Hamilton-Jacobi Equations and Two-Person Zero-Sum Differential Games with Unbounded Controls

Hamilton-Jacobi Equations and Tw o-P erso n Zero-Sum Diﬀeren tial Games with Un b ounded C o n trols ∗ Hong Qiu a,b and Jiongmin Y ong b a Departmen t of Mathematics, Harbin Institute of T echno l ogy , W eihai 264209, Shandong, China b Departmen t of Mathematics, Universit y of Cen tral Florida, Orlando, FL 32816, USA Octob er 13, 2018 Abstract A t wo-person zero-sum d iﬀeren tial game with unb ounded controls is considered. U nder prop er co er- civity co nditions, th e upp er and lo wer v alue functions are chara ct erized as the unique viscosit y solutions to th e correspondin g u pp er and lo wer H amilton–Jacobi– Isaacs equations, respectively . Consequently , when the Isaacs’ cond ition is satisﬁed, the up p er and lo w er v alue functions coincide, leading t o the ex- istence of the value function of the diﬀerential game. Due to the unbound edness of the controls, the correspondin g u p p er and lo wer H amiltonians gro w sup er linearly in the gradient of the upp er and low er v alue functions, resp ectively . A un iqueness theorem of v iscosity sol u tion to Hamilton–Jacobi equations inv olving such k ind of Hamiltonian is p ro ved, without relying on the convexit y/concavit y of the Hamil- tonian. Also, it is shown that the assumed coercivity conditions guaranteeing the ﬁniteness of th e upp er and lo wer v alue fun ctions are sharp in some sense. Keyw ords. Two-perso n zero -sum diﬀerential g ames, un b ounded control, Hamilton-Ja cobi equation, viscosity solution. AMS Mathem atics sub ject class iﬁcation. 49 L25, 49N70, 9 1A23. 1 In tro d uction Let us beg in with the following control system: ( ˙ y ( s ) = f ( s, y ( s ) , u 1 ( s ) , u 2 ( s )) , s ∈ [ t, T ] , y ( t ) = x. (1.1) where f : [0 , T ] × l R n × U 1 × U 2 → l R n is a given map. In the ab ov e, y ( · ) is the state tra jector y ta king v alues in l R n , a nd ( u 1 ( · ) , u 2 ( · )) is the control pair taken fro m the set U σ 1 1 [ t, T ] × U σ 2 2 [ t, T ] of admissible c ontr ols , deﬁned by the following: U σ i i [ t, T ] = n u i : [ t, T ] → U i    k u i ( · ) k L σ i ( t,T ) ≡ h Z T t | u i ( s ) | σ i ds i 1 σ i < ∞ o , i = 1 , 2 , with U i being a closed subset of l R m i and with s o me σ i ≥ 1. W e p oint out that U 1 and U 2 are a llow ed to b e unbounded, and they could even b e l R m 1 and l R m 2 , resp ectively . Hereafter, we suppress l R m i in ∗ This wo rk is supp orted in part by the NSF grant DMS-1007514, the NSFC gran t 11171081, and the P ostgraduate Sc holarship Program of China. 1 k u i ( · ) k L σ i ( t,T ;l R m i ) for no tational simplicit y a nd this will not cause confusion. The p erformanc e functional asso ciated with (1.1) is the following: J ( t, x ; u 1 ( · ) , u 2 ( · )) = Z T t g ( s, y ( s ) , u 1 ( s ) , u 2 ( s )) ds + h ( y ( T )) , (1.2) with g : [0 , T ] × l R n × U 1 × U 2 → l R and h : l R n → l R being some given maps. The ab ov e setting can b e us e d to des crib e a tw o -p erson zer o-sum diﬀeren tia l game: P lay er 1 wan ts to select a control u 1 ( · ) ∈ U σ 1 1 [ t, T ] s o that the functional (1.2) is minimized and Play er 2 w a nt s to select a control u 2 ( · ) ∈ U σ 2 2 [ t, T ] so that the functional (1.2) is maximized. Therefor e, J ( t, x ; u 1 ( · ) , u 2 ( · )) is a c ost functional fo r Play er 1 and a p ayoﬀ functional for Play er 2 , r esp ectively . If U 2 is a singleton, the a b ove is reduced to a s ta ndard o ptimal control problem. Under so me mild co nditions, for an y initial p air ( t, x ) ∈ [0 , T ] × l R n and con trol pair ( u 1 ( · ) , u 2 ( · )) ∈ U σ 1 1 [ t, T ] × U σ 2 2 [ t, T ], the state equation (1.1) admits a unique solution y ( · ) ≡ y ( · ; t, x, u 1 ( · ) , u 2 ( · )), a nd the per formance functional J ( t, x ; u 1 ( · ) , u 2 ( · )) is well-deﬁned. B y adopting the notion of El liott–Kalton str ate gies ([11]), we can deﬁne the upp er and lower value functions V ± : [0 , T ] × l R n → l R (see Section 3 for details). F urther, when V ± ( · , · ) are diﬀeren tia ble, they should satisfy the following upper and lower Hamilton-Jaco bi- Isaacs (HJI, for sho rt) equatio ns, resp ectively: ( V ± t ( t, x ) + H ± ( t, x, V ± x ( t, x )) = 0 , ( t, x ) ∈ [0 , T ] × l R n , V ± ( T , x ) = h ( x ) , x ∈ l R n , (1.3) where H ± ( t, x, p ) ar e the so-called upp er and lower Hamiltonians deﬁned by the following, resp ectively:        H + ( t, x, p ) = inf u 1 ∈ U 1 sup u 2 ∈ U 2 h h p, f ( t, x, u 1 , u 2 ) i + g ( t, x, u 1 , u 2 ) i , H − ( t, x, p ) = sup u 2 ∈ U 2 inf u 1 ∈ U 1 h h p, f ( t, x, u 1 , u 2 ) i + g ( t, x, u 1 , u 2 ) i , ( t, x, p ) ∈ [0 , T ] × l R n × l R n . (1.4) When the se ts U 1 and U 2 are b ounded, the a bove diﬀerential g ame is well-understo o d ([12, 1 6]): Under reasona ble conditions, the upper and lower v alue functions V ± ( · , · ) are the unique v is cosity solutions to the corres p onding upp er and low er HJ I equations , r esp ectively . Consequently , in the case that the following Isaacs c ondition : H + ( t, x, p ) = H − ( t, x, p ) , ∀ ( t, x, p ) ∈ [0 , T ] × l R n × l R n , (1.5) holds, the upper and low er v alue functions coincide and the tw o-p erson zero -sum diﬀerential game admits the v alue function V ( t, x ) = V + ( t, x ) = V − ( t, x ) , ( t, x ) ∈ [0 , T ] × l R n . (1.6) F or compa rison purp oses , let us no w tak e a closer lo o k at the prop erties that the upper and low er v alue functions V ± ( · , · ) and the upp er and low er Hamiltonians H ± ( · , · , · ) hav e, under clas sical assumptions. T o this e nd, let us recall the following c lassical assumption: (B) F unctions f : [0 , T ] × l R n × U 1 × U 2 → l R n , g : [0 , T ] × l R n × U 1 × U 2 → l R, a nd h : l R n → l R are contin uous. There exists a constant L > 0 and a contin uous function ω : [0 , ∞ ) × [0 , ∞ ) → [0 , ∞ ), increasing in each of its arguments and ω ( r , 0) = 0 for a ll r ≥ 0, such that for a ll t, s ∈ [0 , T ] , x, y ∈ l R n , ( u 1 , u 2 ) ∈ U 1 × U 2 ,            | f ( t, x, u 1 , u 2 ) − f ( s, y , u 1 , u 2 ) | ≤ L | x − y | + ω  | x | ∨ | y | , | t − s |  , | g ( t, x, u 1 , u 2 ) − g ( s, y , u 1 , u 2 ) | ≤ ω  | x | ∨ | y | , | x − y | + | t − s |  , | h ( x ) − h ( y ) | ≤ ω  | x | ∨ | y | , | x − y |  , | f ( t, 0 , u 1 , u 2 ) | + | g ( t, 0 , u 1 , u 2 ) | + | h (0) | ≤ L, (1.7) 2 where | x | ∨ | y | = max {| x | , | y |} . Condition (1.7) implies that the cont inu it y and the gr owth o f ( t, x ) 7→ ( f ( t, x, u 1 , u 2 ), g ( t, x, u 1 , u 2 )) are uniform in ( u 1 , u 2 ) ∈ U 1 × U 2 . This essentially will b e the ca se if U 1 and U 2 are b ounded (or compact metric spaces). Let us state the follo wing prop ositio n. Prop ositio n 1.1. Under assumption (B) , one has the following: (i) The upp er and low er v alue functions V ± ( · , · ) are well-deﬁned contin uous functions. Moreov er, they are the unique viscos it y solutions to the upper and low er HJI equations (1 . 3) , resp ectively . In particula r, if Isaacs’ co ndition (1 . 5) holds, the upp er and low er v alue functions coincide. (ii) The upp er and low er Hamiltonians H ± ( · , · , · ) satisfy the following: F o r all t ∈ [0 , T ] , x, y , p, q ∈ l R n , ( | H ± ( t, x, p ) − H ± ( t, y , q ) | ≤ L (1 + | x | ) | p − q | + ω  | x | ∨ | y | , | x − y |  , | H ± ( t, x, p ) | ≤ L (1 + | x | ) | p | + L + ω ( | x | , | x | ) . (1.8) Condition (1.8) pla ys an importa n t role in the pro o f of the unique nes s of viscosity solution to HJI equations (1 .3) ([5, 15]). Note that, in particular, (1.8) implies that p 7→ H ± ( t, x, p ) is a t most of linear growth. Unfortunately , the above prop erty (1.8) fails , in genera l, when the control domains U 1 and/or U 2 is un bo unded. T o make this more convincing, let us lo ok at a o ne-dimensional linear -quadratic (LQ, for short) optimal cont rol problem (which amounts to saying that U 1 = l R and U 2 = { 0 } ). Co nsider the state equation ˙ y ( s ) = y ( s ) + u ( s ) , s ∈ [ t, T ] , with a quadratic cost functional J ( t, x ; u ( · )) = 1 2 h Z T t  | y ( s ) | 2 + | u ( s ) | 2  ds + | y ( T ) | 2 i . Then the Hamiltonian is H ( t, x, p ) = inf u ∈ l R h p ( x + u ) + | x | 2 + | u | 2 2 i = xp + x 2 2 − p 2 2 . Thu s, p 7→ H ( t, x, p ) is of qua dratic g rowth and (1.8) fails. Optimal con trol problems with unbounded co n tr ol do mains were studied in [2, 8]. Uniqueness o f v iscosity solution to the cor resp onding Hamilton-Jaco bi-Bellman equation was proved by some a rguments r e lying on the con vexity/conca vity of the corr esp onding Hamiltonian with resp ect to p . Recently , the ab ov e results w e r e substantially extended to sto chastic optimal control problems ([10]). On the other hand, as an extension of [24], t wo-p erson zero-sum diﬀeren tial games with (only) o ne play er having unbounded control were studied in [20]. So me nonlinear H ∞ problems can also be treated as such kind of diﬀeren tial games [19, 2 1]. F urther, sto chastic t wo-perso n zero -sum diﬀerential g ames were s tudied in [9] with one play er having un bo unded control and with the tw o play ers ’ co nt rols being sepa rated both in the state equation and the per formance functional. The main purpose of this pap er is to study t wo-p e rson zero-sum diﬀere n tia l games