δ 3 ],     1 τ − t Z τ t H ( ξ , y ( · ) , r ) d ξ − H ( t, y ( · ) , r )     ≤ ε 3 . (29) Fix τ ∈ ( t, t + δ 3 ]. W e hav e 1 τ − t Z τ t | H ( ξ , x ( · ) , s ) − H ( ξ , y ( · ) , r ) | d ξ ≤ 1 τ − t Z τ t λ H ( ξ )(1 + ∥ s ∥ ) ∥ x ( · ∧ ξ ) − y ( · ∧ ξ ) ∥ ∞ d ξ + 1 τ − t Z τ t c H ( ξ )(1 + ∥ y ( · ∧ ξ ) ∥ ∞ ) ∥ r − s ∥ d ξ ≤ M (1 + ∥ s ∥ ) ∥ x ( · ) − y ( · ) ∥ ∞ + M (2 + ∥ x ( · ) ∥ ∞ ) ∥ r − s ∥ ≤ ε/ 3 . 12 MIKHAIL GOMOYUNO V Com bining this estimate with ( 28 ) and ( 29 ), w e obtain     1 τ − t Z τ t H ( ξ , x ( · ) , s ) d ξ − H ( t, x ( · ) , s )     ≤ ε. The proof is complete. In addition, we note that Lemma 2.8 allo ws us to obtain stronger v ersions of conditions (A.3) and (A.5) and, in particular, to justify equality ( 7 ). Namely , the follo wing t wo corollaries are v alid under conditions (A.1)–(A.5) on H . Corollary 2.9. L et D ⊂ C ([ − h, T ] , R n ) b e a c omp act set and λ H ( · ) : = λ H ( · ; D ) and m H ( · ) : = m H ( · ; D ) b e the c orr esp onding functions fr om c onditions (A.3) and (A.5) . Then, ther e exists a set E 2 H : = E 2 H ( D ) ⊂ [0 , T ] with µ ( E 2 H ) = T and such that ine qualities ( 4 ) and ( 6 ) hold for al l t ∈ E 2 H , x ( · ) , x 1 ( · ) , x 2 ( · ) ∈ D , and s ∈ R n . Pr o of. Consider the set E 1 H : = E 1 H ( D ) from Lemma 2.8 and let E b e the set of all t ∈ [0 , T ) suc h that t is a Leb esgue point of b oth functions λ H ( · ) and m H ( · ). Deﬁne E 2 H : = E 1 H ∩ E and note that µ ( E 2 H ) = T . F or any t ∈ E 2 H , x 1 ( · ), x 2 ( · ) ∈ D , s ∈ R n , w e ha v e | H ( t, x 1 ( · ) , s ) − H ( t, x 2 ( · ) , s ) | ≤ lim sup τ → t + 1 τ − t Z τ t | H ( ξ , x 1 ( · ) , s ) − H ( ξ , x 2 ( · ) , s ) | d ξ ≤ lim sup τ → t + 1 τ − t Z τ t λ H ( ξ )(1 + ∥ s ∥ ) ∥ x 1 ( · ∧ ξ ) − x 2 ( · ∧ ξ ) ∥ ∞ d ξ ≤ (1 + ∥ s ∥ ) lim τ → t + ∥ x 1 ( · ∧ τ ) − x 2 ( · ∧ τ ) ∥ ∞ lim τ → t + 1 τ − t Z τ t λ H ( ξ ) d ξ = λ H ( t )(1 + ∥ s ∥ ) ∥ x 1 ( · ∧ t ) − x 2 ( · ∧ t ) ∥ ∞ . Similarly , for an y t ∈ E 2 H , x ( · ) ∈ D , we obtain | H ( t, x ( · ) , 0) | ≤ lim sup τ → t + 1 τ − t Z τ t | H ( ξ , x ( · ) , 0) | d ξ ≤ lim τ → t + 1 τ − t Z τ t m H ( ξ ) d ξ = m H ( t ) . The proof is complete. The diﬀerence of the prop erties from Corollary 2.9 compared to conditions (A.3) and (A.5) is that the set of t ∈ [0 , T ] for whic h inequalities ( 4 ) and ( 6 ) hold dep ends only on the compact set D and do es not dep end on the sp eciﬁc c hoice of x 1 ( · ), x 2 ( · ), s in ( 4 ) and of x ( · ) in ( 6 ). Corollary 2.10. F or every c omp act set D ⊂ C ([ − h, T ] , R n ) , ther e exists a set E H : = E H ( D ) ⊂ [0 , T ] with µ ( E H ) = T and such that e quality ( 7 ) holds for al l t ∈ E H , x ( · ) ∈ D , s ∈ R n . Pr o of. Consider the s et D ∗ : = { x ( · ∧ t ) : x ( · ) ∈ D, t ∈ [0 , T ] } , which is compact in C ([ − h, T ] , R n ) b y compactness of D . Let λ H ( · ) : = λ H ( · ; D ∗ ) be the function from condition (A.3) and let E 2 H : = E 2 H ( D ∗ ) b e the set from Corollary 2.9 . Put P A TH-DEPENDENT HJ EQUA TIONS 13 E H : = E 2 H . F or any t ∈ E H , x ( · ) ∈ D , s ∈ R n , noting that x ( · ), x ( · ∧ t ) ∈ D ∗ , we obtain | H ( t, x ( · ) , s ) − H ( t, x ( · ∧ t ) , s ) | ≤ λ H ( t )(1 + ∥ s ∥ ) ∥ x ( · ∧ t ) − x ( · ∧ t ) ∥ ∞ = 0 , whic h completes the pro of. 3. Semiclassical solutions and consistency. A ﬁrst result of this section is that a semiclassical solution of the Cauch y problem ( 1 ), ( 2 ) (if it exists) is a minimax solution of this problem. In order to giv e a deﬁnition of a semiclassical solution, whic h follows [ 38 , Chapter 11], w e need to introduce an appropriate set of functionals φ : [0 , T ] × C ([ − h, T ] , R n ) → R . Denote by Φ the set of all functionals φ : [0 , T ] × C ([ − h, T ] , R n ) → R satisfying the follo wing conditions. (Φ . 1) The functional φ is contin uous and the function τ 7→ φ ( τ , y ( · )), [ t, ϑ ] → R , is absolutely contin uous for all ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), y ( · ) ∈ AC ( t, x ( · )), ϑ ∈ ( t, T ). (Φ . 2) There exists a set E φ ⊂ [0 , T ) with µ ( E φ ) = T and suc h that the functional φ is ci -diﬀerentiable at every p oin t ( t, x ( · )) ∈ E φ × C ([ − h, T ] , R n ). (Φ . 3) The function t 7→ ( ∂ t φ ( t, x ( · )) , ∇ φ ( t, x ( · ))), E φ → R × R n , is measurable for every x ( · ) ∈ C ([ − h, T ] , R n ), and the mapping x ( · ) 7→ ( ∂ t φ ( t, x ( · )) , ∇ φ ( t, x ( · ))), C ([ − h, T ] , R n ) → R × R n , is contin uous for every t ∈ E φ . (Φ . 4) F or every compact set D ⊂ C ([ − h, T ] , R n ) and every ϑ ∈ [0 , T ), there exist a non-negativ e function m φ ( · ) : = m φ ( · ; D, ϑ ) ∈ L 1 ([0 , ϑ ] , R ) and a n umber ℓ φ : = ℓ φ ( D , ϑ ) > 0 such that, for an y t ∈ E φ ∩ [0 , ϑ ], x ( · ) ∈ D , | ∂ t φ ( t, x ( · )) | ≤ m φ ( t ) , ∥∇ φ ( t, x ( · )) ∥ ≤ ℓ φ . Note that (see, e.g., [ 17 , Section 3.1]) it follows from (Φ . 2) that φ ( t, x ( · )) = φ ( t, x ( · ∧ t )) , ∂ t φ ( t, x ( · )) = ∂ t φ ( t, x ( · ∧ t )) , ∇ φ ( t, x ( · )) = ∇ φ ( t, x ( · ∧ t )) (30) for all ( t, x ( · )) ∈ E φ × C ([ − h, T ] , R n ). In particular, by contin uity of φ , w e conclude that φ is automatically non-anticipativ e. F or every functional φ ∈ Φ, the following so-called functional chain rule holds. Prop osition 3.1. L et φ ∈ Φ , ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) , y ( · ) ∈ AC ( t, x ( · )) b e given. Then, for every τ ∈ [ t, T ) , φ ( τ , y ( · )) = φ ( t, x ( · )) + Z τ t  ∂ t φ ( ξ , y ( · )) + ⟨∇ φ ( ξ , y ( · )) , ˙ y ( ξ ) ⟩  d ξ . (31) Pr o of. F or ev ery k ∈ N , b y the Lusin theorem, there exists a closed set F k ⊂ [ t, T ] with µ ( F k ) ≥ T − t − 1 /k and such that ˙ y | F k ( · ) is con tin uous. Consider a function y k ( · ) ∈ AC ( t, x ( · )) suc h that ˙ y k ( τ ) = f k ( τ ) for a.e. τ ∈ [ t, T ], where f k ( τ ) : = ˙ y ( τ ) if τ ∈ F k and f k ( τ ) : = 0 otherwise. By construction, y k | [ t,T ] ( · ) is Lipschitz con tinuous and ∥ ˙ y k ( τ ) ∥ ≤ ∥ ˙ y ( τ ) ∥ for a.e. τ ∈ [ t, T ]. In addition, we hav e ∥ y k ( · ) − y ( · ) ∥ ∞ → 0 as k → ∞ and, by passing to a subsequence if necessary , ˙ y k ( τ ) → ˙ y ( τ ) as k → ∞ for a.e. τ ∈ [ t, T ]. W e ﬁx ϑ ∈ ( t, T ) and prov e equality ( 31 ) for all τ ∈ [ t, ϑ ]. F or ev ery k ∈ N , consider the function ω k ( τ ) : = φ ( τ , y k ( · )), τ ∈ [ t, ϑ ], whic h is absolutely con tinuous by (Φ . 1). Let E k denote the set of all τ ∈ ( t, ϑ ) such that 14 MIKHAIL GOMOYUNO V τ ∈ E φ and b oth deriv atives ˙ ω k ( τ ) and ˙ y k ( τ ) exist. Note that µ ( E k ) = ϑ − t and ﬁx τ ∈ E k . Recalling that the functional φ is ci -diﬀerentiable at the p oin t ( τ , y k ( · )) b y (Φ . 2) and using the inclusion y k ( · ) ∈ AC ( τ , y k ( · )), we obtain that there exists a function o k : (0 , + ∞ ) → R such that ω k ( ξ ) − ω k ( τ ) = φ ( ξ , y k ( · )) − φ ( τ , y k ( · )) = ∂ t φ ( τ , y k ( · ))( ξ − τ ) + ⟨∇ φ ( τ , y k ( · )) , y k ( ξ ) − y k ( τ ) ⟩ + o k ( ξ − τ ) (32) for all ξ ∈ ( τ , ϑ ] and lim ξ → τ + o k ( ξ − τ ) ξ − τ + R ξ τ ∥ ˙ y k ( η ) ∥ d η = 0 . Since y k | [ t,T ] ( · ) is Lipschitz contin uous, lim ξ → τ + o k ( ξ − τ ) / ( ξ − τ ) = 0. Hence, if we divide ( 32 ) by ξ − τ and pass to the limit as ξ → τ + , w e get ˙ ω k ( τ ) = ∂ t φ ( τ , y k ( · )) + ⟨∇ φ ( τ , y k ( · )) , ˙ y k ( τ ) ⟩ . As a result, we conclude that, for an y k ∈ N , τ ∈ [ t, ϑ ], φ ( τ , y k ( · )) − φ ( t, y k ( · )) = Z τ t  ∂ t φ ( ξ , y k ( · )) + ⟨∇ φ ( ξ , y k ( · )) , ˙ y k ( ξ ) ⟩  d ξ . (33) Fix τ ∈ [ t, ϑ ]. Thanks to (Φ . 1), (Φ . 3), and ( 30 ), we derive φ ( τ , y k ( · )) → φ ( τ , y ( · )) as k → ∞ , φ ( t, y k ( · )) = φ ( t, x ( · )) for all k ∈ N , and, for a.e. ξ ∈ [ t, τ ], lim k →∞  ∂ t φ ( ξ , y k ( · )) + ⟨∇ φ ( ξ , y k ( · )) , ˙ y k ( ξ ) ⟩  = ∂ t φ ( ξ , y ( · )) + ⟨∇ φ ( ξ , y ( · )) , ˙ y ( ξ ) ⟩ . Deﬁne the compact set D : = { y k ( · ) : k ∈ N } ∪ { y ( · ) } and tak e m φ ( · ) : = m φ ( · ; D, ϑ ), ℓ φ : = ℓ φ ( D , ϑ ) from (Φ . 4). F or every k ∈ N , we hav e   ∂ t φ ( ξ , y k ( · )) + ⟨∇ φ ( ξ , y k ( · )) , ˙ y k ( ξ ) ⟩   ≤ m φ ( ξ ) + ℓ φ ∥ ˙ y k ( ξ ) ∥ ≤ m φ ( ξ ) + ℓ φ ∥ ˙ y ( ξ ) ∥ for a.e. ξ ∈ [ t, τ ]. Then, by the Lebesgue dominated conv ergence theorem, lim k →∞ Z τ t  ∂ t φ ( ξ , y k ( · )) + ⟨∇ φ ( ξ , y k ( · )) , ˙ y k ( ξ ) ⟩  d ξ = Z τ t  ∂ t φ ( ξ , y ( · )) + ⟨∇ φ ( ξ , y ( · )) , ˙ y ( ξ ) ⟩  d ξ . Consequen tly , by passing to the limit as k → ∞ in ( 33 ), we come to equality ( 31 ) and complete the pro of. Remark 3.2. Supp ose that a functional φ : [0 , T ] × C ([ − h, T ] , R n ) → R has the follo wing prop erties (see also Remark 2.2 ). (i) F or every ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), there exist ∂ ∗ t φ ( t, x ( · )) ∈ R and ∇ ∗ φ ( t, x ( · )) ∈ R n suc h that, for every function y ( · ) ∈ AC ( t, x ( · )) suc h that y | [ t,T ] ( · ) is Lipschitz contin uous, there exists a function o : (0 , + ∞ ) → R such that equality ( 17 ) holds for all τ ∈ ( t, T ] and lim τ → t + o ( τ − t ) / ( τ − t ) = 0. (ii) The functional φ as well as the mappings ∂ ∗ t φ : [0 , T ) × C ([ − h, T ] , R n ) → R , ∇ ∗ φ : [0 , T ) × C ([ − h, T ] , R n ) → R n are con tin uous. Suc h a functional φ is often called c o-invariantly smo oth ( ci -smo oth , for short) in the literature. According to, e.g., [ 17 , Prop osition 1], for every p oin t ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) and ev ery function y ( · ) ∈ AC ( t, x ( · )) such that y | [ t,T ] ( · ) is P A TH-DEPENDENT HJ EQUA TIONS 15 Lipsc hitz con tinuous, the function τ 7→ φ ( τ , y ( · )), [ t, ϑ ] → R , is Lipschitz con tinuous for ev ery ϑ ∈ ( t, T ) and the equalit y φ ( τ , y ( · )) = φ ( t, x ( · )) + Z τ t  ∂ ∗ t φ ( ξ , y ( · )) + ⟨∇ ∗ φ ( ξ , y ( · )) , ˙ y ( ξ ) ⟩  d ξ (34) holds for all τ ∈ [ t, T ). Then, arguing similarly to the pro of of Prop osition 3.1 , w e obtain that equality ( 34 ) actually holds for all ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), y ( · ) ∈ AC ( t, x ( · )), τ ∈ [ t, T ). T ogether with contin uity of ∂ ∗ t φ and ∇ ∗ φ , this fact implies that φ is ci -diﬀerentiable at every p oin t ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) with ∂ t φ ( t, x ( · )) = ∂ ∗ t φ ( t, x ( · )), ∇ φ ( t, x ( · )) = ∇ ∗ φ ( t, x ( · )). As a result, for every functional φ satisfying (i) and (ii), w e deriv e φ ∈ Φ with E φ = [0 , T ). No w, w e are in a p osition to give a deﬁnition of a semiclassical solution. Deﬁnition 3.3. A functional φ ∈ Φ is called a semiclassic al solution of the Cauch y problem ( 1 ), ( 2 ) if it satisﬁes the Hamilton–Jacobi equation ( 1 ) for all ( t, x ( · )) ∈ E φ × C ([ − h, T ] , R n ) and the b oundary condition ( 2 ) for all x ( · ) ∈ C ([ − h, T ] , R n ). The ﬁrst result of this section is the following. Theorem 3.4. Under Assumption 1.1 , every semiclassic al solution φ of the Cauchy pr oblem ( 1 ) , ( 2 ) is a minimax solution of t his pr oblem. Pr o of. T o pro ve the result, it suﬃces to ﬁx ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), s ∈ R n and ﬁnd a function y ( · ) ∈ Y ( t, x ( · ); c H ( · )) for which b oth inequalities ( 23 ) and ( 25 ) take place for all τ ∈ [ t, T ]. Recall that the set Y ( t, x ( · ); c H ( · )) is deﬁned b y ( 19 ) with c ( · ) = c H ( · ), where the function c H ( · ) is tak en from condition (A.4) of Assumption 1.1 . Put E : = [ t, T ) ∩ E (A . 2) ∩ E (A . 4) ∩ E φ , where E (A . 2) , E (A . 4) , E φ are the sets from conditions (A.2), (A.4), (Φ . 2) resp ectiv ely . Note that µ ( E ) = T − t . Let A b e the set of all p oin ts ( τ , y ( · )) ∈ E × C ([ − h, T ] , R n ) such that ∇ φ ( τ , y ( · ∧ τ ))  = s . Consider the function f ( τ , y ( · )) : = H  τ , y ( · ∧ τ ) , ∇ φ ( τ , y ( · ∧ τ ))  − H ( τ , y ( · ∧ τ ) , s ) ∥∇ φ ( τ , y ( · ∧ τ )) − s ∥ 2  ∇ φ ( τ , y ( · ∧ τ )) − s  , where ( τ , y ( · )) ∈ A . It can b e directly veriﬁed that, for every τ ∈ E , the set A τ : = { y ( · ) ∈ C ([ − h, T ] , R n ) : ( τ , y ( · )) ∈ A } is op en, the mapping y ( · ) 7→ f ( τ , y ( · )), A τ → R n , is contin uous, and the inequality b elo w holds for all y ( · ) ∈ A τ : ∥ f ( τ , y ( · )) ∥ ≤ c H ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ); in addition, for every y ( · ) ∈ C ([ − h, T ] , R n ), the set A y ( · ) : = { τ ∈ E : ( τ , y ( · )) ∈ A } and the function τ 7→ f ( τ , y ( · )), A y ( · ) → R n , are measurable. No w, consider a m ultiv alued mapping F : [ t, T ] × C ([ − h, T ] , R n ) ⊸ R n deﬁned for ev ery point ( τ , y ( · )) ∈ [ t, T ] × C ([ − h, T ] , R n ) b y F ( τ , y ( · )) : = ( { f ( τ , y ( · )) } if ( τ , y ( · )) ∈ A, B ( c H ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ )) otherwise , where B ( R ) denotes the closed ball in R n with cen ter at the origin and radius R ≥ 0. By construction, F is non-an ticipativ e and has non-empty , compact, and con vex v alues. F rom the prop erties of the set A and the function f listed ab o v e, it follows that, for every τ ∈ [ t, T ], the multiv alued mapping y ( · ) 7→ F ( τ , y ( · )), C ([ − h, T ] , R n ) ⊸ R n , is upp er semicon tinuous and, for any y ( · ) ∈ C ([ − h, T ] , R n ), max  ∥ f ∥ : f ∈ F ( τ , y ( · ))  ≤ c H ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ); 16 MIKHAIL GOMOYUNO V furthermore, the multiv alued mapping τ 7→ F ( τ , y ( · )), [0 , T ] ⊸ R n , is measurable for every y ( · ) ∈ C ([ − h, T ] , R n ). Therefore, applying a result on the existence of solutions of functional diﬀeren tial inclusions of retarded t yp e (see, e.g., [ 32 , Theorem 2.1]), w e obtain that there exists a function y ( · ) ∈ Y ( t, x ( · ); c H ( · )) satisfying the inclusion ˙ y ( τ ) ∈ F ( τ , y ( · )) for a.e. τ ∈ [ t, T ]. By construction, we hav e ⟨∇ φ ( τ , y ( · ∧ τ )) − s, ˙ y ( τ ) ⟩ = H  τ , y ( · ∧ τ ) , ∇ φ ( τ , y ( · ∧ τ ))  − H ( τ , y ( · ∧ τ ) , s ) for a.e. τ ∈ [ t, T ]. Hence, in view of equalities ( 7 ) and ( 30 ), we get ⟨∇ φ ( τ , y ( · )) − s, ˙ y ( τ ) ⟩ = H  τ , y ( · ) , ∇ φ ( τ , y ( · ))  − H ( τ , y ( · ) , s ) (35) for a.e. τ ∈ [ t, T ]. Consequently , for every τ ∈ [ t, T ] such that equality ( 35 ) holds and φ satisﬁes the Hamilton–Jacobi equation ( 1 ) at the p oin t ( τ , y ( · )), i.e., for a.e. τ ∈ [ t, T ], we derive ∂ t φ ( τ , y ( · )) = − H  τ , y ( · ) , ∇ φ ( τ , y ( · ))  = ⟨ s − ∇ φ ( τ , y ( · )) , ˙ y ( τ ) ⟩ − H ( τ , y ( · ) , s ) . Th us, applying the functional chain rule (see Prop osition 3.1 ), we get φ ( τ , y ( · )) − φ ( t, x ( · )) = Z τ t  ⟨ s, ˙ y ( ξ ) ⟩ − H ( ξ , y ( · ) , s )  d ξ (36) for all τ ∈ [ t, T ). Due to con tinuit y of φ and conditions (A.4) and (A.5), by passing to the limit as τ → T − in ( 36 ), we conclude that ( 36 ) holds for τ = T as w ell. So, inequalities ( 23 ) and ( 25 ) are v alid for all τ ∈ [ t, T ], which completes the pro of. Our next goal is to pro v e that a minimax solution φ of the Cauch y problem ( 1 ), ( 2 ) satisﬁes the Hamilton–Jacobi equation ( 1 ) at p oints of ci -diﬀerentiabilit y . T o this end, we need to hav e stronger prop erties of φ than (U) and (L) (see Deﬁnitions 2.4 and 2.5 ). Lemma 3.5. L et Assumption 1.1 hold and let φ : [0 , T ] × C ([ − h, T ] , R n ) → R b e a non-anticip ative functional. Then, if φ is lower semic ontinuous and p ossesses pr op erty (U) , then, for any ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) , s ∈ R n , ther e exists y ( · ) ∈ Y ( t, x ( · ); c H ( · )) such that ine quality ( 23 ) is valid for al l τ ∈ ( t, T ] . A nal- o gously, if φ is upp er semic ontinuous and p ossesses pr op erty (L) , then, for any ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) , s ∈ R n , ther e exists y ( · ) ∈ Y ( t, x ( · ); c H ( · )) such that ine quality ( 25 ) is valid for al l τ ∈ ( t, T ] . Pr o of. W e prov e the ﬁrst statemen t only , since the pro of for the second one is similar. Fix ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), s ∈ R n . Let k ∈ N and t k,i : = t + ( T − t ) i/k for all i ∈ 0 , k . By prop ert y (U), there are functions y k,i ( · ), i ∈ 0 , k , such that y k, 0 ( · ) = x ( · ) and, for every i ∈ 1 , k , the inclusion y k,i ( · ) ∈ Y ( t k,i − 1 , y k,i − 1 ( · ); c H ( · )) and the inequality b elo w tak e place: φ ( t k,i , y k,i ( · )) − Z t k,i t k,i − 1  ⟨ s, ˙ y k,i ( ξ ) ⟩ − H ( ξ , y k,i ( · ) , s )  d ξ ≤ φ ( t k,i − 1 , y k,i − 1 ( · )) . Denote y k ( · ) : = y k,k ( · ). F or every i ∈ 0 , k , w e hav e y k ( · ∧ t k,i ) = y k,i ( · ∧ t k,i ), and, hence, φ ( t k,i , y k ( · )) = φ ( t k,i , y k,i ( · )) since the functional φ is non-anticipativ e and, moreo ver, H ( ξ , y k,i ( · ) , s ) = H ( ξ , y k ( · ) , s ) for a.e. ξ ∈ [ t, t k,i ] according to ( 7 ). Th us, y k ( · ) ∈ Y ( t, x ( · ); c H ( · )) and, for every i ∈ 0 , k , φ ( t k,i , y k ( · )) − Z t k,i t  ⟨ s, ˙ y k ( ξ ) ⟩ − H ( ξ , y k ( · ) , s )  d ξ ≤ φ ( t, x ( · )) . P A TH-DEPENDENT HJ EQUA TIONS 17 Thanks to compactness of the set Y ( t, x ( · ); c H ( · )), by passing to a subsequence if necessary , we can assume that ∥ y k ( · ) − y ( · ) ∥ ∞ → 0 as k → ∞ for some function y ( · ) ∈ Y ( t, x ( · ); c H ( · )). Now, let τ ∈ ( t, T ] b e ﬁxed. Then, for ev ery k ∈ N , denoting t k : = max { t k,i : t k,i ≤ τ , i ∈ 0 , k } , we get φ ( t k , y k ( · )) − Z t k t  ⟨ s, ˙ y k ( ξ ) ⟩ − H ( ξ , y k ( · ) , s )  d ξ ≤ φ ( t, x ( · )) . (37) W e hav e t k → τ as k → ∞ , and, hence, lim inf k →∞ φ ( t k , y k ( · )) ≥ φ ( τ , y ( · )) b y lo wer semicontin uity of φ and lim k →∞ Z t k t ⟨ s, ˙ y k ( ξ ) ⟩ d ξ = lim k →∞ ⟨ s, y k ( t k ) − y k ( t ) ⟩ = ⟨ s, y ( τ ) − y ( t ) ⟩ = Z τ t ⟨ s, ˙ y ( ξ ) ⟩ d ξ . Moreo ver, taking the corresp onding functions λ H ( · ) : = λ H ( · ; Y ( t, x ( · ); c H ( · ))) and m H ( · ) : = m H ( · ; Y ( t, x ( · ); c H ( · ))) from conditions (A.3) and (A.5) resp ectiv ely and using condition (A.4) as well, we derive     Z t k t H ( ξ , y k ( · ) , s ) d ξ − Z τ t H ( ξ , y ( · ) , s ) d ξ     ≤ Z t k t | H ( ξ , y k ( · ) , s ) − H ( ξ , y ( · ) , s ) | d ξ + Z τ t k | H ( ξ , y ( · ) , s ) − H ( ξ , y ( · ) , 0) | d ξ + Z τ t k | H ( ξ , y ( · ) , 0) | d ξ ≤ (1 + ∥ s ∥ ) ∥ y k ( · ) − y ( · ) ∥ ∞ Z t k t λ H ( ξ ) d ξ + (1 + ∥ y ( · ) ∥ ∞ ) ∥ s ∥ Z τ t k c H ( ξ ) d ξ + Z τ t k m H ( ξ ) d ξ for all k ∈ N , which yields lim k →∞ Z t k t H ( ξ , y k ( · ) , s ) d ξ = Z τ t H ( ξ , y ( · ) , s ) d ξ . Consequen tly , by passing to the inferior limit in ( 37 ) as k → ∞ , we arriv e at ( 23 ) and complete the pro of. The second result of this section is the following. Theorem 3.6. L et Assumption 1.1 hold and let a set S b e the union of c omp act sets D k ⊂ C ([ − h, T ] , R n ) over k ∈ N . Then, ther e exists a set E : = E ( S ) ⊂ [0 , T ) with µ ( E ) = T and such that a minimax solution φ of the Cauchy pr oblem ( 1 ) , ( 2 ) satisﬁes the Hamilton–Jac obi e quation ( 1 ) at every p oint ( t, x ( · )) ∈ E × S wher e φ is ci -diﬀer entiable. Pr o of. F or every k ∈ N , denote by Y k the union of the sets Y ( t, x ( · ); c H ( · )) ov er ( t, x ( · )) ∈ [0 , T ] × D k . Recall that c H ( · ) is the function from condition (A.4). Since D k is compact, it follo ws from Prop osition 2.3 that the set Y k is compact. Let Y b e the union of Y k o ver k ∈ N . Note that S ⊂ Y . By Corollary 2.10 and Lemma 18 MIKHAIL GOMOYUNO V 2.8 , there exists a set E 1 ⊂ [0 , T ) with µ ( E 1 ) = T and such that equalities ( 7 ) and ( 26 ) hold for all t ∈ E 1 , x ( · ) ∈ Y , s ∈ R n . Let E 2 b e the set of all t ∈ [0 , T ) such that t is a Leb esgue p oin t of the function c H ( · ). Put E : = E 1 ∩ E 2 and note that µ ( E ) = T . Fix ( t, x ( · )) ∈ E × S and supp ose that the minimax solution φ is ci -diﬀerentiable at ( t, x ( · )). Since φ is an upp er solution, and thanks to Lemma 3.5 , there exists a function y ( · ) ∈ Y ( t, x ( · ); c H ( · )) suc h that, for all τ ∈ ( t, T ], φ ( τ , y ( · )) − Z τ t  ⟨∇ φ ( t, x ( · )) , ˙ y ( ξ ) ⟩ − H  ξ , y ( · ) , ∇ φ ( t, x ( · ))   d ξ ≤ φ ( t, x ( · )) . Due to ci -diﬀeren tiability of φ at ( t, x ( · )), there exists a function o : (0 , + ∞ ) → R suc h that equality ( 17 ) holds for all τ ∈ ( t, T ] and relation ( 18 ) is v alid. Thus, for ev ery τ ∈ ( t, T ], ∂ t φ ( t, x ( · ))( τ − t ) + Z τ t H  ξ , y ( · ) , ∇ φ ( t, x ( · ))  d ξ ≤ − o ( τ − t ) . (38) Owing to the inclusions t ∈ E 1 and x ( · ), y ( · ) ∈ Y , w e ha v e lim τ → t + 1 τ − t Z τ t H  ξ , y ( · ) , ∇ φ ( t, x ( · ))  d ξ = H  t, y ( · ) , ∇ φ ( t, x ( · ))  = H  t, y ( · ∧ t ) , ∇ φ ( t, x ( · ))  = H  t, x ( · ∧ t ) , ∇ φ ( t, x ( · ))  = H  t, x ( · ) , ∇ φ ( t, x ( · ))  . F urther, in view of the inclusion t ∈ E 2 , there exists M > 0 suc h that the ﬁrst inequalit y in ( 27 ) is v alid for all τ ∈ ( t, T ]. Hence, for every τ ∈ ( t, T ], we derive Z τ t ∥ ˙ y ( ξ ) ∥ d ξ ≤ Z τ t c H ( ξ )(1 + ∥ y ( · ∧ ξ ) ∥ ∞ ) d ξ ≤ (1 + ∥ y ( · ) ∥ ∞ ) Z τ t c H ( ξ ) d ξ ≤ M (1 + ∥ y ( · ) ∥ ∞ )( τ − t ) , whic h implies that lim τ → t + o ( τ − t ) / ( τ − t ) = 0. Thus, if w e divide ( 38 ) by τ − t and then pass to the limit as τ → t + , w e obtain ∂ t φ ( t, x ( · )) + H  t, x ( · ) , ∇ φ ( t, x ( · ))  ≤ 0 . In a similar wa y , we can deduce that ∂ t φ ( t, x ( · )) + H  t, x ( · ) , ∇ φ ( t, x ( · ))  ≥ 0 . Consequen tly , φ satisﬁes the Hamilton–Jacobi equation ( 1 ) at the point ( t, x ( · )), and the pro of is complete. Let us recall that the space C ([ − h, T ] , R n ) can not b e represen ted as a countable union of its compact subsets. That is why a set S app ears in Theorem 3.6 instead of the whole space C ([ − h, T ] , R n ). Ho wev er, if w e strengthen condition (A.3) of Assumption 1.1 accordingly , we can obtain a more con ven tional consistency result, whic h corresp onds to, e.g., [ 38 , Theorem 12.5, ii)] in the non-path-dep enden t case. Namely , let us consider the following assumption. P A TH-DEPENDENT HJ EQUA TIONS 19 Assumption 3.7. F or every bounded set D ⊂ C ([ − h, T ] , R n ), there exists a non- negativ e function λ H ( · ) : = λ H ( · ; D ) ∈ L 1 ([0 , T ] , R ) with the following prop ert y: for an y x 1 ( · ), x 2 ( · ) ∈ D , s ∈ R n , there exists a set E : = E ( D , x 1 ( · ) , x 2 ( · ) , s ) ⊂ [0 , T ] with µ ( E ) = T and such that inequalit y ( 4 ) holds for all t ∈ E . In this case, the following theorem is v alid. Theorem 3.8. Under c onditions (A.1) , (A.2) , (A.4) – (A.6) of Assumption 1.1 and Assumption 3.7 , ther e exists a set E ⊂ [0 , T ) with µ ( E ) = T and such that a minimax solution φ of the Cauchy pr oblem ( 1 ) , ( 2 ) satisﬁes the Hamilton–Jac obi e quation ( 1 ) at every p oint ( t, x ( · )) ∈ E × C ([ − h, T ] , R n ) wher e φ is ci -diﬀer entiable. The pro of of Theorem 3.8 is completely analogous to that of Theorem 3.6 and is therefore omitted. 