with b oth players hav- ing unbounded controls, a nd the cont r ols of t wo players are not necessar ily se pa rated. One motiv ation comes from the problem of what we call the a ﬃne-quadratic (AQ, for short) t wo-p erson zero- sum diﬀerential games , by which we mean that the right hand side of the state equation is a ﬃne in the controls, and the integrand of the p erforma nce functional is quadra tic in the controls (see Sectio n 2). This is a na tural gener alization of the cla ssical LQ problems. F or g eneral tw o-p erson zer o-sum diﬀerential g ames with (b oth play er s having) un bo unded controls, under so me mild co ercivity conditions, the upper and low er Hamilto nians H ± ( t, x, p ) 3 are prov ed to b e well-deﬁned, contin uous, and lo cally Lipsc hitz in p . Therefore, the upp er and low er HJI equations can b e formulated. Then w e will esta blis h the uniqueness of viscosity so lutio ns to a general ﬁrst order Hamilton- Jacobi equa tio n which includes our upp er and lo wer HJI equations of the diﬀerential game. Comparing w ith a relev an t result found in [6], the conditions we ass umed here are a little diﬀeren t from theirs and we present a detailed pro o f for reader’s c onv enience. By as suming a little stronger co erc ivit y conditions, together with some additional conditions (guaranteeing the well-posedness of the state equation, etc.), w e show tha t the upp er a nd low er v alue functions can b e well-deﬁned and ar e contin uous. Combining the ab ov e results, one obtains a characterizatio n of the upp er and lo wer v alue functions of the diﬀerential ga me as the unique v iscosity s olutions to the corresp onding upp er and lower HJI equations. Then if in addition, the Isaacs’ condition holds, the upp er and low er v alue functions coincide which yields the existence of the v alue function o f the diﬀerent ial ga me. W e w o uld like to mention her e that due to the unboundedness of the c ontrols, the contin uity of the upper and lower v alue functions V ± ( t, x ) in t is quite subtle. T o prov e that, we need to esta blish a modiﬁed principle of optimalit y and fully use the co erciv it y conditions. It is in teres ting to indicate that the assumed co ercivity conditions that ens uring the ﬁniteness of the upp er and low er v alue functions are actually sharp in s ome sense, whic h was illustr ated b y a one-dimensional LQ situation. F or some other rele v ant works in the literature, we would like to mention [18, 14, 13, 1, 25, 22], and references cited therein. The rest of the paper is org anized as follo ws . In Section 2, we make some brief obser v ations on an AQ t wo-perso n diﬀere ntial game, for which w e have a s ituation that the Isaacs’ condition holds and the upp er and lo wer Hamiltonians H ± ( t, x, p ) are q uadratic in p but ma y b e neither conv ex nor c oncav e. Section 3 is devoted to a study of upp er and lo wer Hamiltonians. The uniqueness of viscosity solutions to a class o f HJ equations will b e proved in Section 4. In Section 5, w e will s how that under cer tain conditions, the upp er and lo wer v alue functions are well-deﬁned a nd co n tin uous. Finally , in Section 6, we show that the assumed co ercivity conditions ensuring the upp er and low er v alue functions to be well-deﬁned ar e sha rp in so me sense. 2 An Aﬃne-Quadratic Tw o-P erson Diﬀeren tial Game T o better unders tand tw o- per son zero-s um diﬀeren tia l ga mes with unbounded controls, in this sectio n, w e lo ok a t a nontrivial spe cial case which is a main motiv ation of this paper . Consider the following s ta te equation: ( ˙ y ( s ) = A ( s, y ( s )) + B 1 ( s, y ( s )) u 1 ( s ) + B 2 ( s, y ( s )) u 2 ( s ) , s ∈ [ t, T ] , y ( t ) = x, (2.1) for some suitable matrix v alued functions A ( · , · ), B 1 ( · , · ), and B 2 ( · , · ). The state y ( · ) takes v alues in l R n and the cont rol u i ( · ) takes v alues in U i = l R m i ( i = 1 , 2). The p erfor ma nce functional is given b y J ( t, x ; u 1 ( · ) , u 2 ( · )) = Z T t h Q ( s, y ( s )) + 1 2 h R 1 ( s, y ( s )) u 1 ( s ) , u 1 ( s ) i + h S ( s, y ( s )) u 1 ( s ) , u 2 ( s ) i − 1 2 h R 2 ( s, y ( s )) u 2 ( s ) , u 2 ( s ) i + h θ 1 ( s, y ( s )) , u 1 ( s ) i + h θ 2 ( s, y ( s )) , u 2 ( s ) i i ds + G ( y ( T )) , (2.2) for so me scalar functions Q ( · , · ) and G ( · ), s ome vector v alued functions θ 1 ( · , · ) and θ 2 ( · , · ), and some matrix v alued functions R 1 ( · , · ), R 2 ( · , · ), a nd S ( · , · ). Note that the rig h t hand s ide of the state equa tion is a ﬃne in the controls u 1 ( · ) and u 2 ( · ), and the integrand in the per formance functional is up to quadratic in u 1 ( · ) and u 2 ( · ). Ther efore, we r efer to such a problem as an aﬃne-quadr atic (A Q, for short) two-p erson zer o-sum diﬀer ential game . W e also note that due to the pr esence o f the term h S ( s, y ( s )) u 1 ( s ) , u 2 ( s ) i , con tr o ls u 1 ( · ) 4 and u 2 ( · ) cannot be completely separa ted. Let us no w in tro duce the following basic hypotheses concerning the a bove AQ tw o-p ers on zero-s um diﬀeren tial game. (A Q1) The maps A : [0 , T ] × l R n → l R n , B 1 : [0 , T ] × l R n → l R n × m 1 , B 2 : [0 , T ] × l R n → l R n × m 2 , are con tinuous. (A Q2) The maps Q : [0 , T ] × l R n → l R , G : l R n → l R , R 1 : [0 , T ] × l R n → S m 1 , R 2 : [0 , T ] × l R n → S m 2 , S : [0 , T ] × l R n → l R m 2 × m 1 , θ 1 : [0 , T ] × l R n → l R m 1 , θ 2 : [0 , T ] × l R n → l R m 2 are cont inu ous (wher e S m stands for the set of all ( m × m ) symmetric matrices), a nd R 1 ( t, x ) and R 2 ( t, x ) are positive deﬁnite for all ( t, x ) ∈ [0 , T ] × l R n . With the above hypo theses, we let l H( t, x, p, u 1 , u 2 ) = h p, A ( t, x ) + B 1 ( t, x ) u 1 + B 2 ( t, x ) u 2 i + Q ( t, x ) + 1 2 h R 1 ( t, x ) u 1 , u 1 i + h S ( t, x ) u 1 , u 2 i − 1 2 h R 2 ( t, x ) u 2 , u 2 i + h θ 1 ( t, x ) , u 1 i + h θ 2 ( t, x ) , u 2 i . (2.3) Our r esult concer ning the ab ove-deﬁned function is the following prop osition. Prop ositio n 2.1. Let (A Q1)–(AQ2) hold. Then the matrix  R 1 ( t, x ) S ( t, x ) T S ( t, x ) − R 2 ( t, x )  is inv er tible, and l H( t, x, p, u 1 , u 2 ) = 1 2 h R 1 ( t, x )( u 1 − ¯ u 1 ) , u 1 − ¯ u 1 i + h S ( t, x )( u 1 − ¯ u 1 ) , u 2 − ¯ u 2 i − 1 2 h R 2 ( t, x )( u 2 − ¯ u 2 ) , u 2 − ¯ u 2 i + Q 0 ( t, x, p ) , (2.4) where  ¯ u 1 ¯ u 2  = −  R 1 ( t, x ) S ( t, x ) T S ( t, x ) − R 2 ( t, x )  − 1  B 1 ( t, x ) T p + θ 1 ( t, x ) B 2 ( t, x ) T p + θ 2 ( t, x )  , (2.5) and Q 0 ( t, x, p ) = Q ( t, x ) + h p, A ( t, x ) i − 1 2  B 1 ( t, x ) T p + θ 1 ( t, x ) B 2 ( t, x ) T p + θ 2 ( t, x )  T  R 1 ( t, x ) S ( t, x ) T S ( t, x ) − R 2 ( t, x )  − 1  B 1 ( t, x ) T p + θ 1 ( t, x ) B 2 ( t, x ) T p + θ 2 ( t, x )  , (2.6) F urther, ( ¯ u 1 , ¯ u 2 ) g iven by (2.5) is the unique sa ddle p oint of ( u 1 , u 2 ) 7→ l H( t, x, p, u 1 , u 2 ), na mely , l H( t, x, p, ¯ u 1 , u 2 ) ≤ l H( t, x, p, ¯ u 1 , ¯ u 2 ) ≤ l H( t, x, p, u 1 , ¯ u 2 ) , ∀ ( u 1 , u 2 ) ∈ U 1 × U 2 , (2.7) and cons equently , the Isa acs’ condition is satisﬁed: H + ( t, x, p ) ≡ inf u 1 ∈ U 1 sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) = sup u 2 ∈ U 2 inf u 1 ∈ U 1 l H( t, x, p, u 1 , u 2 ) ≡ H − ( t, x, p ) = Q 0 ( t, x, p ) , ∀ ( t, x, p ) ∈ [0 , T ] × l R n × l R n . (2.8) Pr o of. F or s implicit y of notation, let us suppress ( t, x ) b e low. W e ma y write l H( p, u 1 , u 2 ) = 1 2 h R 1 ( u 1 − ¯ u 1 ) , u 1 − ¯ u 1 i + h S ( u 1 − ¯ u 1 ) , u 2 − ¯ u 2 i − 1 2 h R 2 ( u 2 − ¯ u 2 ) , u 2 − ¯ u 2 i + Q 0 , 5 with ¯ u 1 , ¯ u 2 , a nd Q 0 undetermined. Then h p, A i + Q + h B T 1 p + θ 1 , u 1 i + h B T 2 p + θ 2 , u 2 i + 1 2 h R 1 u 1 , u 1 i + h S u 1 , u 2 i − 1 2 h R 2 u 2 , u 2 i = l H( p, u 1 , u 2 ) = 1 2 h R 1 u 1 , u 1 i + h S u 1 , u 2 i − 1 2 h R 2 u 2 , u 2 i − h R 1 ¯ u 1 , u 1 i − h S T ¯ u 2 , u 1 i − h S ¯ u 1 , u 2 i + h R 2 ¯ u 2 , u 2 i + 1 2 h R 1 ¯ u 1 , ¯ u 1 i + h S ¯ u 1 , ¯ u 2 i − 1 2 h R 2 ¯ u 2 , ¯ u 2 i + Q 0 . Hence, we must hav e    B T 1 p + θ 1 = − R 1 ¯ u 1 − S T ¯ u 2 , B T 2 p + θ 2 = − S ¯ u 1 + R 2 ¯ u 2 , h p, A i + Q = 1 2 h R 1 ¯ u 1 , ¯ u 1 i + h S ¯ u 1 , ¯ u 2 i − 1 2 h R 2 ¯ u 2 , ¯ u 2 i + Q 0 . (2.9) Consequently , from the ﬁrst tw o equations in (2.9), we ha ve  R 1 S T S − R 2   ¯ u 1 ¯ u 2  = −  B T 1 p + θ 1 B T 2 p + θ 2  . Note that det  R 1 S T S − R 2  = det  R 1 0 0 − ( R 2 + S R − 1 1 S T )  = ( − 1) m 2 det( R 1 ) det( R 2 + S R − 1 1 S T ) 6 = 0 . Thu s,  R 1 S T S − R 2  is in vertible, which yields  ¯ u 1 ¯ u 2  = −  R 1 S T S − R 2  − 1  B T 1 p + θ 1 B T 2 p + θ 2  . Then fr om the last equality in (2.9), one has Q 0 = h p, A i + Q − 1 2  ¯ u 1 ¯ u 2  T  R 1 S T S − R 2   ¯ u 1 ¯ u 2  = h p, A i + Q − 1 2  B T 1 p + θ 1 B T 2 p + θ 2  T  R 1 S T S − R 2  − 1  B T 1 p + θ 1 B T 2 p + θ 2  , proving (3.5). No w, w e see that l H( p, ¯ u 1 , u 2 ) = − 1 2 h R 2 ( u 2 − ¯ u 2 ) , u 2 − ¯ u 2 i + Q 0 ≤ Q 0 = l H( p, ¯ u 1 , ¯ u 2 ) ≤ 1 2 h R 1 ( u 1 − ¯ u 1 ) , u 1 − ¯ u 1 i + Q 0 ( t, x, p ) = l H( p, u 1 , ¯ u 2 ) , which means that ( ¯ u 1 , ¯ u 2 ) is a saddle p oint of l H( t, x, p, u 1 , u 2 ). Then the Isaacs condition (2.8) follo ws easily . Finally , since R 1 and R 2 are positive deﬁnite, the saddle point must b e uniq ue. W e see that in the curr en t cas e , p 7→ H ± ( t, x, p ) is quadratic, and is neither co n vex nor concav e in general. As a matter of fact, the Hessian H ± pp ( t, x, p ) of H ± ( t, x, p ) is given by the following: H ± pp ( t, x, p ) = − 1 2  B 1 ( t, x ) T B 2 ( t, x ) T  T  R 1 ( t, x ) S ( t, x ) T S ( t, x ) − R 2 ( t, x )  − 1  B 1 ( t, x ) T B 2 ( t, x ) T  . which is indeﬁnite in gener al. W e have s een fro m the ab ove that in order the upp er and lower Hamiltonians to b e well-deﬁned, the only crucial a s sumption that we made is the p ositive deﬁniteness of the matrix-v alued maps R 1 ( · , · ) and R 2 ( · , · ). Whereas, in order to study the AQ t wo-per son zero-sum diﬀeren tial g ames, we nee d a little stronger hypotheses. F or ex a mple, in order the state equatio n to b e well-posed, we need the right hand side of the state equation is Lipsc hitz c o nt inu ous in the s tate v a r iable, for any given pair of controls, etc. W e will lo o k at the general situation a little later. 6 3 Upp er and Lo w er Hamiltonia ns In this section, we will car efully lo ok a t the upper and lower Hamiltonia ns ass o ciated with gener al tw o- per son zero-sum diﬀerential games with unbounded con tr ols. First of all, we introduce the following standing assumptions. (H0) F o r i = 1 , 2, the set U i ⊆ l R m i is clos ed and 0 ∈ U i , i = 1 , 2 . (3.1) The time horizon T > 0 is ﬁxed. Note that b oth U 1 and U 2 could b e unbounded and may even be equal to l R m 1 and l R m 2 , resp ectively . Condition (3.1) is for conv enience. W e may make a tr a nslation o f the con tr ol domains and mak e corre s po nding changes in the cont rol systems and p erfor mance functional to a chieve this. Inspired b y the A Q t wo-p erson zero-sum diﬀerential ga mes, let us now intro duce the following as sumptions for the in volv ed functions f and g in the state equation (1.1) and the perfo rmance functional (1 .2 ). W e denote h x i = p 1 + | x | 2 . (H1) Map f : [0 , T ] × l R n × U 1 × U 2 → l R n is contin uo us and there ar e constants σ 1 , σ 2 ≥ 0 such that | f ( t, x, u 1 , u 2 ) | ≤ L  h x i + | u 1 | σ 1 + | u 2 | σ 2  , ∀ ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . (3.2) (H2) Map g : [0 , T ] × l R n × U 1 × U 2 → l R is contin uous and there exis t c o nstants L, c, ρ 1 , ρ 2 > 0 and µ ≥ 1 such that c | u 1 | ρ 1 − L  h x i µ + | u 2 | ρ 2  ≤ g ( t, x, u 1 , u 2 ) ≤ L  h x i µ + | u 1 | ρ 1  − c | u 2 | ρ 2 , ∀ ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . (3.3) F urther, w e intro duce the following compatibility condition w hich will be crucial b elow. (H3) The constants σ 1 , σ 2 , ρ 1 , ρ 2 in (H1)– (H2) sa tisfy the follo wing: σ i < ρ i , i = 1 , 2 . (3.4) It is not hard to see that the ab ov e (H1)–(H3) inc ludes the AQ tw o-p erson zero-sum diﬀerential game describ ed in the pr e vious se ction as a sp ecial case. Now, we let l H( t, x, p, u 1 , u 2 ) = h p, f ( t, x, u 1 , u 2 ) i + g ( t, x, u 1 , u 2 ) , ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . (3.5) Then the upp er a nd lower Hamiltonians are deﬁned as follows:      H + ( t, x, p ) = inf u 1 ∈ U 1 sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) , H − ( t, x, p ) = sup u 2 ∈ U 2 inf u 1 ∈ U 1 l H( t, x, p, u 1 , u 2 ) , ( t, x, p ) ∈ [0 , T ] × l R n × l R n , (3.6) provided the inv olved inﬁmu m and supremum exist. Note that the upper and lower Hamiltonians are nothing to do with the function h ( · ) (a pp ea r s as the terminal cost/pay oﬀ in (1.2 )). The main r esult of this section is the following. Prop ositio n 3.1 . Under (H1)–(H3) , the upp er and low e r Hamitonians H ± ( · , · , · ) are w e ll-deﬁned and contin uous. Moreov er, there are consta nts C > 0 , λ i , ν i ≥ 0 , ( i = 1 , 2 , · · · , k ) such tha t − L h x i µ − L h x i | p | − C | p | ρ 1 ρ 1 − σ 1 ≤ H ± ( t, x, p ) ≤ L h x i µ + L h x i | p | + C | p | ρ 2 ρ 2 − σ 2 , ∀ ( t, x, p ) ∈ [0 , T ] × l R n × l R n , (3.7) 7 and | H ± ( t, x, p ) − H ± ( t, x, q ) | ≤ C k X i =1 h x i λ i  | p | ∨ | q |  ν i | p − q | , ∀ ( t, x ) ∈ [0 , T ] × l R n , p, q ∈ l R n . (3.8) T o pr ov e the above, we will use the follo wing lemma. Lemma 3. 