4. Comparison principle and uniqueness. Supp ose that Hamiltonians H 1 and H 2 satisfy conditions (A.1)–(A.5) and a boundary functional σ satisﬁes condition (A.6) (see Assumption 1.1 ). Consider a low er solution φ 1 of the Cauc hy problem ( 1 ), ( 2 ) with the Hamiltonian H = H 1 and an upp er solution φ 2 of the Cauch y problem ( 1 ), ( 2 ) with the Hamiltonian H = H 2 . Fix ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) and deﬁne the set Y ( t, x ( · ); c H 1 ( · )) according to ( 19 ) with c ( · ) = c H 1 ( · ), where the function c H 1 ( · ) is taken from condition (A.4) for H = H 1 . Theorem 4.1. L et the ab ove assumptions b e fulﬁl le d. Supp ose t hat, for every y ( · ) ∈ Y ( t, x ( · ); c H 1 ( · )) , ther e exists a set E : = E ( y ( · )) ⊂ [ t, T ] with µ ( E ) = T − t and such that, for any τ ∈ E and s ∈ R n , H 1 ( τ , y ( · ) , s ) ≤ H 2 ( τ , y ( · ) , s ) . (39) Then, the fol lowing ine quality is valid: φ 1 ( t, x ( · )) ≤ φ 2 ( t, x ( · )) . (40) Pr o of. The pro of follows the scheme of the proof of [ 17 , Lemma 1]. F or con venience, w e split the pro of in to four steps. Step 1. Let c H 2 ( · ) b e the function from condition (A.4) for H = H 2 and consider the set Y ( t, x ( · ); c H 2 ( · )). Denote Y 1 : = Y ( t, x ( · ); c H 1 ( · )), Y 2 : = Y ( t, x ( · ); c H 2 ( · )), D : = Y 1 ∪ Y 2 , R : = max {∥ y ( · ) ∥ ∞ : y ( · ) ∈ D } . T ake the corresponding functions λ H 1 ( · ) : = λ H 1 ( · ; D ) and λ H 2 ( · ) : = λ H 2 ( · ; D ) from condition (A.3). Let E 1 b e the set of all τ ∈ [0 , T ) such that d d τ R τ 0 λ H 2 ( ξ ) d ξ = λ H 2 ( τ ) and note that µ ( E 1 ) = T . Using condition (A.4) and Corollary 2.9 , c ho ose a set E 2 ⊂ [ t, T ] with µ ( E 2 ) = T − t and suc h that, for all τ ∈ E 2 , y 1 ( · ), y 2 ( · ) ∈ D , s 1 , s 2 ∈ R n , i ∈ { 1 , 2 } , | H i ( τ , y 1 ( · ) , s 1 ) − H i ( τ , y 2 ( · ) , s 2 ) | ≤ λ H i ( τ )(1 + ∥ s 2 ∥ ) ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ + c H i ( τ )(1 + ∥ y 1 ( · ∧ τ ) ∥ ∞ ) ∥ s 1 − s 2 ∥ . (41) F ollowing, e.g., [ 40 ] (see also [ 18 ]), consider the mappings γ ( τ , w ( · )) : =       ∥ w ( · ∧ τ ) ∥ 2 ∞ − ∥ w ( τ ) ∥ 2  2 ∥ w ( · ∧ τ ) ∥ 2 ∞ + ∥ w ( τ ) ∥ 2 if ∥ w ( · ∧ τ ) ∥ ∞ > 0 , 0 otherwise (42) 20 MIKHAIL GOMOYUNO V and q ( τ , w ( · )) : =     2 − 4 ∥ w ( · ∧ τ ) ∥ 2 ∞ − ∥ w ( τ ) ∥ 2 ∥ w ( · ∧ τ ) ∥ 2 ∞  w ( τ ) if ∥ w ( · ∧ τ ) ∥ ∞ > 0 , 0 otherwise , (43) where ( τ , w ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ). In accordance with [ 18 , App endix B] and Remark 3.2 , the mapping q : [0 , T ] × C ([ − h, T ] , R n ) → R n is con tin uous, the functional γ b elongs to the set Φ (see Section 3 ) with E γ = [0 , T ), and its ci - deriv atives are giv en b y ∂ t γ ( τ , w ( · )) = 0 , ∇ γ ( τ , w ( · )) = q ( τ , w ( · )) (44) for all ( τ , w ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ). Moreov er (see, e.g., [ 40 , Lemma 2.3] and [ 18 , Section 4.1]), γ and q satisfy the inequalities γ ( τ , w ( · )) ≥ κ ∥ w ( · ∧ τ ) ∥ 2 ∞ , κ : = (3 − √ 5) / 2 , ∥ q ( τ , w ( · )) ∥ ≤ 2 ∥ w ( τ ) ∥ (45) for all ( τ , w ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ). No w, tak e the num b er κ from ( 45 ) and put ε 0 : = 1 √ κ exp  − ∥ λ H 2 ( · ) ∥ 1 κ  . F or every ε ∈ (0 , ε 0 ], consider the Lyapuno v–Krasovskii functional ν ε ( τ , w ( · )) : = p ε 4 + γ ( τ , w ( · )) ε  exp  − 1 κ Z τ 0 λ H 2 ( ξ ) d ξ  − ε √ κ  (46) and the auxiliary mapping s ε ( τ , w ( · )) : = q ( τ , w ( · )) 2 ε p ε 4 + γ ( τ , w ( · ))  exp  − 1 κ Z τ 0 λ H 2 ( ξ ) d ξ  − ε √ κ  , (47) where ( τ , w ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ). Then, s ε : [0 , T ] × C ([ − h, T ] , R n ) → R n is con tinuous, ν ε is non-negativ e, and ν ε ( t, w ( · ) ≡ 0) ≤ ε. (48) Moreo ver, o wing to con tinuit y of the functional σ (see condition (A.6)), com- pactness of the set D , and the ﬁrst inequalit y in ( 45 ), the relation b elo w holds for ev ery K ≥ 0: lim ε → 0 + max  | σ ( y 1 ( · )) − σ ( y 2 ( · )) | : y 1 ( · ) , y 2 ( · ) ∈ D , ν ε ( T , y 1 ( · ) − y 2 ( · )) ≤ K  = 0 . (49) F urther, we hav e ν ε ∈ Φ with E ν ε = E 1 and ∂ t ν ε ( τ , w ( · )) = − λ H 2 ( τ ) p ε 4 + γ ( τ , w ( · )) ε κ exp  − 1 κ Z τ 0 λ H 2 ( ξ ) d ξ  , ∇ ν ε ( τ , w ( · )) = s ε ( τ , w ( · )) (50) for all ( τ , w ( · )) ∈ E ν ε × C ([ − h, T ] , R n ). Finally , according to ( 45 ), for any τ ∈ E 2 ∩ E ν ε , y 1 ( · ), y 2 ( · ) ∈ D , we derive ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ ≤ p ε 4 + γ ( τ , y 1 ( · ) − y 2 ( · )) / √ κ , ∥ q ( τ , y 1 ( · ) − y 2 ( · )) ∥ ≤ 2 ∥ y 1 ( τ ) − y 2 ( τ ) ∥ ≤ 2 p ε 4 + γ ( τ , y 1 ( · ) − y 2 ( · )) / √ κ , P A TH-DEPENDENT HJ EQUA TIONS 21 and, therefore (see also ( 41 )), H 2  τ , y 1 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))  − H 2  τ , y 2 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))  ≤ λ H 2 ( τ )  1 + ∥ q ( τ , y 1 ( · ) − y 2 ( · )) ∥ 2 ε p ε 4 + γ ( τ , y 1 ( · ) − y 2 ( · ))  exp  − 1 κ Z τ 0 λ H 2 ( ξ ) d ξ  − ε √ κ  × ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ ≤ λ H 2 ( τ ) √ κ  1 + 1 ε √ κ  exp  − 1 κ Z τ 0 λ H 2 ( ξ ) d ξ  − ε √ κ  × p ε 4 + γ ( τ , y 1 ( · ) − y 2 ( · )) = − ∂ t ν ε ( τ , y 1 ( · ) − y 2 ( · )) . (51) Step 2. Fix ε ∈ (0 , ε 0 ] and denote by W ε the set of all triples ( y 1 ( · ) , y 2 ( · ) , z ( · )) suc h that y 1 ( · ) ∈ Y 1 , y 2 ( · ) ∈ Y 2 , z : [ t, T ] → R is absolutely contin uous, z ( t ) = φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) , (52) and, for a.e. τ ∈ [ t, T ],   ˙ z ( τ ) + ⟨ s ε ( τ , y 1 ( · ) − y 2 ( · )) , ˙ y 1 ( τ ) − ˙ y 2 ( τ ) ⟩ − H 1  τ , y 1 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))  + H 2  τ , y 2 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))    ≤ ε ( c H 1 ( τ ) + c H 2 ( τ )) . Giv en an y y 1 ( · ) ∈ Y 1 , y 2 ( · ) ∈ Y 2 , w e ha v e ( y 1 ( · ) , y 2 ( · ) , z ( · )) ∈ W ε with z ( τ ) : = φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) − Z τ t  ⟨ s ε ( ξ , y 1 ( · ) − y 2 ( · )) , ˙ y 1 ( ξ ) − ˙ y 2 ( ξ ) ⟩ − H 1  ξ , y 1 ( · ) , s ε ( ξ , y 1 ( · ) − y 2 ( · ))  + H 2  ξ , y 2 ( · ) , s ε ( ξ , y 1 ( · ) − y 2 ( · ))   d ξ for all τ ∈ [ t, T ]. In particular, W ε  = ∅ . In the remainder of this step, w e prov e that the set W ε is compact in C ([ − h, T ] , R n ) × C ([ − h, T ] , R n ) × C ([ t, T ] , R ). Consider a sequence ( y [ k ] 1 ( · ) , y [ k ] 2 ( · ) , z [ k ] ( · )) ∈ W ε , k ∈ N . Since the sets Y 1 and Y 2 are compact, by passing to a subsequence if necessary , we can assume that ∥ y [ k ] 1 ( · ) − y [0] 1 ( · ) ∥ ∞ → 0 and ∥ y [ k ] 2 ( · ) − y [0] 2 ( · ) ∥ ∞ → 0 as k → ∞ for some functions y [0] 1 ( · ) ∈ Y 1 and y [0] 2 ( · ) ∈ Y 2 . Denote s [ k ] ( τ ) : = s ε ( τ , y [ k ] 1 ( · ) − y [ k ] 2 ( · )) , H [ k ] ( τ ) : = H 1 ( τ , y [ k ] 1 ( · ) , s [ k ] ( τ )) − H 2 ( τ , y [ k ] 2 ( · ) , s [ k ] ( τ )) , where τ ∈ [ t, T ], k ∈ N ∪ { 0 } . By contin uity of s ε , w e ha v e lim k →∞ max τ ∈ [ t,T ] ∥ s [ k ] ( τ ) − s [0] ( τ ) ∥ = 0 . (53) In particular, there exists M > 0 such that ∥ s [ k ] ( τ ) ∥ ≤ M for all τ ∈ [ t, T ] and k ∈ N ∪ { 0 } . F urthermore, for ev ery τ ∈ E 2 , using ( 41 ), w e deriv e | H [ k ] ( τ ) − H [0] ( τ ) | ≤ ( λ H 1 ( τ ) + λ H 2 ( τ ))(1 + M )( ∥ y [ k ] 1 ( · ) − y [0] 1 ( · ) ∥ ∞ + ∥ y [ k ] 2 ( · ) − y [0] 2 ( · ) ∥ ∞ ) + ( c H 1 ( τ ) + c H 1 ( τ ))(1 + R ) ∥ s [ k ] ( τ ) − s [0] ( τ ) ∥ . (54) 22 MIKHAIL GOMOYUNO V Let δ ≥ 0. F or every τ ∈ [ t, T ], denote by F δ ( τ ) the set consisting of all triples ( f 1 , f 2 , h ) ∈ R n × R n × R suc h that ∥ f 1 ∥ ≤ c H 1 ( τ )(1 + R ) , ∥ f 2 ∥ ≤ c H 2 ( τ )(1 + R ) , (55) and | h + ⟨ s [0] ( τ ) , f 1 − f 2 ⟩ − H [0] ( τ ) | ≤ ε ( c H 1 ( τ ) + c H 2 ( τ )) + δ ( λ H 1 ( τ ) + λ H 2 ( τ ) + c H 1 ( τ ) + c H 2 ( τ )) . (56) Note that, for every τ ∈ [ t, T ], the set F δ ( τ ) is non-empty , conv ex, and compact, and the inequality ∥ f 1 ∥ + ∥ f 2 ∥ + | h | ≤ (1 + R )(1 + M )( c H 1 ( τ ) + c H 2 ( τ )) + | H [0] ( τ ) | + ε ( c H 1 ( τ ) + c H 2 ( τ )) + δ ( λ H 1 ( τ ) + λ H 2 ( τ ) + c H 1 ( τ ) + c H 2 ( τ )) holds for all ( f 1 , f 2 , h ) ∈ F δ ( τ ). F urthermore, it can b e veriﬁed directly that the m ultiv alued mapping τ 7→ F δ ( τ ), [ t, T ] ⊸ R n × R n × R , is measurable. Fix δ > 0. Based on ( 53 ) and ( 54 ), tak e k ∗ ∈ N such that, for every k ≥ k ∗ , ev ery τ ∈ E 2 , and any f 1 , f 2 ∈ R n satisfying ( 55 ), the inequalit y below is v alid: |⟨ s [0] ( τ ) − s [ k ] ( τ ) , f 1 − f 2 ⟩ + H [ k ] ( τ ) − H [0] ( τ ) | ≤ δ ( λ H 1 ( τ ) + λ H 2 ( τ ) + c H 1 ( τ ) + c H 2 ( τ )) . Then, for every k ≥ k ∗ , w e obtain ( ˙ y [ k ] 1 ( τ ) , ˙ y [ k ] 2 ( τ ) , ˙ z [ k ] ( τ )) ∈ F δ ( τ ) for a.e. τ ∈ [ t, T ]. Consequen tly , applying a result on the compactness of solution sets of (ordinary) diﬀeren tial inclusions (see, e.g., [ 6 , Theorem 6.3.3]), we derive that, by passing to a subsequence if necessary , max τ ∈ [ t,T ] | z [ k ] ( τ ) − z [0] ( τ ) | → 0 as k → ∞ for some absolutely con tinuous function z [0] : [ t, T ] → R suc h that the initial condition ( 52 ) is satisﬁed and the inclusion ( ˙ y [0] 1 ( τ ) , ˙ y [0] 2 ( τ ) , ˙ z [0] ( τ )) ∈ F 0 ( τ ) is v alid for a.e. τ ∈ [ t, T ]. So, ( y [0] 1 ( · ) , y [0] 2 ( · ) , z [0] ( · )) ∈ W ε , and the pro of of compactness of W ε is complete. Step 3. Let us show that a triple ( y ( ε ) 1 ( · ) , y ( ε ) 2 ( · ) , z ( ε ) ( · )) ∈ W ε exists suc h that z ( ε ) ( T ) ≥ φ 2 ( T , y ( ε ) 2 ( · )) − φ 1 ( T , y ( ε ) 1 ( · )) . (57) F or every τ ∈ [ t, T ], consider the set M ε ( τ ) : =  ( y 1 ( · ) , y 2 ( · ) , z ( · )) ∈ W ε : z ( τ ) ≥ φ 2 ( τ , y 2 ( · )) − φ 1 ( τ , y 1 ( · ))  . Due to the initial condition ( 52 ) and since φ 1 and φ 2 are non-anticipativ e, we ha ve M ε ( t ) = W ε  = ∅ . Put τ ε : = max  τ ∈ [ t, T ] : M ε ( τ )  = ∅  . (58) The maximum in ( 58 ) is attained by compactness of W ε , upp er semicontin uity of φ 1 , and low er semicontin uity of φ 2 . In order to prov e the statement, it suﬃces to v erify that τ ε = T . Arguing b y con tradiction, assume that τ ε < T . T ake arbitrarily ( ˆ y 1 ( · ) , ˆ y 2 ( · ) , ˆ z ( · )) ∈ M ε ( τ ε ) and denote ˆ s : = s ε ( τ ε , ˆ y 1 ( · ) − ˆ y 2 ( · )). In accordance with Lemma 3.5 , there are functions y 1 ( · ) ∈ Y ( τ ε , ˆ y 1 ( · ); c H 1 ( · )) ⊂ Y 1 , y 2 ( · ) ∈ Y ( τ ε , ˆ y 2 ( · ); c H 2 ( · )) ⊂ Y 2 suc h that φ 2 ( τ , y 2 ( · )) − φ 1 ( τ , y 1 ( · )) ≤ φ 2 ( τ ε , ˆ y 2 ( · )) − φ 1 ( τ ε , ˆ y 1 ( · )) + z ( τ ) , where τ ∈ [ τ ε , T ] and z ( τ ) : = Z τ τ ε  −⟨ ˆ s, ˙ y 1 ( ξ ) − ˙ y 2 ( ξ ) ⟩ + H 1 ( ξ , y 1 ( · ) , ˆ s ) − H 2 ( ξ , y 2 ( · ) , ˆ s )  d ξ . P A TH-DEPENDENT HJ EQUA TIONS 23 Since s ε is contin uous and non-anticipativ e, w e obtain s ε ( τ , y 1 ( · ) − y 2 ( · )) → ˆ s as τ → τ + ε . Hence, and using ( 41 ), we conclude that there exists δ ∈ (0 , T − τ ε ] such that, for a.e. τ ∈ [ τ ε , τ ε + δ ],   ˙ z ( τ ) + ⟨ s ε ( τ , y 1 ( · ) − y 2 ( · )) , ˙ y 1 ( τ ) − ˙ y 2 ( τ ) ⟩ − H 1  τ , y 1 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))  + H 2  τ , y 2 ( · ) , s ε ( τ , y 1 ( · ) − y 2 ( · ))    ≤ ε ( c H 1 ( τ ) + c H 2 ( τ )) . No w, tak e y ∗ 1 ( · ) ∈ Y ( τ ε + δ, y 1 ( · ); c H 1 ( · )) ⊂ Y 1 , y ∗ 2 ( · ) ∈ Y ( τ ε + δ, y 2 ( · ); c H 2 ( · )) ⊂ Y 2 and consider a function z ∗ : [ t, T ] → R that is deﬁned by z ∗ ( τ ) : = ˆ z ( τ ) if τ ∈ [ t, τ ε ], z ∗ ( τ ) : = ˆ z ( τ ε ) + z ( τ ) if τ ∈ [ τ ε , τ ε + δ ], and z ∗ ( τ ) : = ˆ z ( τ ε ) + z ( τ ε + δ ) − Z τ τ ε  ⟨ s ε ( ξ , y ∗ 1 ( · ) − y ∗ 2 ( · )) , ˙ y ∗ 1 ( ξ ) − ˙ y ∗ 2 ( ξ ) ⟩ − H 1  ξ , y ∗ 1 ( · ) , s ε ( ξ , y ∗ 1 ( · ) − y ∗ 2 ( · ))  + H 2  ξ , y ∗ 2 ( · ) , s ε ( ξ , y ∗ 1 ( · ) − y ∗ 2 ( · ))   d ξ if τ ∈ ( τ ε + δ , T ]. Then, taking equality ( 7 ) in to accoun t and recalling that s ε is non-an ticipative, w e derive ( y ∗ 1 ( · ) , y ∗ 2 ( · ) , z ∗ ( · )) ∈ W ε . F urthermore, since φ 1 and φ 2 are non-an ticipativ e, w e ha ve φ 2 ( τ ε + δ, y ∗ 2 ( · )) − φ 1 ( τ ε + δ, y ∗ 1 ( · )) = φ 2 ( τ ε + δ, y 2 ( · )) − φ 1 ( τ ε + δ, y 1 ( · )) ≤ φ 2 ( τ ε , ˆ y 2 ( · )) − φ 1 ( τ ε , ˆ y 1 ( · )) + z ( τ ε + δ ) ≤ ˆ z ( τ ε ) + z ( τ ε + δ ) = z ∗ ( τ ε + δ ) . Therefore, the inclusion ( y ∗ 1 ( · ) , y ∗ 2 ( · ) , z ∗ ( · )) ∈ M ε ( τ ε + δ ) is v alid, whic h con tradicts the deﬁnition of τ ε (see ( 58 )). Step 4. Consider the auxiliary function ω ( τ ) : = ν ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + z ( ε ) ( τ ), where τ ∈ [ t, T ]. Fix ϑ ∈ ( t, T ). Since ν ε ∈ Φ, the restriction ω | [ t,ϑ ] ( · ) of the function ω ( · ) to the in terv al [ t, ϑ ] is absolutely con tinuous and, b y Proposition 3.1 , ˙ ω ( τ ) = ∂ t ν ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + ⟨∇ ν ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) , ˙ y ( ε ) 1 ( τ ) − ˙ y ( ε ) 2 ( τ ) ⟩ + ˙ z ( ε ) ( τ ) for a.e. τ ∈ [ t, ϑ ]. Due to the inclusion ( y ( ε ) 1 ( · ) , y ( ε ) 2 ( · ) , z ( ε ) ( · )) ∈ W ε and relations ( 39 ) and ( 51 ), we hav e ˙ z ( ε ) ( τ ) ≤ −⟨ s ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) , ˙ y ( ε ) 1 ( τ ) − ˙ y ( ε ) 2 ( τ ) ⟩ + H 1  τ , y ( ε ) 1 ( · ) , s ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · ))  − H 2  τ , y ( ε ) 2 ( · ) , s ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · ))  + ε ( c H 1 ( τ ) + c H 2 ( τ )) ≤ −⟨ s ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) , ˙ y ( ε ) 1 ( τ ) − ˙ y ( ε ) 2 ( τ ) ⟩ − ∂ t ν ε ( τ , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + ε ( c H 1 ( τ ) + c H 2 ( τ )) for a.e. τ ∈ [ t, ϑ ]. Hence, and thanks to ( 50 ), w e obtain ˙ ω ( τ ) ≤ ε ( c H 1 ( τ ) + c H 2 ( τ )) for a.e. τ ∈ [ t, ϑ ], which yields ω ( ϑ ) ≤ ω ( t ) + ε Z ϑ t ( c H 1 ( τ ) + c H 2 ( τ )) d τ . 