2. Let 0 < σ < ρ and c, N > 0 . Let θ ( r ) = N r σ − c r ρ , r ∈ [0 , ∞ ) . Then max r ∈ [0 , ∞ ) θ ( r ) = max r ∈ [0 , ¯ r ] θ ( r ) = θ ( ¯ r ) = ( ρ − σ )  σ σ N ρ ρ ρ c σ  1 ρ − σ , (3.9) with ¯ r =  σ N ρc  1 ρ − σ . (3.10) Pr o of. F rom θ (0 ) = 0 , lim r →∞ θ ( r ) = − ∞ , we s ee that the maximum of θ ( · ) on [0 , ∞ ) is achieved at some point ¯ r ∈ (0 , ∞ ). Set 0 = θ ′ ( r ) = N σ r σ − 1 − c ρr ρ − 1 . Then r ρ − σ = N σ cρ > 0 , which implies that the maximum is ac hieved at ¯ r g iven by (3.1 0), and max r ∈ [0 , ∞ ) θ ( r ) = max r ∈ [0 , ¯ r ] θ ( r ) = θ ( ¯ r ) = N  N σ cρ  σ ρ − σ − c  N σ cρ  ρ ρ − σ = h σ cρ  σ ρ − σ − c  σ cρ  ρ ρ − σ i N ρ ρ − σ = ( ρ − σ ) σ σ ρ − σ c σ ρ − σ ρ ρ ρ − σ N ρ ρ − σ . This pro ves our co nc lus ion. Pr o of of Pr op osition 3.1. Let us lo ok a t H + ( t, x, p ) carefully ( H − ( t, x, p ) can b e treated similarly). First, by o ur as sumption, we hav e l H( t, x, p, u 1 , u 2 ) ≤ | p | | f ( t, x, u 1 , u 2 ) | + g ( t, x, u 1 , u 2 ) ≤ L  h x i + | u 1 | σ 1 + | u 2 | σ 2  | p | + L  h x i µ + | u 1 | ρ 1  − c | u 2 | ρ 2 = L  h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1  + L | p | | u 2 | σ 2 − c | u 2 | ρ 2 , (3.11) and l H( t, x, p, u 1 , u 2 ) ≥ −| p | | f ( t, x, u 1 , u 2 ) | + g ( t, x, u 1 , u 2 ) ≥ − L  h x i + | u 1 | σ 1 + | u 2 | σ 2  | p | − L  h x i µ + | u 2 | ρ 2  + c | u 1 | ρ 2 = − L  h x i µ + h x i | p | + | p | | u 2 | σ 2 + | u 2 | ρ 2  − L | p | | u 1 | σ 1 + c | u 1 | ρ 1 . (3.12) Noting σ 1 < ρ 1 , from (3.11), we se e that for any ﬁxe d ( t, x, p, u 1 ) ∈ [0 , T ] × l R n × l R n × U 1 , the map u 2 7→ l H( t, x, p, u 1 , u 2 ) is co er cive fro m ab ov e. Cons equently , since U 2 is closed, for any given ( t, x, p, u 1 ) ∈ 8 [0 , T ] × l R n × l R n × U 1 , ther e exists a ¯ u 2 ≡ ¯ u 2 ( t, x, p, u 1 ) ∈ U 2 such tha t H + ( t, x, p, u 1 ) ≡ sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) = sup u 2 ∈ U 2 , | u 2 |≤| ¯ u 2 | l H( t, x, p, u 1 , u 2 ) = l H( t, x, p, u 1 , ¯ u 2 ) ≤ L  h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1  + L | p | | ¯ u 2 | σ 2 − c | ¯ u 2 | ρ 2 ≤ L  h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1  + ( ρ 2 − σ 2 )  σ σ 2 2  L | p |  ρ 2 ρ ρ 2 2 c σ 2  1 ρ 2 − σ 2 ≤ L  h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1  + K 2 | p | ρ 2 ρ 2 − σ 2 , (3.13) where K 2 = ( ρ 2 − σ 2 )  σ σ 2 2 L ρ 2 ρ ρ 2 2 c σ 2  1 ρ 2 − σ 2 . Here, w e hav e used Lemma 3.2 . On the other hand, from (3.12), for an y ( t, x, p, u 1 ) ∈ [0 , T ] × l R n × l R n × U 1 , we have H + ( t, x, p, u 1 ) = sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) ≥ l H( t, x, p, u 1 , 0) ≥ − L  h x i µ + h x i | p |  − L | p | | u 1 | σ 1 + c | u 1 | ρ 1 . (3.14) By Y o ung ’s inequality , w e ha ve L | p | | u i | σ i ≤ c 2 | u i | ρ i + ¯ K i | p | ρ i ρ i − σ i , i = 1 , 2 , for so me abso lute constants ¯ K i (depending on L, c, ρ i , σ i only), whic h leads to c 2 | u i | ρ i ≤ c | u i | ρ i − L | p | | u i | σ i + ¯ K i | p | ρ i ρ i − σ i , i = 1 , 2 . (3.15) Hence, c ombinin g the ﬁrst inequality in (3.13) and (3.14), we obtain c 2 | ¯ u 2 | ρ 2 ≤ c | ¯ u 2 | ρ 2 − L | p | | ¯ u 2 | σ 2 + ¯ K 2 | p | ρ 2 ρ 2 − σ 2 ≤ L  h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1  − H + ( t, x, u 1 ) + ¯ K 2 | p | ρ 2 ρ 2 − σ 2 ≤ 2 L  h x i µ + h x i | p | + | p | | u 1 | σ 1  + ( L − c ) | u 1 | ρ 1 + ¯ K 2 | p | ρ 2 ρ 2 − σ 2 ≡ b K 2  | x | , | p | , | u 1 |  . (3.16) The ab ove implies that for any compact set G ⊆ [0 , T ] × l R n × l R n × U 1 , there exists a compact set b U 2 ( G ) ⊆ U 2 , depe nding on G , such that H + ( t, x, p, u 1 ) = sup u 2 ∈ b U 2 ( G ) l H( t, x, p, u 1 , u 2 ) , ∀ ( t, x, p, u 1 ) ∈ G. Hence, H + ( · , · , · , · ) is contin uous. Next, fro m (3.14), noting σ 1 < ρ 1 , we ha ve that for any ﬁxed ( t, x, p ) ∈ [0 , T ] × l R n × l R n , the ma p u 1 7→ H + ( t, x, p, u 1 ) is co erc ive from below. Therefore , using the contin uity of H + ( · , · , · , · ), o ne can ﬁnd a ¯ u 1 ≡ ¯ u 1 ( t, x, p ) such that (note (3.15 )) H + ( t, x, p ) = inf u 1 ∈ U 1 sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) = inf u 1 ∈ U 1 H + ( t, x, p, u 1 ) = H + ( t, x, p, ¯ u 1 ) ≥ inf u 1 ∈ U 1 l H( t, x, p, u 1 , 0) ≥ inf u 1 ∈ U 1 n − L  h x i µ + h x i | p |  − L | p | | u 1 | σ 1 + c | u 1 | ρ 1 o ≥ − L  h x i µ + h x i | p |  − ¯ K 1 | p | ρ 1 ρ 1 − σ 1 . (3.17) This mea ns that H + ( t, x, p ) is well-deﬁned for all ( t, x, p ) ∈ [0 , T ] × l R n × l R n , and it is locally b ounded from below. Also, from (3.13), we obtain H + ( t, x, p ) = inf u 1 ∈ U 1 sup u 2 ∈ U 2 l H( t, x, p, u 1 , u 2 ) ≤ sup u 2 ∈ U 2 l H( t, x, p, 0 , u 2 ) ≡ H + ( t, x, p, 0) ≤ L  h x i µ + h x i | p |  + K 2 | p | ρ 2 ρ 2 − σ 2 . (3.18) 9 This pro ves (3.7) for H + ( · , · , · ). Next, we wan t to get the lo cal Lipsc hitz con tinuit y of the map p 7→ H + ( t, x, p ). T o this end, we ﬁrst let U 1 ( | x | , | p | ) = n u 1 ∈ U 1   c 2 | u 1 | ρ 1 ≤ 2 L  h x i µ + h x i | p |  + ¯ K 1 | p | ρ 1 ρ 1 − σ 1 + K 2 | p | ρ 2 ρ 2 − σ 2 + 1 o , ∀ x, p ∈ l R n , which, for any given x, p ∈ l R n , is a co mpact se t. Clea rly , for any u 1 ∈ U 1 \ U 1 ( | x | , | p | ), one has (note (3.15)) c | u 1 | ρ 1 − L | p | | u 