24 MIKHAIL GOMOYUNO V Since this inequality holds for all ϑ ∈ ( t, T ) and ω ( · ) is con tinuou s by contin uity of ν ε , w e ev en tually get ω ( T ) ≤ ω ( t ) + ε ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 . In view of the deﬁnition of ω ( · ), the initial condition ( 52 ), and inequality ( 48 ), recalling that ν ε is non-an ticipativ e, w e deriv e ν ε ( T , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + z ( ε ) ( T ) ≤ ν ε ( t, y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + z ( ε ) ( t ) + ε ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 ≤ φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) + ε (1 + ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 ) . Since φ 1 and φ 2 satisfy the b oundary conditions ( 24 ) and ( 22 ) resp ectiv ely , we ha ve z ( ε ) ( T ) ≥ σ ( y ( ε ) 2 ( · )) − σ ( y ( ε ) 1 ( · )) according to ( 57 ). As a result, ν ε ( T , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) + σ ( y ( ε ) 2 ( · )) − σ ( y ( ε ) 1 ( · )) ≤ φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) + ε (1 + ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 ) . (59) Note that inequality ( 59 ) holds for every ε ∈ (0 , ε 0 ]. Owing to compactness of D and contin uity of σ , there exists K > 0 suc h that | σ ( y ( · )) | ≤ K for all y ( · ) ∈ D . Consequen tly , it follows from ( 59 ) that ν ε ( T , y ( ε ) 1 ( · ) − y ( ε ) 2 ( · )) ≤ 2 K + φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) + ε 0 (1 + ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 ) for all ε ∈ (0 , ε 0 ]. Therefore, | σ ( y ( ε ) 1 ( · )) − σ ( y ( ε ) 2 ( · )) | → 0 as ε → 0 + b y ( 49 ). Finally , using ( 59 ) and the fact that ν ε is non-negativ e, w e derive σ ( y ( ε ) 2 ( · )) − σ ( y ( ε ) 1 ( · )) ≤ φ 2 ( t, x ( · )) − φ 1 ( t, x ( · )) + ε (1 + ∥ c H 1 ( · ) + c H 2 ( · ) ∥ 1 ) for all ε ∈ (0 , ε 0 ]. So, by passing to the limit as ε → 0 + , we arrive at the desired inequalit y ( 40 ), which completes the proof. The follo wing result, symmetric to Theorem 4.1 in some sense, is also v alid. Theorem 4.2. L et Hamiltonians H 1 and H 2 satisfy c onditions (A.1) – (A.5) and a b oundary functional σ satisfy c ondition (A.6) . L et φ 1 b e an upp er solution of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H 1 , φ 2 b e a lower solution of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H 2 , and let a p oint ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) b e ﬁxe d. Supp ose that, for every y ( · ) ∈ Y ( t, x ( · ); c H 1 ( · )) , ther e is a set E : = E ( y ( · )) ⊂ [ t, T ] with µ ( E ) = T − t and such that H 1 ( τ , y ( · ) , s ) ≥ H 2 ( τ , y ( · ) , s ) for al l τ ∈ E and s ∈ R n . Then, the ine quality φ 1 ( t, x ( · )) ≥ φ 2 ( t, x ( · )) is valid. The pro of of Theorem 4.2 is completely analogous to that of Theorem 4.1 and is therefore omitted. Recalling Deﬁnition 2.6 of a minimax solution of the Cauch y problem ( 1 ), ( 2 ), w e deriv e from Theorems 4.1 and 4.2 the result b elow. Corollary 4.3. L et Hamiltonians H 1 and H 2 satisfy c onditions (A.1) – (A.5) and a b oundary functional σ satisfy c ondition (A.6) . L et φ 1 b e a minimax solution of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H 1 , φ 2 b e a minimax solution of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H 2 , and let a p oint ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ) b e ﬁxe d. Supp ose that, for every y ( · ) ∈ Y ( t, x ( · ); c H 1 ( · )) , ther e exists a set E : = E ( y ( · )) ⊂ [ t, T ] with µ ( E ) = T − t and such that H 1 ( τ , y ( · ) , s ) = H 2 ( τ , y ( · ) , s ) for al l τ ∈ E and s ∈ R n . Then, the e quality φ 1 ( t, x ( · )) = φ 2 ( t, x ( · )) is valid. P A TH-DEPENDENT HJ EQUA TIONS 25 In particular, we get the following uniqueness theorem for a minimax solution of the Cauc h y problem ( 1 ), ( 2 ). Theorem 4.4. Supp ose that a Hamiltonian H and a b oundary functional σ satisfy Assumption 1.1 . Then, ther e exists at most one minimax solution φ of the Cauchy pr oblem ( 1 ) , ( 2 ) . Remark 4.5. Deﬁnition 2.6 of a minimax solution φ of the Cauc hy problem ( 1 ), ( 2 ) in volv es a function c H ( · ), which can be c hosen from condition (A.4) of Assumption 1.1 in an ambiguous w ay . Nevertheless, it follows directly from Theorem 4.4 that, if Assumption 1.1 is satisﬁed, then a minimax solution φ do es not dep end on this c hoice (in this connection, see also [ 17 , p. 270]). 5. Stabilit y. Let Hamiltonians H , H k , k ∈ N , and boundary functionals σ , σ k , k ∈ N , satisfy Assumption 1.1 . Supp ose that lim k →∞ ∥ c H k ( · ) − c H ( · ) ∥ 1 = 0 , (60) where c H ( · ), c H k ( · ), k ∈ N , are the corresp onding functions from condition (A.4) (in this connection, see also Remark 4.5 ), and lim k →∞ max y ( · ) ∈ D Z T 0 | H k ( t, y ( · ) , s ) − H ( t, y ( · ) , s ) | d t = 0 , lim k →∞ max y ( · ) ∈ D | σ k ( y ( · )) − σ ( y ( · )) | = 0 (61) for ev ery compact set D ⊂ C ([ − h, T ] , R n ) and every s ∈ R n . Suppose also that, for ev ery k ∈ N , a minimax solution φ k of the Cauch y problem ( 1 ), ( 2 ) with H = H k and σ = σ k exists. Theorem 5.1. L et the ab ove assumptions b e fulﬁl le d. Then, the Cauchy pr oblem ( 1 ) , ( 2 ) has at le ast one minimax solution φ . In addition, it holds that lim k →∞ max ( t,x ( · )) ∈ [0 ,T ] × D | φ k ( t, x ( · )) − φ ( t, x ( · )) | = 0 (62) for every c omp act set D ⊂ C ([ − h, T ] , R n ) . Pr o of. W e follow the sc heme of the pro of of [ 29 , Theorem 9.1] (see also, e.g., [ 3 , Theorem 6.1]). W e split the pro of in to ﬁv e steps for conv enience. Step 1. F or every p oint ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), deﬁne φ − ( t, x ( · )) : = lim δ → 0 + inf  φ k ( t ′ , x ′ ( · )) : k ∈ N , k ≥ 1 /δ, ( t ′ , x ′ ( · )) ∈ O δ ( t, x ( · ∧ t ))  , (63) where O δ ( t, x ( · ∧ t )) : =  ( t ′ , x ′ ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) : | t ′ − t | + ∥ x ′ ( · ) − x ( · ∧ t ) ∥ ∞ ≤ δ  . Note that the limit in ( 63 ) exists by monotonicity (but may take inﬁnite v alues, in general) and φ − is non-an ticipativ e. According to ( 63 ), for every p oin t ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), there exist a subsequence φ k i , i ∈ N , and a sequence ( t i , x i ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), i ∈ N , suc h that lim i →∞ φ k i ( t i , x i ( · )) = φ − ( t, x ( · )) , lim i →∞  | t i − t | + ∥ x i ( · ) − x ( · ∧ t ) ∥ ∞  = 0 . (64) 26 MIKHAIL GOMOYUNO V Step 2. Let us ﬁx x ( · ) ∈ C ([ − h, T ] , R n ) and prov e the inequality φ − ( T , x ( · )) ≥ σ ( x ( · )) . (65) Let φ k i , ( t i , x i ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), i ∈ N , b e such that lim i →∞ φ k i ( t i , x i ( · )) = φ − ( T , x ( · )) , lim i →∞  | t i − T | + ∥ x i ( · ) − x ( · ) ∥ ∞  = 0 . (66) F or every i ∈ N , since φ k i is an upp er solution of the Cauch y problem ( 1 ), ( 2 ) with H = H k i and σ = σ k i (see Deﬁnition 2.4 ), there exists y i ( · ) ∈ Y ( t i , x i ( · ); c H k i ( · )) suc h that φ k i ( t i , x i ( · )) ≥ σ k i ( y i ( · )) + Z T t i H k i ( ξ , y i ( · ) , 0) d ξ . (67) In view of ( 60 ), ( 66 ), Prop osition 2.3 , and the equality Y ( T , x ( · ); c H ( · )) = { x ( · ) } , b y passing to a subsequence if necessary , w e can assume that ∥ y i ( · ) − x ( · ) ∥ ∞ → 0 as i → ∞ . Consider the compact set D ∗ : = { y i ( · ) : i ∈ N } ∪ { x ( · ) } and take the corresp onding function m H ( · ) : = m H ( · ; D ∗ ) from condition (A.5). Then, owing to the second equality in ( 61 ) and contin uity of σ , we hav e σ k i ( y i ( · )) → σ ( x ( · )) as i → ∞ . F urthermore, for every i ∈ N , we derive Z T t i | H k i ( ξ , y i ( · ) , 0) | d ξ ≤ max z ( · ) ∈ D ∗ Z T 0 | H k i ( ξ , z ( · ) , 0) − H ( ξ , z ( · ) , 0) | d ξ + Z T t i m H ( ξ ) d ξ . Th us, b y passing to the limit in ( 67 ) as i → ∞ and using the ﬁrst equality in ( 61 ), w e arriv e at the desired inequality ( 65 ). Step 3. Let us verify that φ − has prop erty (U) from Deﬁnition 2.4 . Namely , let us ﬁx ( t, x ( · )) ∈ [0 , T ) × C ([ − h, T ] , R n ), s ∈ R n , τ ∈ ( t, T ] and show that there exists y ( · ) ∈ Y ( t, x ( · ); c H ( · )) for which inequality ( 23 ) with φ = φ − holds. Consider φ k i , ( t i , x i ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), i ∈ N , satisfying ( 64 ). Since t i → t as i → ∞ and t < τ , by passing to a subsequence if necessary , w e can assume that t i < τ for all i ∈ N . F or every i ∈ N , taking into account that φ k i p ossesses prop ert y (U) with H = H k i , c ho ose y i ( · ) ∈ Y ( t i , x i ( · ); c H k i ( · )) suc h that φ k i ( τ , y i ( · )) − Z τ t i  ⟨ s, ˙ y i ( ξ ) ⟩ − H k i ( ξ , y i ( · ) , s )  d ξ ≤ φ k i ( t i , x i ( · )) . (68) Due to Prop osition 2.3 , by passing to a subsequence if necessary , we can assume that ∥ y i ( · ) − y ( · ) ∥ ∞ → 0 as i → ∞ for some function y ( · ) ∈ Y ( t, x ( · ); c H ( · )). In particular, w e ha v e lim i →∞ Z τ t i ⟨ s, ˙ y i ( ξ ) ⟩ d ξ = lim i →∞ ⟨ s, y i ( τ ) − y i ( t i ) ⟩ = ⟨ s, y ( τ ) − y ( t ) ⟩ = Z τ t ⟨ s, ˙ y ( ξ ) ⟩ d ξ . (69) In addition, lim inf i →∞ φ k i ( τ , y i ( · )) = lim inf i →∞ φ k i ( τ , y i ( · ∧ τ )) ≥ φ − ( τ , y ( · )) (70) since φ k i , i ∈ N , are non-an ticipative and thanks to the deﬁnition of φ − ( τ , y ( · )). F urther, consider the compact set D ∗ : = { y i ( · ) : i ∈ N } ∪ { y ( · ) } and tak e the cor- resp onding functions λ ∗ H ( · ) : = λ H ( · ; D ∗ ) and m ∗ H ( · ) : = m H ( · ; D ∗ ) from conditions P A TH-DEPENDENT HJ EQUA TIONS 27 (A.3) and (A.5) resp ectiv ely . Let i ∈ N and assume that t i ≤ t for deﬁniteness. Then, w e deriv e     Z τ t i H k i ( ξ , y i ( · ) , s ) d ξ − Z τ t H ( ξ , y ( · ) , s ) d ξ     ≤ Z τ t i | H k i ( ξ , y i ( · ) , s ) − H ( ξ , y i ( · ) , s ) | d ξ + Z τ t i | H ( ξ , y i ( · ) , s ) − H ( ξ , y ( · ) , s ) | d ξ + Z t t i | H ( ξ , y ( · ) , s ) − H ( ξ , y ( · ) , 0) | d ξ + Z t t i | H ( ξ , y ( · ) , 0) | d ξ ≤ max z ( · ) ∈ D ∗ Z T 0 | H k i ( ξ , z ( · ) , s ) − H ( ξ , z ( · ) , s ) | d