1 | σ 1 ≥ c 2 | u 1 | ρ 1 − ¯ K 1 | p | ρ 1 ρ 1 − σ 1 > 2 L  h x i µ + h x i | p |  + K 2 | p | ρ 2 ρ 2 − σ 2 + 1 . Thu s, for suc h a u 1 , by (3.14 ) and (3.18), H + ( t, x, p, u 1 ) ≥ − L  h x i µ + h x i | p |  − L | p | | u 1 | σ 1 + c | u 1 | ρ 1 > L  h x i µ + h x i | p |  + K 2 | p | ρ 2 ρ 2 − σ 2 + 1 ≥ H + ( t, x, p ) + 1 = inf u 1 ∈ U 1 H + ( t, x, p, u 1 ) + 1 . (3.19) Hence, inf u 1 ∈ U 1 H + ( t, x, p, u 1 ) = inf u 1 ∈ U 1 ( | x | , | p | ) H + ( t, x, p, u 1 ) . (3.20) Now, fo r any u 1 ∈ U 1 ( | x | , | p | ), b y (3.16), we hav e c 2 | ¯ u 2 | ρ 2 ≤ b K 2  | x | , | p | , | u 1 |  ≤ e K 2 ( | x | , | p | ) , (3.21) for so me e K 2 ( | x | , | p | ). Hence, if w e let U 2 ( | x | , | p | ) = n u 2 ∈ U 2   c 2 | u 2 | ρ 2 ≤ e K 2 ( | x | , | p | ) o , which is a compact set (for any given x, p ∈ l R n ), then for any ( t, x, p ) ∈ [0 , T ] × l R n × l R n , H + ( t, x, p ) = inf u 1 ∈ U 1 ( | x | , | p | ) sup u 2 ∈ U 2 ( | x | , | p | ) l H( x, p, u 1 , u 2 ) . (3.22) This implies that H + ( · , · , · ) is co ntin uo us. Next, we lo o k at some estimates . By deﬁnition, for any u 1 ∈ U 1 ( | x | , | p | ), w e ha ve | u 1 | ρ 1 ≤ C n h x i µ + h x i | p | + | p | ρ 1 ρ 1 − σ 1 + | p | ρ 2 ρ 2 − σ 2 o . Therefore, | u 1 | σ 1 ≤ C n h x i µ + h x i | p | + | p | ρ 1 ρ 1 − σ 1 + | p | ρ 2 ρ 2 − σ 2 o σ 1 ρ 1 ≤ C n h x i σ 1 µ ρ 1 + h x i σ 1 ρ 1 | p | σ 1 ρ 1 + | p | σ 1 ρ 1 − σ 1 + | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) o . Also, by (3.1 6), one has | ¯ u 2 | ρ 2 ≤ C n h x i µ + h x i | p | + | p | | u 1 | σ 1 + | u 1 | ρ 1 + | p | ρ 2 ρ 2 − σ 2 o ≤ C n h x i µ + h x i | p | + h h x i σ 1 µ ρ 1 + h x i σ 1 ρ 1 | p | σ 1 ρ 1 + | p | σ 1 ρ 1 − σ 1 + | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) i | p | + h x i µ + h x i | p | + | p | ρ 1 ρ 1 − σ 1 + | p | ρ 2 ρ 2 − σ 2 o ≤ C n h x i µ + h x i | p | + h x i σ 1 µ ρ 1 | p | + h x i σ 1 ρ 1 | p | σ 1 + ρ 1 ρ 1 + | p | ρ 1 ρ 1 − σ 1 + | p | ρ 2 ρ 2 − σ 2 + | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) +1 o . (3.23) Hence, | ¯ u 2 | σ 2 ≤ C n h x i µ + h x i | p | + h x i σ 1 µ ρ 1 | p | + h x i σ 1 ρ 1 | p | σ 1 + ρ 1 ρ 1 + | p | ρ 1 ρ 1 − σ 1 + | p | ρ 2 ρ 2 − σ 2 + | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) +1 o σ 2 ρ 2 ≤ C n h x i σ 2 µ ρ 2 + h x i σ 2 ρ 2 | p | σ 2 ρ 2 + h x i σ 1 σ 2 µ ρ 1 ρ 2 | p | σ 2 ρ 2 + h x i σ 1 σ 2 ρ 1 ρ 2 | p | σ 2 ( σ 1 + ρ 1 ) ρ 1 ρ 2 + | p | σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) + | p | σ 2 ρ 2 − σ 2 + | p | σ 1 σ 2 ρ 1 ( ρ 2 − σ 2 ) + σ 2 ρ 2 o . 10 Consequently , for any ( t, x ) ∈ [0 , T ] × l R n , p, q ∈ l R n and u i ∈ U i ( | x | , | p | ∨ | q | ) ( i = 1 , 2 ), we ha ve (without loss o f genera lity , let | q | ≤ | p | ) | l H( t, x, p, u 1 , u 2 ) − l H( t, x, q , u 1 , u 2 ) | ≤ | p − q | | f ( t, x, u 1 , u 2 ) | ≤ L  h x i + | u 1 | σ 1 + | u 2 | σ 2  | p − q | ≤ C n h x i + h x i σ 1 µ ρ 1 + h x i σ 2 µ ρ 2 + h x i σ 1 ρ 1 | p | σ 1 ρ 1 + h x i σ 2 ρ 2 | p | σ 2 ρ 2 + | p | σ 1 ρ 1 − σ 1 + | p | σ 2 ρ 2 − σ 2 + | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) + | p | σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) + h x i σ 1 σ 2 µ ρ 1 ρ 2 | p | σ 2 ρ 2 + h x i σ 1 σ 2 ρ 1 ρ 2 | p | σ 2 ( σ 1 + ρ 1 ) ρ 1 ρ 2 + | p | σ 1 σ 2 ρ 1 ( ρ 2 − σ 2 ) + σ 2 ρ 2 o | p − q | ≡ C 12 X i =1 h x i λ i  | p | ∨ | q |  ν i | p − q | . (3.24) Due to the fact tha t the inﬁmum and supre m um in (3.22) can be ta ken on compact sets, w e can pr ov e the contin uity of ( t, x ) 7→ H + ( t, x, p ). A similar result as ab ov e can b e prov ed under some muc h weaker conditions. In fact, we ca n relax (H1)–(H2) to the following. (H1) ∗ Map f : [0 , T ] × l R n × U 1 × U 2 → l R n is contin uous and ther e are constant s σ 1 , σ 2 ≥ 0 and µ 0 , µ 1 , µ 2 ∈ l R suc h that | f ( t, x, u 1 , u 2 ) | ≤ L  h x i µ 0 + h x i µ 1 | u 1 | σ 1 + h x i µ 2 | u 2 | σ 2  , ∀ ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . (3.25) (H2) ∗ Map g : [0 , T ] × l R n × U 1 × U 2 → l R is contin uous and there exist constants L, c, ρ 1 , ρ 2 > 0 and ¯ µ 0 , ¯ µ 1 , ¯ µ 2 ∈ l R suc h that c h x i ¯ µ 1 | u 1 | ρ 1 − L  h x i ¯ µ 0 + h x i ¯ µ 2 | u 2 | ρ 2  ≤ g ( t, x, u 1 , u 2 ) ≤ L  h x i ¯ µ 0 + h x i ¯ µ 1 | u 1 | ρ 1  − c h x i ¯ µ 2 | u 2 | ρ 2 , ∀ ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . (3.26) The following res ult can b e prov ed in the sa me wa y as Prop os ition 3.1. Prop ositio n 3.1 ∗ . Under (H1) ∗ –(H2) ∗ and (H3) , the upp er and low er Hamitonia ns H ± ( · , · , · ) are well- deﬁned a nd contin uous . Moreover, there a re consta n ts C > 0 , ν i ≥ 0 , and λ i ∈ l R ( i = 1 , 2 , · · · , k ) s uch that − L h x