ξ + (1 + ∥ s ∥ ) ∥ λ ∗ H ( · ) ∥ 1 ∥ y i ( · ) − y ( · ) ∥ ∞ + (1 + ∥ y ( · ) ∥ ∞ ) ∥ s ∥ Z t t i c H ( ξ ) d ξ + Z t t i m ∗ H ( ξ ) d ξ . Therefore, lim i →∞ Z τ t i H k i ( ξ , y i ( · ) , s ) d ξ = Z τ t H ( ξ , y ( · ) , s ) d ξ . (71) Th us, by passing to the inferior limit in ( 68 ) as i → ∞ and taking ( 69 )–( 71 ) into accoun t, w e conclude the v alidity of ( 23 ) with φ = φ − . Step 4. F or any ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), we obtain φ − ( t, x ( · )) > −∞ . Indeed, if t = T , we hav e φ − ( T , x ( · )) ≥ σ ( x ( · )) (see ( 65 )), and if t ∈ [0 , T ), o wing to property (U) and using ( 65 ), we derive φ − ( t, x ( · )) ≥ φ − ( T , y ( · )) + Z T t H ( ξ , y ( · ) , 0) d ξ ≥ σ ( y ( · )) + Z T t H ( ξ , y ( · ) , 0) d ξ for some function y ( · ) ∈ Y ( t, x ( · ); c H ( · )). Step 5. Now, for every ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), deﬁne φ + ( t, x ( · )) : = lim δ → 0 + sup  φ k ( t ′ , x ′ ( · )) : k ∈ N , k ≥ 1 /δ, ( t ′ , x ′ ( · )) ∈ O δ ( t, x ( · ∧ t ))  . (72) Then, rep eating the ab o ve reasoning with clear changes, w e can conclude that φ + is non-anticipativ e, φ + ( T , x ( · )) ≤ σ ( x ( · )) for all x ( · ) ∈ C ([ − h, T ] , R n ), φ + satisﬁes prop ert y (L) from Deﬁnition 2.5 , and the inequality φ + ( t, x ( · )) < + ∞ holds for all ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ). According to ( 63 ) and ( 72 ), for ev ery ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), −∞ < φ − ( t, x ( · )) ≤ φ + ( t, x ( · )) < + ∞ . (73) Hence, it follo ws directly from ( 63 ) and ( 72 ) that φ − : [0 , T ] × C ([ − h, T ] , R n ) → R is lo wer semicontin uous and φ + : [0 , T ] × C ([ − h, T ] , R n ) → R is upp er semicon tinuous. Consequen tly , φ − is an upp er solution of the Cauch y problem ( 1 ), ( 2 ) and φ + is a lo wer solution of this problem. Therefore, we ha ve φ + ( t, x ( · )) ≤ φ − ( t, x ( · )) for all ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) by Theorem 4.1 . Thanks to ( 73 ), we obtain that the functionals φ − and φ + are equal, which means that φ : = φ − = φ + is a minimax solution of the Cauch y problem ( 1 ), ( 2 ). 28 MIKHAIL GOMOYUNO V It remains to observe that the v alidity of the equalit y ( 62 ) for ev ery compact set D ⊂ C ([ − h, T ] , R n ) follo ws from the equalities φ = φ − = φ + and deﬁnitions ( 63 ) and ( 72 ) of the functionals φ − and φ + (in this connection, see, e.g., [ 13 , Remark 6.4]). The pro of is complete. 6. Existence. In this section, w e apply Theorem 5.1 in order to pro ve the existence of a minimax solution of the Cauc hy problem ( 1 ), ( 2 ) under Assumption 1.1 on the Hamiltonian H and the boundary functional σ . F or ev ery k ∈ N , deﬁne a Hamiltonian H k : [0 , T ] × C ([ − h, T ] , R n ) × R n → R using the Steklov transformation of H with resp ect to the ﬁrst v ariable t : for an y t ∈ [0 , T ], x ( · ) ∈ C ([ − h, T ] , R n ), s ∈ R n , H k ( t, x ( · ) , s ) : = k 2 Z t +1 /k t − 1 /k H ( τ , x ( · ∧ t ) , s ) d τ . (74) In the case where τ ∈ [ − 1 , 0) ∪ ( T , T + 1], we put formally H ( τ , x ( · ) , s ) : = 0 for all x ( · ) ∈ C ([ − h, T ] , R n ), s ∈ R n . Lemma 6.1. F or every k ∈ N , the Hamiltonian H k satisﬁes c onditions (B.1) – (B.3) of Assumption 1.2 . Pr o of. Let us sho w that H k satisﬁes (B.2). Fix a compact set D ⊂ C ([ − h, T ] , R n ). Consider the set D ∗ : = { x ( · ∧ t ) : x ( · ) ∈ D , t ∈ [0 , T ] } , which is compact by com- pactness of D , take the corresp onding function λ H ( · ) : = λ H ( · ; D ∗ ) from condition (A.3) of Assumption 1.1 , and deﬁne λ H k : = max t ∈ [0 ,T ] k 2 Z t +1 /k t − 1 /k λ H ( τ ) d τ , where w e put formally λ H ( τ ) : = 0 for all τ ∈ [ − 1 , 0) ∪ ( T , T + 1]. Then, for an y t ∈ [0 , T ], x 1 ( · ), x 2 ( · ) ∈ D , s ∈ R n , w e obtain | H k ( t, x 1 ( · ) , s ) − H k ( t, x 2 ( · ) , s ) | ≤ k 2 Z t +1 /k t − 1 /k | H ( τ , x 1 ( · ∧ t ) , s ) − H ( τ , x 2 ( · ∧ t ) , s ) | d τ ≤ (1 + ∥ s ∥ ) ∥ x 1 ( · ∧ t ) − x 2 ( · ∧ t ) ∥ ∞ k 2 Z t +1 /k t − 1 /k λ H ( τ ) d τ ≤ λ H k (1 + ∥ s ∥ ) ∥ x 1 ( · ∧ t ) − x 2 ( · ∧ t ) ∥ ∞ . F urther, let us pro ve that H k satisﬁes (B.3). By the function c H ( · ) from condition (A.4) of Assumption 1.1 , deﬁne c H k : = max t ∈ [0 ,T ] k 2 Z t +1 /k t − 1 /k c H ( τ ) d τ , where w e put formally c H ( τ ) : = 0 for all τ ∈ [ − 1 , 0) ∪ ( T , T + 1]. Then, for any t ∈ [0 , T ], x ( · ) ∈ C ([ − h, T ] , R n ), s 1 , s 2 ∈ R n , w e ha v e | H k ( t, x ( · ) , s 1 ) − H k ( t, x ( · ) , s 2 ) | ≤ k 2 Z t +1 /k t − 1 /k | H ( τ , x ( · ∧ t ) , s 1 ) − H ( τ , x ( · ∧ t ) , s 2 ) | d τ ≤ (1 + ∥ x ( · ∧ t ) ∥ ∞ ) ∥ s 1 − s 2 ∥ k 2 Z t +1 /k t − 1 /k c H ( τ ) d τ P A TH-DEPENDENT HJ EQUA TIONS 29 ≤ c H k (1 + ∥ x ( · ∧ t ) ∥ ∞ ) ∥ s 1 − s 2 ∥ . It remains to v erify that H k is contin uous, i.e., satisﬁes (B.1). Thanks to (B.2) and (B.3), it suﬃces to ﬁx x ( · ) ∈ C ([ − h, T ] , R n ) and s ∈ R n and pro ve that the function t 7→ H k ( t, x ( · ) , s ), [0 , T ] → R , is con tinuous. Consider the compact set D ∗ : = { x ( · ∧ t ) : t ∈ [0 , T ] } , take the corresp onding function λ ∗ H ( · ) : = λ H ( · ; D ∗ ) from condition (A.3), and put formally λ ∗ H ( τ ) : = 0 for all τ ∈ [ − 1 , 0) ∪ ( T , T + 1]. F or any t , t ′ ∈ [0 , T ], we derive | H k ( t ′ , x ( · ) , s ) − H k ( t, x ( · ) , s ) | ≤ k 2 Z t ′ +1 /k t ′ − 1 /k | H ( τ , x ( · ∧ t ′ ) , s ) − H ( τ , x ( · ∧ t ) , s ) | d τ + k 2     Z t ′ +1 /k t ′ − 1 /k H ( τ , x ( · ∧ t ) , s ) d τ − Z t +1 /k t − 1 /k H ( τ , x ( · ∧ t ) , s ) d τ     ≤ (1 + ∥ s ∥ ) ∥ x ( · ∧ t ) − x ( · ∧ t ′ ) ∥ ∞ k 2 Z t ′ +1 /k t ′ − 1 /k λ ∗ H ( τ ) d τ + k 2     Z t ′ +1 /k t ′ − 1 /k H ( τ , x ( · ∧ t ) , s ) d τ − Z t +1 /k t − 1 /k H ( τ , x ( · ∧ t ) , s ) d τ     , whic h yields H k ( t ′ , x ( · ) , s ) → H k ( t, x ( · ) , s ) as t ′ → t . The pro of is complete. In particular, w e obtain that, for e v ery k ∈ N , the Hamiltonian H k satisﬁes conditions (A.1)–(A.5) of Assumption 1.1 and, moreov er, condition (A.4) is fulﬁlled with the function ˜ c H k ( t ) : = k 2 Z t +1 /k t − 1 /k c H ( τ ) d τ for all t ∈ [0 , T ]. Recall that c H ( · ) is the function from condition (A.4) for the Hamiltonian H and c H ( τ ) : = 0 for all τ ∈ [ − 1 , 0) ∪ ( T , T + 1]. Note that, according to, e.g., [ 31 , Section XVI II.3, Lemma 4], we ha ve lim k →∞ ∥ ˜ c H k ( · ) − c H ( · ) ∥ 1 = 0 . (75) Lemma 6.2. F or every c omp act set D ⊂ C ([ − h, T ] , R n ) and every s ∈ R n , the ﬁrst r elation in ( 61 ) is fulﬁl le d for H k , k ∈ N , given by ( 74 ) . Pr o of. Let us consider a mapping H : C ([ − h, T ] , R n ) → L 1 ([0 , T ] , R ) that to every function y ( · ) ∈ C ([ − h, T ] , R n ) assigns the function t 7→ H ( t, y ( · ) , s ), [0 , T ] → R . Here, L 1 ([0 , T ] , R ) is the Banach space of all (equiv alence classes of ) functions from L 1 ([0 , T ] , R ) with the standard norm ∥ · ∥ 1 . Let us v erify that H is contin uous. Let a function y ( · ) ∈ C ([ − h, T ] , R n ) and a sequence y i ( · ) ∈ C ([ − h, T ] , R n ), i ∈ N , b e such that ∥ y i ( · ) − y ( · ) ∥ ∞ → 0 as i → ∞ . Deﬁne the compact set D ∗ : = { y i ( · ) : i ∈ N } ∪ { y ( · ) } and c ho ose the corresp onding function λ H ( · ) : = λ H ( · ; D ∗ ) according to condition (A.3) of Assumption 1.1 . Then, for ev ery i ∈ N , we hav e Z T 0 | H ( t, y i ( · ) , s ) − H ( t, y ( · ) , s ) | d t ≤ (1 + ∥ s ∥ ) ∥ y i ( · ) − y ( · ) ∥ ∞ Z T 0 λ H ( t ) d t, whic h implies that lim i →∞ Z T 0 | H ( t, y i ( · ) , s ) − H ( t, y ( · ) , s ) | d t = 0 . 30 MIKHAIL GOMOYUNO V No w, ﬁx a compact set D ⊂ C ([ − h, T ] , R n ) and s ∈ R n . Owing to contin uity of H , the set H ( D ) of all functions t 7→ H ( t, y ( · ) , s ), [0 , T ] → R , where y ( · ) ∈ D , is compact in the space L 1 ([0 , T ] , R ). Therefore, according to the Kolmorogov criterion for compactness in this space (see, e.g., the pro of of the necessity part of [ 31 , Section XVI I I.3, Theorem 6]), w e get lim k →∞ max y ( · ) ∈ D Z T 0     k 2 Z t +1 /k t − 1 /k H ( τ , y ( · ) , s ) d τ − H ( t, y ( · ) , s )     d t = 0 . (76) F urther, consider the compact set D ∗ : = { y ( · ∧ t ) : y ( · ) ∈ D , t ∈ [0 , T ] } , tak e the corresp onding function λ ∗ H ( · ) : = λ H ( · ; D ∗ ) from condition (A.3), and put formally λ ∗ H ( τ ) : = 0 for all τ ∈ [ − 1 , 0) ∪ ( T , T + 1]. F or an y k ∈ N , y ( · ) ∈ D , w e deriv e     H k ( t, y ( · ) , s ) − k 2 Z t +1 /k t − 1 /k H ( τ , y ( · ) , s ) d τ     ≤ k 2 Z t +1 /k t − 1 /k | H ( τ , y ( · ∧ t ) , s ) − H ( τ , y ( · ) , s ) | d τ ≤ (1 + ∥ s ∥ ) max τ ∈ [ t, min { t +1 /k,T } ] ∥ y ( t ) − y ( τ ) ∥ k 2 Z t +1 /k t − 1 /k λ ∗ H ( τ ) d τ for all t ∈ [0 , T ] and, therefore, using [ 31 , Section XVI II.3, Lemma 2], w e obtain Z T 0     H k ( t, y ( · ) , s ) − k 2 Z t +1 /k t − 1 /k H ( τ , y ( · ) , s ) d τ     d t ≤ (1 + ∥ s ∥ ) max t ∈ [0 ,T ] max τ ∈ [ t, min { t +1 /k,T } ] ∥ y ( t ) − y ( τ ) ∥ Z T 0 k 2 Z t +1 /k t − 1 /k λ ∗ H ( τ ) d τ d t ≤ (1 + ∥ s ∥ ) ∥ λ ∗ H ( · ) ∥ 1 max t ∈ [0 ,T ] max τ ∈ [ t, min { t +1 /k,T } ] ∥ y ( t ) − y ( τ ) ∥ . Consequen tly , thanks to the Arzel` a–Ascoli theorem, lim k →∞ max y ( · ) ∈ D Z T 0     H k ( t, y ( · ) , s ) − k 2 Z t +1 /k t − 1 /k H ( τ , y ( · ) , s ) d τ     d t = 0 . (77) The ﬁrst relation in ( 61 ) follows from ( 76 ) and ( 77 ). F or ev ery k ∈ N , by Lemma 6.1 and [ 18 , Theorem 1], there exists a (unique) minimax solution φ k of the Cauch y problem ( 1 ), ( 2 ) with H = H k . Hence, owing to equalit y ( 75 ) and Lemma 6.2 , w e derive from Theorem 5.1 that there exists a minimax solution of the original Cauch y problem ( 1 ), ( 2 ). Th us, w e get the following result. Theorem 6.3. Supp ose that a Hamiltonian H and a b oundary functional σ satisfy Assumption 1.1 . Then, ther e exists at le ast one minimax solution of the Cauchy pr oblem ( 1 ) , ( 2 ) . 7. Applications to zero-sum diﬀerential games for time-delay systems. W e b egin by giving a formalization of the diﬀerential game ( 8 ), ( 9 ). Let an initial p oin t ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) b e ﬁxed. The sets U [ t, T ] and V [ t, T ] of players’ admissible c ontr ols on the time interv al [ t, T ] consist of all measurable functions u : [ t, T ] → P and v : [ t, T ] → Q resp ectiv ely . A motion of system ( 8 ) corresp onding to ( t, x ( · )) and u ( · ) ∈ U [ t, T ], v ( · ) ∈ V [ t, T ] is deﬁned as a function y ( · ) ∈ AC ( t, x ( · )) (see Section 2 ) that satisﬁes the dynamic equation P A TH-DEPENDENT HJ EQUA TIONS 31 ( 8 ) for a.e. τ ∈ [ t, T ]. Thanks to conditions (C.1)–(C.4) (see Assumption 1.3 ), suc h a motion y ( · ) : = y ( · ; t, x ( · ) , u ( · ) , v ( · )) exists, is unique, and b elongs to the compact set Y ( t, x ( · ); c f ( · )) deﬁned b y ( 19 ) with c ( · ) = c f ( · ) (note that the existence can b e derived from, e.g., [ 32 , Theorem 2.1], while the uniqueness follows directly from condition (C.3); in this connection, see also, e.g., [ 20 , Section 2.6] and [ 5 , Section I I.4, Theorem 3.1]). The corresp onding v alue J ( t, x ( · ) , u ( · ) , v ( · )) of the cost functional ( 9 ) is well-deﬁned owing to conditions (C.1), (C.2), and (C.5). Let us deﬁne the low er and upp er v alues of the game. F or the ﬁrst pla yer, a non-anticip ative str ate gy is a mapping α : V [ t, T ] → U [ t, T ] with the following prop ert y: for an y τ ∈ [ t, T ] and any con trols v ( · ), v ′ ( · ) ∈ V [ t, T ] of the second pla yer, if the equality v ( ξ ) = v ′ ( ξ ) holds for a.e. ξ ∈ [ t, τ ], then the corresp onding con trols u ( · ) : = α [ v ( · )]( · ) and u ′ ( · ) : = α [ v ′ ( · )]( · ) of the ﬁrst pla yer satisfy the equality u ( ξ ) = u ′ ( ξ ) for a.e. ξ ∈ [ t, τ ]. Then, the lower value of the game is given by ρ − ( t, x ( · )) : = inf α ∈A [ t,T ] sup v ( · ) ∈V [ t,T ] J  t, x ( · ) , α [ v ( · )]( · ) , v ( · )  , (78) where A [ t, T ] is the set of all ﬁrst pla yer’s non-anticipativ e strategies α . In a similar w ay , a second play er’s non-anticip ative str ate gy is a mapping β : U [ t, T ] → V [ t, T ] suc h that, for any τ ∈ [ t, T ] and u ( · ), u ′ ( · ) ∈ U [ t, T ], if u ( ξ ) = u ′ ( ξ ) for a.e. ξ ∈ [ t, τ ], then v ( ξ ) = v ′ ( ξ ) for a.e. ξ ∈ [ t, τ ], where v ( · ) : = β [ u ( · )]( · ) and v ′ ( · ) : = β [ u ′ ( · )]( · ). So, the upp er value of the game is ρ + ( t, x ( · )) : = sup β ∈B [ t,T ] inf u ( · ) ∈U [ t,T ] J  t, x ( · ) , u ( · ) , β [ u ( · )]( · )  , (79) where B [ t, T ] denotes the set of all second pla y er’s non-an ticipative strategies β . In addition, the corresp onding functionals ρ − : [0 , T ] × C ([ − h, T ] , R n ) → R and ρ + : [0 , T ] × C ([ − h, T ] , R n ) → R are called the lower and upp er value functionals resp ectiv ely . If ρ − and ρ + coincide, it is said that the game ( 8 ), ( 9 ) has the value ρ ( t, x ( · )) : = ρ − ( t, x ( · )) = ρ + ( t, x ( · )) , (80) where ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ), and ρ : [0 , T ] × C ([ − h, T ] , R n ) → R is called the value functional . Remark 7.1. Let mappings ( f 1 , χ 1 ) and ( f 2 , χ 2 ) satisfy conditions (C.1)–(C.5) and a functional σ satisfy condition (C.6) (see Assumption 1.3 ). Supp ose that, for every y ( · ) ∈ C ([ − h, T ] , R n ), there exists a set E : = E ( y ( · )) ⊂ [0 , T ] with µ ( E ) = T and suc h that, for any τ ∈ E , u ∈ P , v ∈ Q , f 1 ( τ , y ( · ) , u, v ) = f 2 ( τ , y ( · ) , u, v ) , χ 1 ( τ , y ( · ) , u, v ) = χ 2 ( τ , y ( · ) , u, v ) . Consider the low er ρ − 1 (resp ectiv ely , ρ − 2 ) and upp er ρ + 1 (resp ectiv ely , ρ + 2 ) v alue functionals of the diﬀerential game ( 8 ), ( 9 ) with ( f , χ ) = ( f 1 , χ 1 ) (resp ectiv ely , with ( f , χ ) = ( f 2 , χ 2 )). Then, it follows directly from the abov e deﬁnitions that ρ − 1 = ρ − 2 and ρ + 1 = ρ + 2 . In this regard, we note that conditions (C.2) and (C.4) can be somewhat strengthened when dealing with lo w er or upp er v alue functionals. Namely , we can assume that mapping ( 10 ) is contin uous for all τ ∈ [0 , T ] and that inequalit y ( 12 ) holds for all τ ∈ [0 , T ], y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . Consider the low er H − and upp er H + Hamiltonians that corresp ond to the diﬀeren tial game ( 8 ), ( 9 ) and are deﬁned by ( 15 ) and ( 16 ). It can b e directly v eriﬁed that H − and H + satisfy conditions (A.1)–(A.5) of Assumption 1.1 and condition (A.4) is fulﬁlled with c H − ( · ) = c f ( · ) and c H + ( · ) = c f ( · ). The cen tral result of this section is the following. 32 MIKHAIL GOMOYUNO V Theorem 7.2. L et Assumption 1.3 b e fulﬁl le d in the diﬀer ential game ( 8 ) , ( 9 ) . Then, the lower value functional ρ − ( r esp e ctively, upp er value functional ρ + ) c o- incides with the minimax solution φ − ( r esp e ctively, minimax solution φ + ) of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H − ( r esp e ctively, with H = H + ) . Recall that the statement of Theorem 7.2 tak es place under Assumption 1.4 b y virtue of [ 3 , Theorem 7.1], where a more general case of inﬁnite dimensional path- dep enden t Hamilton–Jacobi equations is considered (in this connection, see also, e.g., [ 27 , 28 ]). Using this fact, we ﬁrst establish a weak er version of Theorem 7.2 . Lemma 7.3. Supp ose that c onditions (C.1) and (C.6) of Assumption 1.3 ar e ful- ﬁl le d, mapping ( 10 ) is c ontinuous for al l τ ∈ [0 , T ] , and ther e exists a numb er Λ > 0 such that, for any τ ∈ [0 , T ] , y ( · ) , y 1 ( · ) , y 2 ( · ) ∈ C ([ − h, T ] , R n ) , u ∈ P , v ∈ Q , ∥ f ( τ , y 1 ( · ) , u, v ) − f ( τ , y 2 ( · ) , u, v ) ∥ + | χ ( τ , y 1 ( · ) , u, v ) − χ ( τ , y 2 ( · ) , u, v ) | ≤ Λ ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ (81) and ∥ f ( τ , y ( · ) , u, v ) ∥ + | χ ( τ , y ( · ) , u, v ) | ≤ Λ(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) . (82) Then, the e qualities ρ − = φ − and ρ + = φ + ar e valid. Pr o of. Let ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) b e ﬁxed. Below, we prov e the equality ρ − ( t, x ( · )) = φ − ( t, x ( · )) only , since the pro of of the equality ρ + ( t, x ( · )) = φ + ( t, x ( · )) is similar. W e split the pro of in to three steps for conv enience. Step 1. Fix k ∈ N . By the Scorza Dragoni theorem, there exists a closed set A k ⊂ [0 , T ] with µ ([0 , T ] \ A k ) ≤ T /k and suc h that the restriction of ( f , χ ) to the set A k × C ([ − h, T ] , R n ) × P × Q is contin uous. Put F k : = A k ∪ { 0 , T } and note that F k is closed with µ ([0 , T ] \ F k ) ≤ T /k and the restriction of ( f , χ ) to the set F k × C ([ − h, T ] , R n ) × P × Q is con tinuous. W e can assume that F k  = [0 , T ] since, otherwise, we can replace F k b y F k \ (0 , T /k ) without any changes. Denote G k : = [0 , T ] \ F k  = ∅ . Since F k is closed and 0, T ∈ F k , the set G k is op en (in R ). Hence (see, e.g., [ 30 , Section I I.5, Theorem 3]), G k is a union of a family of pairwise disjoin t interv als ( a i k , b i k ), where a i k , b i k ∈ F k , a i k < b i k , i ∈ I k , and I k ⊂ N is a set of indices. Consider a mapping ( f k , χ k ) : [0 , T ] × C ([ − h, T ] , R n ) × P × Q → R n × R deﬁned for an y y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q by f k ( τ , y ( · ) , u, v ) : = f ( τ , y ( · ) , u, v ) , χ k ( τ , y ( · ) , u, v ) : = χ ( τ , y ( · ) , u, v ) (83) if τ ∈ F k and b y f k ( τ , y ( · ) , u, v ) : = b i k − τ b i k − a i k f ( a i k , y ( · ) , u, v ) + τ − a i k b i k − a i k f ( b i k , y ( · ∧ τ ) , u, v ) , χ k ( τ , y ( · ) , u, v ) : = b i k − τ b i k − a i k χ ( a i k , y ( · ) , u, v ) + τ − a i k b i k − a i k χ ( b i k , y ( · ∧ τ ) , u, v ) (84) if τ ∈ ( a i k , b i k ) and i ∈ I k . Thanks to ( 81 ) and ( 82 ), we hav e ∥ f k ( τ , y 1 ( · ) , u, v ) − f k ( τ , y 2 ( · ) , u, v ) ∥ + | χ k ( τ , y 1 ( · ) , u, v ) − χ k ( τ , y 2 ( · ) , u, v ) | ≤ Λ ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ (85) and ∥ f k ( τ , y ( · ) , u, v ) ∥ + | χ k ( τ , y ( · ) , u, v ) | ≤ Λ(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) (86) for all τ ∈ [0 , T ], y ( · ), y 1 ( · ), y 2 ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . P A TH-DEPENDENT HJ EQUA TIONS 33 No w, let us verify that ( f k , χ k ) is contin uous. W e deal with the mapping f k only , since the pro of for χ k is similar. Note that, in view of ( 85 ), it suﬃces to ﬁx y ( · ) ∈ C ([ − h, T ] , R n ), tak e ( τ , u, v ), ( τ j , u j , v j ) ∈ [0 , T ] × P × Q , j ∈ N , suc h that lim j →∞  | τ j − τ | + ∥ u j − u ∥ + ∥ v j − v ∥  = 0 , and pro ve that f k ( τ j , y ( · ) , u j , v j ) → f k ( τ , y ( · ) , u, v ) as j → ∞ . Directly from the deﬁnition of f k and con tin uity of the restriction of f to F k × C ([ − h, T ] , R n ) × P × Q , it follows that only the case where τ ∈ F k and τ j ∈ G k for all j ∈ N requires sp ecial analysis. Moreov er, we can limit ourselv es to considering only t wo situations: either τ j > τ for all j ∈ N or τ j < τ for all j ∈ N . Supp ose that the ﬁrst situation tak es place. F or every j ∈ N , consider i ( j ) ∈ I k suc h that τ j ∈ ( a i ( j ) k , b i ( j ) k ). Observ e that τ j > a i ( j ) k ≥ τ for all j ∈ N and, therefore, a i ( j ) k → τ as j → ∞ . Fix ε > 0. Denote D ∗ : = { y ( · ∧ ¯ τ ) : ¯ τ ∈ [0 , T ] } . The restriction of f to the compact set F k × D ∗ × P × Q is uniformly contin uous and b ounded. Hence, ﬁrst, there exists δ > 0 such that, for ev ery j ∈ N , if b i ( j ) k − a i ( j ) k ≤ δ , then ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ ≤ ε/ 2 , and, second, there exists j ∗ ∈ N such that, for every j ≥ j ∗ , ∥ f ( a i ( j ) k , y ( · ) , u j , v j ) − f ( τ , y ( · ) , u, v ) ∥ ≤ ε/ 2 and τ j − a i ( j ) k δ  ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ + ∥ f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥  ≤ ε 2 . Let j ≥ j ∗ . In accordance with ( 83 ) and ( 84 ), noting that f ( a i ( j ) k , y ( · ) , u j , v j ) = f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) thanks to ( 81 ) and since τ j > a i ( j ) k , w e deriv e ∥ f k ( τ j , y ( · ) , u j , v j ) − f k ( τ , y ( · ) , u, v ) ∥ ≤ τ j − a i ( j ) k b i ( j ) k − a i ( j ) k ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ) , u j , v j ) ∥ + ∥ f ( a i ( j ) k , y ( · ) , u j , v j ) − f ( τ , y ( · ) , u, v ) ∥ ≤ τ j − a i ( j ) k b i ( j ) k − a i ( j ) k ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ + ε 2 . F urther, in the case where b i ( j ) k − a i ( j ) k ≤ δ , w e obtain τ j − a i ( j ) k b i ( j ) k − a i ( j ) k ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ ≤ ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ ≤ ε/ 2 . Otherwise, w e ha v e b i ( j ) k − a i ( j ) k > δ , and, therefore, τ j − a i ( j ) k b i ( j ) k − a i ( j ) k ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) − f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ 34 MIKHAIL GOMOYUNO V ≤ τ j − a i ( j ) k δ  ∥ f ( b i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥ + ∥ f ( a i ( j ) k , y ( · ∧ τ j ) , u j , v j ) ∥  ≤ ε/ 2 . Consequen tly , w e arrive at the inequality ∥ f k ( τ j , y ( · ) , u j , v j ) − f k ( τ , y ( · ) , u, v ) ∥ ≤ ε and complete the pro of of the desired conv ergence. The situation where τ j < τ for all j ∈ N is handled in a similar manner. In particular, we conclude that ( f k , χ k ) satisﬁes conditions (D.1)–(D.3) of As- sumption 1.4 . Step 2. F or ev ery k ∈ N , consider the diﬀerential game ( 8 ), ( 9 ) with ( f , χ ) = ( f k , χ k ) and the low er v alue of this game ρ − k ( t, x ( · )) (see ( 78 )). Let us pro v e that lim k →∞ ρ − k ( t, x ( · )) = ρ − ( t, x ( · )) . (87) Let ε > 0 b e ﬁxed. Deﬁne the compact set Y ( t, x ( · ); c Λ ( · )) according to ( 19 ) with c ( · ) = c Λ ( · ), where c Λ ( τ ) : = Λ for all τ ∈ [0 , T ], and c ho ose R > 0 such that ∥ y ( · ) ∥ ∞ ≤ R for all y ( · ) ∈ Y ( t, x ( · ); c Λ ( · )). Since the functional σ is con tinuous b y condition (C.6), there exists ζ > 0 such that | σ ( y 1 ( · )) − σ ( y 2 ( · )) | ≤ ε/ 2 for all y 1 ( · ), y 2 ( · ) ∈ Y ( t, x ( · ); c Λ ( · )) satisfying the inequalit y ∥ y 1 ( · ) − y 2 ( · ) ∥ ∞ ≤ ζ . Cho ose k ∗ ∈ N from the conditions 2Λ(1 + R ) T e Λ T /k ∗ ≤ ζ , 2Λ(1 + R ) T (Λ T e Λ T + 1) /k ∗ ≤ ε/ 2 . Let k ≥ k ∗ , u ( · ) ∈ U [ t, T ], v ( · ) ∈ V [ t, T ], and let y ( · ) : = y ( · ; t, x ( · ) , u ( · ) , v ( · )) b e the motion of system ( 8 ) and y k ( · ) : = y k ( · ; t, x ( · ) , u ( · ) , v ( · )) be the motion of system ( 8 ) with f = f k . Note that y ( · ), y k ( · ) ∈ Y ( t, x ( · ); c Λ ( · )). Due to ( 83 ), we hav e ∥ y ( τ ) − y k ( τ ) ∥ ≤ Z τ t ∥ f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f k ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ ≤ Z τ t ∥ f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ + Z [ t,τ ] \ F k ∥ f ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ + Z [ t,τ ] \ F k ∥ f k ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ for all τ ∈ [ t, T ]. Hence, using ( 81 ), ( 82 ), and ( 86 ) and taking the inequality µ ([0 , T ] \ F k ) ≤ T /k into account, we obtain ∥ y ( τ ) − y k ( τ ) ∥ ≤ Λ Z τ t ∥ y ( · ∧ ξ ) − y k ( · ∧ ξ ) ∥ ∞ d ξ + 2Λ(1 + R ) T /k for all τ ∈ [ t, T ], whic h yields ∥ y ( · ∧ τ ) − y k ( · ∧ τ ) ∥ ∞ ≤ Λ Z τ t ∥ y ( · ∧ ξ ) − y k ( · ∧ ξ ) ∥ ∞ d ξ + 2Λ(1 + R ) T /k for all τ ∈ [ t, T ]. Then, applying the Gronw all inequality , w e derive ∥ y ( · ∧ τ ) − y k ( · ∧ τ ) ∥ ∞ ≤ 2Λ(1 + R ) T e Λ T /k for all τ ∈ [ t, T ]. In particular, ∥ y ( · ) − y k ( · ) ∥ ∞ ≤ ζ , whic h implies the inequalit y | σ ( y ( · )) − σ ( y k ( · )) | ≤ ε/ 2. In addition, arguing similarly to the ab o ve, we get     Z T t χ ( τ , y ( · ) , u ( τ ) , v ( τ )) d τ − Z T t χ k ( τ , y k ( · ) , u ( τ ) , v ( τ )) d τ     P A TH-DEPENDENT HJ EQUA TIONS 35 ≤ Z T t | χ ( τ , y ( · ) , u ( τ ) , v ( τ )) − χ ( τ , y k ( · ) , u ( τ ) , v ( τ )) | d τ + Z [ t,T ] \ F k | χ ( τ , y k ( · ) , u ( τ ) , v ( τ )) | d τ + Z [ t,T ] \ F k | χ k ( τ , y k ( · ) , u ( τ ) , v ( τ )) | d τ ≤ Λ Z T t ∥ y ( · ∧ τ ) − y k ( · ∧ τ ) ∥ ∞ d τ + 2Λ(1 + R ) T /k ≤ 2Λ 2 (1 + R ) T 2 e Λ T /k + 2Λ(1 + R ) T /k ≤ ε/ 2 . Th us, w e come to the inequality | J ( t, x ( · ) , u ( · ) , v ( · )) − J k ( t, x ( · ) , u ( · ) , v ( · )) | ≤ ε, (88) where J k ( t, x ( · ) , u ( · ) , v ( · )) is the v alue of the cost functional ( 9 ) with ( f , χ ) = ( f k , χ k ). Since inequalit y ( 88 ) holds for all u ( · ) ∈ U [ t, T ] and v ( · ) ∈ V [ t, T ], | ρ − ( t, x ( · )) − ρ − k ( t, x ( · )) | ≤ ε, (89) whic h completes the pro of of equality ( 87 ). Step 3. Let k ∈ N . Consider the Hamiltonian H − k ( τ , y ( · ) , s ) : = max v ∈ Q min u ∈ P  ⟨ s, f k ( τ , y ( · ) , u, v ) ⟩ − χ k ( τ , y ( · ) , u, v )  , (90) where τ ∈ [0 , T ], y ( · ) ∈ C ([ − h, T ] , R n ), and s ∈ R . Note that H − k satisﬁes conditions (B.1)–(B.3) of Assumption 1.2 and condition (B.3) is fulﬁlled with c H − k = Λ. Let φ − k b e the minimax solution of the Cauch y problem ( 1 ), ( 2 ) with H = H − k . Due to [ 3 , Theorem 7.1], we hav e ρ − k ( t, x ( · )) = φ − k ( t, x ( · )) . (91) Let a compact set D ⊂ C ([ − h, T ] , R n ) and s ∈ R n b e ﬁxed. F or every y ( · ) ∈ D , noting that H − k ( τ , y ( · ) , s ) = H − ( τ , y ( · ) , s ) for all τ ∈ F k thanks to ( 83 ), taking into accoun t that (see ( 15 ) and ( 82 )) | H − ( τ , y ( · ) , s ) | ≤ | H − ( τ , y ( · ) , s ) − H − ( τ , y ( · ) , 0) | + | H − ( τ , y ( · ) , 0) | ≤ Λ(1 + ∥ s ∥ )(1 + max z ( · ) ∈ D ∥ z ( · ) ∥ ∞ ) for all τ ∈ [0 , T ] and, similarly (see ( 86 ) and ( 90 )), | H − k ( τ , y ( · ) , s ) | ≤ Λ(1 + ∥ s ∥ )(1 + max z ( · ) ∈ D ∥ z ( · ) ∥ ∞ ) for all τ ∈ [0 , T ], and recalling that µ ([0 , T ] \ F k ) ≤ T /k , we derive Z T 0 | H − k ( τ , y ( · ) , s ) − H − ( τ , y ( · ) , s ) | d τ ≤ Z [0 ,T ] \ F k | H − k ( τ , y ( · ) , s ) | d τ + Z [0 ,T ] \ F k | H − ( τ , y ( · ) , s ) | d τ ≤ 2Λ(1 + ∥ s ∥ )(1 + max z ( · ) ∈ D ∥ z ( · ) ∥ ∞ ) T /k . Consequen tly , the ﬁrst equalit y in ( 61 ) with H = H − and H k = H − k is v alid. Th us, φ − k ( t, x ( · )) → φ − ( t, x ( · )) as k → ∞ by Theorem 5.1 . Therefore, o wing to ( 87 ) and ( 91 ), we get ρ − ( t, x ( · )) = φ − ( t, x ( · )) and complete the pro of. 36 MIKHAIL GOMOYUNO V Remark 7.4. Even if the mapping ( f , χ ) satisﬁes Assumption 1.5 , the mappings ( f k , χ k ), k ∈ N , deﬁned according to ( 83 ) and ( 84 ) may not satisfy this assumption. In particular, the corresp onding approximating diﬀerential games may not hav e a v alue, and if we follow the same reasoning as in the proof of Lemma 7.3 , w e still need to deal with the low er and upper v alue functionals separately . Remark 7.5. As in the pro of of Theorem 6.3 , we can deﬁne approximating map- pings ( f k , χ k ), k ∈ N , in the pro of of Lem ma 7.3 using the Steklov transformation of ( f , χ ) with resp ect to the ﬁrst v ariable τ (for ﬁxed y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , and v ∈ Q ). How ever, in this case, diﬃculties arise in justifying equality ( 87 ), for example, when obtaining estimates for the v alues lik e Z τ t ∥ f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ = Z τ t     k 2 Z ξ +1 /k ξ − 1 /k f ( η , y ( · ) , u ( ξ ) , v ( ξ )) d η − f ( ξ , y ( · ) , u ( ξ ) , v ( ξ ))     d ξ . The problem is that diﬀerent times η and ξ are inv olved in f ( η , y ( · ) , u ( ξ ) , v ( ξ )). No w, relying on Lemma 7.3 , we prov e Theorem 7.2 . Pr o of of The or em 7.2 . Let us ﬁx ( t, x ( · )) ∈ [0 , T ] × C ([ − h, T ] , R n ) and prov e only that ρ − ( t, x ( · )) = φ − ( t, x ( · )) (the pro of of the equalit y ρ + ( t, x ( · )) = φ + ( t, x ( · )) is similar). F or conv enience, w e split the pro of into three steps, which are in natural agreemen t with the corresp onding steps of the pro of of Lemma 7.3 . Step 1. T ake the function c f ( · ) from condition (C.4) (see Assumption 1.3 ) and deﬁne the set Y : = Y ( t, x ( · ); c f ( · )) according to ( 19 ) with c ( · ) = c f ( · ). Recall that Y is compact by Prop osition 2.3 and consider the functions λ f ,χ ( · ) : = λ f ,χ ( · ; Y ) and m χ ( · ) : = m χ ( · ; Y ) from conditions (C.3) and (C.5) resp ectiv ely . T aking Remarks 2.7 and 7.1 into account, w e can assume in the pro of that mapping ( 10 ) is contin uous for all τ ∈ [0 , T ] and that inequalities ( 11 )–( 13 ) hold for all τ ∈ [0 , T ], y ( · ), y 1 ( · ), y 2 ( · ) ∈ Y , u ∈ P , v ∈ Q . F or ev ery τ ∈ [0 , T ], denote λ ∗ ( τ ) : = (1 + √ n ) λ f ,χ ( τ ) , c ∗ ( τ ) : = ( c f ( τ ) + λ ∗ ( τ ))(1 + ∥ x ( · ) ∥ ∞ ) , m ∗ ( τ ) : = m χ ( τ ) + λ ∗ ( τ )(1 + ∥ x ( · ) ∥ ∞ ) . (92) Fix k ∈ N . Applying the Scorza Dragoni theorem to the mapping ( f , χ ) and the Lusin theorem to the functions λ ∗ ( · ), c ∗ ( · ), m ∗ ( · ), we c ho ose a closed set F k ⊂ [0 , T ] with µ ([0 , T ] \ F k ) ≤ 1 /k and such that the restriction of ( f , χ ) to F k × Y × P × Q and the restrictions of λ ∗ ( · ), c ∗ ( · ), m ∗ ( · ) to F k are con tin uous. Then, in accordance with the McShane–Whitney extension theorem, we deﬁne a mapping ( ¯ f k , ¯ χ k ) : F k × C ([ − h, T ] , R n ) × P × Q → R n × R b y ¯ f i k ( τ , y ( · ) , u, v ) : = m ax z ( · ) ∈ Y  f i ( τ , z ( · ) , u, v ) − λ f ,χ ( τ ) ∥ z ( · ∧ τ ) − y ( · ∧ τ ) ∥ ∞  , ¯ χ k ( τ , y ( · ) , u, v ) : = m ax z ( · ) ∈ Y  χ ( τ , z ( · ) , u, v ) − λ f ,χ ( τ ) ∥ z ( · ∧ τ ) − y ( · ∧ τ ) ∥ ∞  (93) for all τ ∈ F k , y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , and v ∈ Q . Here, ¯ f i k ( τ , y ( · ) , u, v ) (resp ectiv ely , f i ( τ , z ( · ) , u, v )) is the i -th co ordinate of the vector ¯ f k ( τ , y ( · ) , u, v ) (resp ectiv ely , f ( τ , z ( · ) , u, v )) and i ∈ 1 , n . The restrictions of ( ¯ f k , ¯ χ k ) and ( f , χ ) to P A TH-DEPENDENT HJ EQUA TIONS 37 F k × Y × P × Q coincide and ∥ ¯ f k ( τ , y 1 ( · ) , u, v ) − ¯ f k ( τ , y 2 ( · ) , u, v ) ∥ + | ¯ χ k ( τ , y 1 ( · ) , u, v ) − ¯ χ k ( τ , y 2 ( · ) , u, v ) | ≤ λ ∗ ( τ ) ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ (94) for all τ ∈ F k , y 1 ( · ), y 2 ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . In addition, the mapping ( ¯ f k , ¯ χ k ) is con tin uous and, for any τ ∈ F k , y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q , thanks to the inclusion x ( · ∧ t ) ∈ Y , w e deriv e ∥ ¯ f k ( τ , y ( · ) , u, v ) ∥ ≤ ∥ ¯ f k ( τ , y ( · ) , u, v ) − ¯ f k ( τ , x ( · ∧ t ) , u, v ) ∥ + ∥ f ( τ , x ( · ∧ t ) , u, v ) ∥ ≤ λ ∗ ( τ ) ∥ y ( · ∧ τ ) − x ( · ) ∥ ∞ + c f ( τ )(1 + ∥ x ( · ) ∥ ∞ ) ≤ c ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) (95) and, similarly , | ¯ χ k ( τ , y ( · ) , u, v ) | ≤ m ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) . (96) Consider a mapping ( f k , χ k ) : [0 , T ] × C ([ − h, T ] , R n ) × P × Q → R n × R deﬁned for all y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q by f k ( τ , y ( · ) , u, v ) : = ¯ f k ( τ , y ( · ) , u, v ) , χ k ( τ , y ( · ) , u, v ) : = ¯ χ k ( τ , y ( · ) , u, v ) (97) if τ ∈ F k and f k ( τ , y ( · ) , u, v ) : = 0 , χ k ( τ , y ( · ) , u, v ) : = 0 (98) otherwise. Owing to con tinuit y of the mapping ( ¯ f k , ¯ χ k ), the mapping ( f k , χ k ) sat- isﬁes condition (C.1) and the mapping ( y ( · ) , u, v ) 7→ ( f k ( τ , y ( · ) , u, v ) , χ k ( τ , y ( · ) , u, v )) , C ([ − h, T ] , R n ) × P × Q → R n × R , is con tin uous for all τ ∈ [0 , T ]. Thanks to ( 94 )–( 96 ), we hav e ∥ f k ( τ , y 1 ( · ) , u, v ) − f k ( τ , y 2 ( · ) , u, v ) ∥ + | χ k ( τ , y 1 ( · ) , u, v ) − χ k ( τ , y 2 ( · ) , u, v ) | ≤ λ ∗ ( τ ) ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ (99) and ∥ f k ( τ , y ( · ) , u, v ) ∥ ≤ c ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) , | χ k ( τ , y ( · ) , u, v ) | ≤ m ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) (100) for all τ ∈ [0 , T ], y ( · ), y 1 ( · ), y 2 ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . Recalling that the set F k is compact and the restrictions of the functions λ ∗ ( · ), c ∗ ( · ), m ∗ ( · ) to F k are conctinuous, w e conclude that inequalities ( 81 ) and ( 82 ) with ( f , χ ) = ( f k , χ k ) and Λ = Λ k : = max τ ∈ F k max { λ ∗ ( τ ) , c ∗ ( τ ) + m ∗ ( τ ) } hold for all τ ∈ [0 , T ], y ( · ), y 1 ( · ), y 2 ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . Th us, the mapping ( f k , χ k ) satisﬁes the assumptions of Lemma 7.3 . Step 2. F or ev ery k ∈ N , consider the diﬀerential game ( 8 ), ( 9 ) with ( f , χ ) = ( f k , χ k ) and deﬁne the corresp onding lo wer game v alue ρ − k ( t, x ( · )). Let us pro v e that equalit y ( 87 ) tak es place. Let ε > 0. Cho ose R > 0 such that ∥ y ( · ) ∥ ∞ ≤ R for all y ( · ) ∈ Y . Deﬁne the compact set Y ( t, x ( · ); c ∗ ( · )). By condition (C.6), there exists ζ > 0 suc h that | σ ( y 1 ( · )) − σ ( y 2 ( · )) | ≤ ε/ 2 for all y 1 ( · ), y 2 ( · ) ∈ Y ( t, x ( · ); c ∗ ( · )) satisfying the inequal- it y ∥ y 1 ( · ) − y 2 ( · ) ∥ ∞ ≤ ζ . T aking into account that µ ([0 , T ] \ F k ) → 0 as k → ∞ , c ho ose k ∗ ∈ N from the conditions 2(1 + R ) e ∥ λ ∗ ( · ) ∥ 1 Z [0 ,T ] \ F k c ∗ ( ξ ) d ξ ≤ ζ , 38 MIKHAIL GOMOYUNO V 2(1 + R )  Z [0 ,T ] \ F k m ∗ ( τ ) d τ + ∥ λ ∗ ( · ) ∥ 1 e ∥ λ ∗ ( · ) ∥ 1 Z [0 ,T ] \ F k c ∗ ( τ ) d τ  ≤ ε 2 . Let k ≥ k ∗ , u ( · ) ∈ U [ t, T ], v ( · ) ∈ V [ t, T ], and let y ( · ) : = y ( · ; t, x ( · ) , u ( · ) , v ( · )) b e the motion of system ( 8 ) and y k ( · ) : = y k ( · ; t, x ( · ) , u ( · ) , v ( · )) be the motion of system ( 8 ) with f = f k . Note that y ( · ) ∈ Y ⊂ Y ( t, x ( · ); c ∗ ( · )), y k ( · ) ∈ Y ( t, x ( · ); c ∗ ( · )), and (see also ( 97 )) f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) = ¯ f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) = f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) for all ξ ∈ [ t, T ] ∩ F k . Then, using ( 99 ) and ( 100 ), we obtain ∥ y ( τ ) − y k ( τ ) ∥ ≤ Z τ t ∥ f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f k ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ ≤ Z τ t ∥ f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ + Z τ t ∥ f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) − f k ( ξ , y k ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ ≤ Z [ t,τ ] \ F k ∥ f ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ + Z [ t,τ ] \ F k ∥ f k ( ξ , y ( · ) , u ( ξ ) , v ( ξ )) ∥ d ξ + Z τ t λ ∗ ( ξ ) ∥ y ( · ∧ ξ ) − y k ( · ∧ ξ ) ∥ ∞ d ξ ≤ Z τ t λ ∗ ( ξ ) ∥ y ( · ∧ ξ ) − y k ( · ∧ ξ ) ∥ ∞ d ξ + 2(1 + R ) Z [0 ,T ] \ F k c ∗ ( ξ ) d ξ for all τ ∈ [ t, T ]. Therefore, based on the Gronw all inequality , we derive ∥ y ( · ∧ τ ) − y k ( · ∧ τ ) ∥ ∞ ≤ 2(1 + R ) e ∥ λ ∗ ( · ) ∥ 1 Z [0 ,T ] \ F k c ∗ ( ξ ) d ξ for all τ ∈ [ t, T ]. In particular, ∥ y ( · ) − y k ( · ) ∥ ∞ ≤ ζ , whic h implies the inequalit y | σ ( y ( · )) − σ ( y k ( · )) | ≤ ε/ 2. In a similar wa y , we hav e     Z T t χ ( τ , y ( · ) , u ( τ ) , v ( τ )) d τ − Z T t χ k ( τ , y k ( · ) , u ( τ ) , v ( τ )) d τ     ≤ Z [ t,T ] \ F k | χ ( τ , y ( · ) , u ( τ ) , v ( τ )) | d τ + Z [ t,T ] \ F k | χ k ( τ , y ( · ) , u ( τ ) , v ( τ )) | d τ + Z T t λ ∗ ( τ ) ∥ y ( · ∧ τ ) − y k ( · ∧ τ ) ∥ ∞ d τ ≤ 2(1 + R ) Z [0 ,T ] \ F k m ∗ ( τ ) d τ + 2(1 + R ) ∥ λ ∗ ( · ) ∥ 1 e ∥ λ ∗ ( · ) ∥ 1 Z [0 ,T ] \ F k c ∗ ( τ ) d τ ≤ ε/ 2 . Th us, inequality ( 88 ) is satisﬁed for all u ( · ) ∈ U [ t, T ] and v ( · ) ∈ V [ t, T ], which yields ( 89 ) and completes the pro of of ( 87 ). Step 3. F or every k ∈ N , consider the Hamiltonian H − k giv en b y ( 90 ). Note that H − k satisﬁes conditions (A.1)–(A.5) of Assumption 1.1 and condition (A.4) is fulﬁlled with c H − k ( · ) = c ∗ ( · ) owing to the ﬁrst inequalit y in ( 100 ). Denote b y φ − k the minimax solution of the Cauch y problem ( 1 ), ( 2 ) with H = H − k . By Lemma P A TH-DEPENDENT HJ EQUA TIONS 39 7.3 , equality ( 91 ) tak es place. Thus, φ − k ( t, x ( · )) → ρ − ( t, x ( · )) as k → ∞ thanks to ( 87 ), and it remains to show that φ − k ( t, x ( · )) → φ − ( t, x ( · )) as k → ∞ . Let F ∗ b e the union of the sets F k o ver k ∈ N . Note that µ ( F ∗ ) = T . Consider a mapping ( f ∗ , χ ∗ ) : [0 , T ] × C ([ − h, T ] , R n ) × P × Q → R n × R deﬁned for all y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q by f i ∗ ( τ , y ( · ) , u, v ) : = max z ( · ) ∈ Y  f i ( τ , z ( · ) , u, v ) − λ f ,χ ( τ ) ∥ z ( · ∧ τ ) − y ( · ∧ τ ) ∥ ∞  , χ ∗ ( τ , y ( · ) , u, v ) : = max z ( · ) ∈ Y  χ ( τ , z ( · ) , u, v ) − λ f ,χ ( τ ) ∥ z ( · ∧ τ ) − y ( · ∧ τ ) ∥ ∞  if τ ∈ F ∗ and f ∗ ( τ , y ( · ) , u, v ) : = 0 , χ ∗ ( τ , y ( · ) , u, v ) : = 0 otherwise. Here, f i ∗ ( τ , y ( · ) , u, v ) (resp ectiv ely , f i ( τ , z ( · ) , u, v )) is the i -th co ordinate of the vector f ∗ ( τ , y ( · ) , u, v ) (resp ectiv ely , f ( τ , z ( · ) , u, v )) and i ∈ 1 , n . By construc- tion, w e ha v e f ∗ ( τ , y ( · ) , u, v ) = f ( τ , y ( · ) , u, v ) , χ ∗ ( τ , y ( · ) , u, v ) = χ ( τ , y ( · ) , u, v ) (101) for all τ ∈ F ∗ , y ( · ) ∈ Y , u ∈ P , v ∈ Q . In addition, for ev ery k ∈ N , taking ( 93 ) and ( 97 ) into account, we obtain f ∗ ( τ , y ( · ) , u, v ) = f k ( τ , y ( · ) , u, v ) , χ ∗ ( τ , y ( · ) , u, v ) = χ k ( τ , y ( · ) , u, v ) (102) for all τ ∈ F k , y ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . This implies that the mapping ( f ∗ , χ ∗ ) satisﬁes conditions (C.1) and (C.2). Moreov er, owing to ( 99 ) and ( 100 ), we ha ve ∥ f ∗ ( τ , y 1 ( · ) , u, v ) − f ∗ ( τ , y 2 ( · ) , u, v ) ∥ + | χ ∗ ( τ , y 1 ( · ) , u, v ) − χ ∗ ( τ , y 2 ( · ) , u, v ) | ≤ λ ∗ ( τ ) ∥ y 1 ( · ∧ τ ) − y 2 ( · ∧ τ ) ∥ ∞ (103) and ∥ f ∗ ( τ , y ( · ) , u, v ) ∥ ≤ c ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) , | χ ∗ ( τ , y ( · ) , u, v ) | ≤ m ∗ ( τ )(1 + ∥ y ( · ∧ τ ) ∥ ∞ ) (104) for all τ ∈ F ∗ , y ( · ), y 1 ( · ), y 2 ( · ) ∈ C ([ − h, T ] , R n ), u ∈ P , v ∈ Q . Therefore, the mapping ( f ∗ , χ ∗ ) satisﬁes conditions (C.3)–(C.5). Consider a Hamiltonian H − ∗ : [0 , T ] × C ([ − h, T ] , R n ) × R n → R deﬁned for any y ( · ) ∈ C ([ − h, T ] , R n ) and s ∈ R n b y H − ∗ ( τ , y ( · ) , s ) : = max v ∈ Q min u ∈ P  ⟨ s, f ∗ ( τ , y ( · ) , u, v ) ⟩ − χ ∗ ( τ , y ( · ) , u, v )  if τ ∈ F ∗ and b y H − ∗ ( τ , y ( · ) , s ) : = 0 otherwise. Let φ − ∗ b e the minimax solution of the Cauch y problem ( 1 ), ( 2 ) with H = H − ∗ . Owing to ( 101 ), for any τ ∈ F ∗ , y ( · ) ∈ Y , s ∈ R n , H − ( τ , y ( · ) , s ) = H − ∗ ( τ , y ( · ) , s ) , and, therefore, φ − ( t, x ( · )) = φ − ∗ ( t, x ( · )) by Corollary 4.3 . Let a compact set D ⊂ C ([ − h, T ] , R n ) and s ∈ R n b e ﬁxed. F or any k ∈ N and y ( · ) ∈ D , taking into account that H − k ( τ , y ( · ) , s ) = H − ∗ ( τ , y ( · ) , s ) for all τ ∈ F k b y ( 102 ), H − k ( τ , y ( · ) , s ) = 0 for all τ ∈ [0 , T ] \ F k b y ( 98 ), and | H − ∗ ( τ , y ( · ) , s ) | ≤ | H − ∗ ( τ , y ( · ) , s ) − H − ∗ ( τ , y ( · ) , 0) | + | H − ∗ ( τ , y ( · ) , 0) | ≤ ( c ∗ ( τ ) ∥ s ∥ + m ∗ ( τ ))(1 + max z ( · ) ∈ D ∥ z ( · ) ∥ ∞ ) 40 MIKHAIL GOMOYUNO V for a.e. τ ∈ [0 , T ] by ( 104 ), we obtain Z T 0 | H − k ( τ , y ( · ) , s ) − H − ∗ ( τ , y ( · ) , s ) | d τ ≤ Z [0 ,T ] \ F k | H − k ( τ , y ( · ) , s ) | d τ + Z [0 ,T ] \ F k | H − ∗ ( τ , y ( · ) , s ) | d τ ≤ (1 + max z ( · ) ∈ D ∥ z ( · ) ∥ ∞ ) Z [0 ,T ] \ F k ( c ∗ ( τ ) ∥ s ∥ + m ∗ ( τ )) d τ . Hence, and due to the conv ergence µ ([0 , T ] \ F k ) → 0 as k → ∞ , w e conclude that the ﬁrst equality in ( 61 ) with H = H − ∗ and H k = H − k is v alid. Consequen tly , applying Theorem 5.1 , w e get φ − k ( t, x ( · )) → φ − ∗ ( t, x ( · )) as k → ∞ , whic h completes the pro of. Theorem 7.2 and Corollary 4.3 immediately imply the following result. Theorem 7.6. Under Assumptions 1.3 and 1.5 , the diﬀer ential game ( 8 ) , ( 9 ) has a value, and the value functional ρ c oincides with the minimax solution φ of the Cauchy pr oblem ( 1 ) , ( 2 ) with H = H − ( or with H = H + ) . REFERENCES [1] E. Bandini and C. Keller, Path-dependent Hamilton–Jacobi equations with u -dependence and time-measurable Hamiltonians, Appl. Math. Optim. , 91 (2) (2025), art. 34. [2] E. N. Barron and R. Jensen, Generalized viscosity solutions for Hamilton–Jacobi equations with time-measurable Hamiltonians, J. Diﬀer ential Equations , 68 (1) (1987), 10–21. [3] E. Bayrakta r, M. I. Gomoyuno v, and C. Keller, Viscosity and minimax solutions for path- dependent Hamilton–Jacobi equations in inﬁnite dimensions and related diﬀerential games, preprint, 2025, . [4] E. Bayraktar and C. Keller, Path-dependent Hamilton–Jacobi equations in inﬁnite dimen- sions, J. F unct. Anal. , 275 (8) (2018), 2096–2161. [5] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, R epr esentation and Contr ol of Inﬁnite Dimensional Systems , 2 nd edition, Birkh¨ auser, Boston, 2006. [6] P . Bettiol and R. B. Vinter, Principles of Dynamic Optimization . Springer, Cham, 2024. [7] V. I. Bogachev, Me asure The ory. V ol. I , Springer, Berlin, 2007. [8] A. Briani and F. Rampazzo, A densit y approach to Hamilton–Jacobi equations with t -meas- urable Hamiltonians, NoDEA Nonline ar Diﬀer ential Equations Appl. , 12 (1) (2005), 71–91. [9] S ¸ . Cobza ¸ s, R. Miculescu, and A. Nicolae, Lipschitz F unctions , Springer, Cham, 2019. [10] A. Cosso, F. Gozzi, M. Rosestolato, and F. Russo, Path-dependent Hamilton–Jacobi–Bellman equation: Uniqueness of Crandall–Lions viscosity solutions, preprint, 2021, arXiv:2107.05959 . [11] A. Cosso and F. Russo, Crandall–Lions viscosity solutions for path-dep enden t PDEs: The case of heat equation, Bernoul li , 28 (1) (2022), 481–503. [12] M. G. Crandall, L. C. Ev ans, and P .-L. Lions, Some prop erties of viscosit y solutions of Hamilton–Jacobi equations, T rans. Amer. Math. So c. , 282 (2) (1984), 487–582. [13] M. G. Crandall, H. Ishii, and P .-L. Lions, User’s guide to viscosity solutions of second order partial diﬀerential equations, Bul l. Amer. Math. So c. , 27 (1) (1992), 1–67. [14] M. G. Crandall and P .-L. Lions, Viscosity solutions of H amilton–Jacobi equations, T r ans. Amer. Math. So c. , 277 (1) (1983), 1–42. [15] I. Ekren, C. Keller, N. T ouzi, and J. Zhang, On viscosity solutions of path dependent PDEs, Ann. Pr ob ab. , 42 (1) (2014), 204–236. [16] I. Ekren, N. T ouzi, and J. Zhang, Viscosity solutions of fully nonlinear parab olic path dep en- dent PDEs: Part I, Ann. Pr ob ab. , 44 (2) (2016), 1212–1253. [17] M. I. Gomo yunov and N. Y u. Luko yano v, Minimax solutions of Hamilton–Jacobi equations in dynamic optimization problems for hereditary systems, Russ. Math. Surv. , 79 (2) (2024), 229–324. P A TH-DEPENDENT HJ EQUA TIONS 41 [18] M. I. Gomoyuno v, N. Y u. Luko yano v, and A. R. Plaksin, Path-dependent Hamilton–Jacobi equations: the minimax solutions revised, Appl. Math. Optim. , 84 (1) (2021), S1087–S1117. [19] M. I. Gomoyunov and A. R. Plaksin, Equiv alence of minimax and viscosity solutions of path- dependent Hamilton–Jacobi equations, J. F unct. Anal. , 285 (11) (2023), art. 110155. [20] J. K. Hale and S. M. V. Lunel, Intr o duction to F unctional Diﬀer ential Equations , Springer, New Y ork, 1993. [21] H. Ishii, Hamilton–Jacobi equations with discontin uous Hamiltonians on arbitrary op en sets, Bul l. F ac. Sci. Eng. Chuo Univ. , 28 (1985), 33–77. [22] H. Kaise, Conv ergence of discrete-time deterministic games to path-dep endent Isaacs partial diﬀerential equations under quadratic growth conditions, Appl. Math. Optim. , 86 (1) (2022), art. 13. [23] A. Kucia, Scorza Dragoni type theorems, F und. Math. , 138 (3) (1991), 197–203. [24] P .-L. Lions and B. Perthame, Remarks on Hamilton–Jacobi equations with measurable time- dependent Hamiltonians, Nonline ar Anal. , 11 (5) (1987), 613–621. [25] N. Y u. Luko yano v, Minimax solutions of functional equations of the Hamilton–Jacobi type for hereditary systems, Diﬀer. Equ. , 37 (2) (2001), 246–256. [26] N. Y u. Lukoy anov, On viscosity solution of functional Hamilton–Jacobi type equations for hereditary systems, Pr o c. Steklov Inst. Math. , 259 (Suppl. 2) (2007), S190–S200. [27] N. Y u. Luko yano v, Minimax and viscosity solutions in optimization problems for hereditary systems, Pr o c. Steklov Inst. Math. , 269 (Suppl. 2) (2010), S214–S225. [28] N. Y u. Luko yano v, On Hamilton–Jacobi formalism in time-delay control systems, T r. Inst. Mat. Mekh. , 16 (5) (2010), 269–277. [29] N. Y u. Luko yano v, F unctional Hamilton–Jacobi Equations and Contr ol Pr oblems with Her e d- itary Information , Ural F ederal Univ ersity Publishing, Ek aterinburg, 2011. (in Russian). [30] I. P . Natanson, The ory of F unctions of a R e al V ariable. V ol. 1 , F rederic k Ungar Publishing Co., New Y ork, 1964. [31] I. P . Natanson, The ory of F unctions of a R e al V ariable. V ol. 2 , F rederic k Ungar Publishing Co., New Y ork, 1960. [32] V. V. Obukhovskii, Semilinear functional-diﬀerential inclusions in a Banach space and con- trolled parab olic systems, Soviet J. Automat. Inform. Sci. , 24 (3) (1992), 71–79. [33] A. R. Plaksin, Viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations for time- delay systems, SIAM J. Contr ol Optim. , 59 (3) (2021), 1951–1972. [34] Z. Ren, N. T ouzi, and J. Zhang, An ov erview of viscosity solutions of path-dep endent PDEs, Sto chastic Analysis and Applications, 2014 , Springer Proc. Math. Stat., 100, Springer, Cham, 2014, 397–453. [35] A. I. Subb otin, Generalization of the main equation of diﬀerential game theory, J. Optim. The ory Appl. , 43 (1) (1984), 103–133. [36] A. I. Subb otin, Minimax Ine qualities and Hamilton–Jac obi Equations , Nauk a Publ., Moscow, 1991. (in Russian) [37] A. I. Subbotin, Gener alize d Solutions of First Or der PDEs: The Dynamical Optimization Persp e ctive , Birkh¨ auser, Basel, 1995. [38] D. V. T ran, T. Mikio, and D. T. S. Nguyen, The Char acteristic Metho d and its Gener aliza- tions for First-Or der Nonline ar Partial Diﬀer ential Equations , Chapman & Hall/CRC, Boca Raton, 2000. [39] R. B. Vin ter and P . W olenski, Hamilton–Jacobi theory for optimal control problems with data measurable in time, SIAM J. Contr ol Optim. , 28 (6) (1990), 1404–1419. [40] J. Zhou, Viscosity solutions to ﬁrst order path-dep endent HJB equations, preprint, 2020, arXiv:2004.02095 . [41] J. Zhou, Viscosity solutions to ﬁrst order path-dep enden t Hamilton–Jacobi–Bellman equations in Hilb ert space, Automatic a , 142 (2022), art. 110347. Receiv ed xxxx 20xx; revised xxxx 20xx; early access xxxx 20xx.

Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games

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