i ¯ µ 0 − L h x i µ 0 | p | − C h x i µ 1 ρ 1 − ¯ µ 1 σ 1 ρ 1 − σ 1 | p | ρ 1 ρ 1 − σ 1 ≤ H ± ( t, x, p ) ≤ L h x i ¯ µ 0 + L h x i µ 0 | p | + C h x i µ 2 ρ 2 − ¯ µ 2 σ 2 ρ 2 − σ 2 | p | ρ 2 ρ 2 − σ 2 , ∀ ( t, x, p ) ∈ [0 , T ] × l R n × l R n , (3.27) and | H ± ( t, x, p ) − H ± ( t, x, q ) | ≤ C k X i =1 h x i λ i  | p | ∨ | q |  ν i | p − q | , ∀ ( t, x ) ∈ [0 , T ] × l R n , p, q ∈ l R n . (3.28) W e p oint out that diﬀerent from Prop osition 3.1, there ar e more terms in (3.2 8) than in (3.8), and the expressions of λ i and ν i are a little more complicated. In fact, instead of (3.24) we can pro ve the follo wing: 11 (for nota tional simplicity , w e le t | q | ≤ | p | )) | l H( t, x, p, u 1 , u 2 ) − l H( t, x, q , u 1 , u 2 ) | ≤ | p − q | | f ( t, x, u 1 , u 2 ) | ≤ L  h x i µ 0 + h x i µ 1 | u 1 | σ 1 + h x i µ 2 | u 2 | σ 2  | p − q | ≤ C n h x i µ 0 + h x i µ 1 − σ 1 ¯ µ 1 ρ 1 + h x i µ 1 + σ 1 ( ¯ µ 0 − ¯ µ 1 ) ρ 1 + h x i µ 1 + σ 1 ( µ 0 − ¯ µ 1 ) ρ 1 | p | σ 1 ρ 1 + h x i µ 1 + σ 1 ( µ 1 − ¯ µ 1 ) ρ 1 − σ 1 | p | σ 1 ρ 1 − σ 1 + h x i µ 1 + σ 1 [ ρ 2 ( µ 2 − ¯ µ 1 ) − σ 2 ( ¯ µ 2 − ¯ µ 1 )] ρ 1 ( ρ 2 − σ 2 ) | p | σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) + h x i µ 2 − σ 2 ¯ µ 2 ρ 2 + h x i µ 2 + σ 2 ( ¯ µ 0 − ¯ µ 2 ) ρ 2 + h x i µ 2 + σ 2 ( µ 0 − ¯ µ 2 ) ρ 2 | p | σ 2 ρ 2 + h x i µ 2 + σ 2 ( µ 1 − ¯ µ 2 ) ρ 2 − σ 1 σ 2 ¯ µ 1 ρ 1 ρ 2 | p | σ 2 ρ 2 + h x i µ 2 + σ 2 ( µ 1 − ¯ µ 2 ) ρ 2 + σ 1 σ 2 ( ¯ µ 0 − ¯ µ 1 ) ρ 1 ρ 2 | p | σ 2 ρ 2 + h x i µ 2 + σ 2 ( µ 1 − ¯ µ 2 ) ρ 2 + σ 1 σ 2 ( µ 0 − ¯ µ 1 ) ρ 1 ρ 2 | p | σ 2 ( σ 1 + ρ 1 ) ρ 1 ρ 2 + h x i µ 2 + σ 2 ( µ 1 − ¯ µ 2 ) ρ 2 + σ 1 σ 2 ( µ 1 − ¯ µ 1 ) ρ 2 ( ρ 1 − σ 1 ) | p | σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) + h x i µ 2 + σ 2 ( µ 1 − ¯ µ 2 ) ρ 2 + σ 1 σ 2 [ ρ 2 ( µ 2 − ¯ µ 1 ) − σ 2 ( ¯ µ 2 − ¯ µ 1 )] ρ 1 ρ 2 ( ρ 2 − σ 2 ) | p | σ 1 σ 2 ρ 1 ( ρ 2 − σ 2 ) + σ 2 ρ 2 + h x i µ 2 + σ 2 ( µ 1 ρ 1 − ¯ µ 1 σ 1 ) ρ 2 ( ρ 1 − σ 1 ) − σ 2 ¯ µ 2 ρ 2 | p | σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) + h x i µ 2 + σ 2 ( µ 2 ρ 2 − ¯ µ 2 σ 2 ) ρ 2 ( ρ 2 − σ 2 ) − σ 2 ¯ µ 2 ρ 2 | p | σ 2 ρ 2 − σ 2 o | p − q | ≡ C 16 X i =1 h x i λ i  | p | ∨ | q |  ν i | p − q | . Note that (3.24) is a sp ecia l case of the above with: µ 0 = 1 , ¯ µ 0 = µ, µ 1 = µ 2 = ¯ µ 1 = ¯ µ 2 = 0 . 4 Uniqueness of V iscosit y Solution Consider the following HJ ineq ua lities: ( V t ( t, x ) + H ( t, x, V x ( t, x )) ≥ 0 , ( t, x ) ∈ [0 , T ] × l R n , V ( T , x ) ≤ h ( x ) , x ∈ l R n , (4.1) and ( V t ( t, x ) + H ( t, x, V x ( t, x )) ≤ 0 , ( t, x ) ∈ [0 , T ] × l R n , V ( T , x ) ≥ h ( x ) , x ∈ l R n , (4.2) as well as the following HJ equation: ( V t ( t, x ) + H ( t, x, V x ( t, x )) = 0 , ( t, x ) ∈ [0 , T ] × l R n , V ( T , x ) = h ( x ) , x ∈ l R n . (4.3) W e recall the follo w ing deﬁnition. Deﬁnition 4. 1. (i) A contin uo us function V ( · , · ) is called a visc osity sub-solution of (4.1) if V ( T , x ) ≤ h ( x ) , ∀ x ∈ l R n , and for a n y con tin uous diﬀerent ia ble function ϕ ( · , · ), if ( t 0 , x 0 ) ∈ [0 , T ) × l R n is a lo cal maximu m o f ( t, x ) 7→ V ( t, x ) − ϕ ( t, x ), then ϕ t ( t 0 , x 0 ) + H ( t 0 , x 0 , ϕ x ( t 0 , x 0 )) ≥ 0 . (ii) A contin uous function V ( · , · ) is ca lle d a visc osity s up er-solution of (4.2) if V ( T , x ) ≥ h ( x ) , ∀ x ∈ l R n , 12 and for any contin uous diﬀerentiable function ϕ ( · , · ), if ( t 0 , x 0 ) ∈ [0 , T ) × l R n is a lo cal minimum of ( t, x ) 7→ V ( t, x ) − ϕ ( t, x ), then ϕ t ( t 0 , x 0 ) + H ( t 0 , x 0 , ϕ x ( t 0 , x 0 )) ≤ 0 . (iii) A co n tin uo us function V ( · , · ) is called a visc osity solution of (4.3) if it is a viscosity sub-so lution of (4.1) and a vis cosity sup er- solution o f (4.2). The following lemma is taken from [6]. Lemma 4.2. Suppos e H : [0 , T ] × l R n × l R n → l R is contin uous. Let V ( · , · ) a nd b V ( · , · ) b e a viscosity sub- and sup er-solutions of (4 . 1) and (4 . 2) , respectively . Then W ( t, x, y ) = V ( t, x ) − b V ( t, y ) , ( t, x, y ) ∈ [0 , T ] × l R n × l R n is a viscosity sub-solution o f the follo wing : ( W t ( t, x, y ) + H ( t, x, W x ( t, x, y )) − H ( t, y , − W x ( t, x, y )) ≥ 0 , ( t, x, y ) ∈ [0 , T ] × l R n × l R n , W ( T , x, y ) ≤ 0 , ( x, y ) ∈ l R n × l R n . Now fo r HJ equation (4.3), w e ass ume the follo wing . (HJ) The maps H : [0 , T ] × l R n × l R n → l R and h : l R n → l R ar e co nt inu ous and there a r e constants K 0 > 0, µ ≥ 1, and λ i , ν i ≥ 0 ( i = 1 , 2 , · · · , k ) with λ i + ( µ − 1) ν i ≤ 1 , 1 ≤ i ≤ k , (4.4) and a contin uous function ω : [0 , ∞ ) 3 → [0 , ∞ ) with prop erty ω ( r , s , 0) = 0, such that | H ( t, x, p ) − H ( t, y , p ) | ≤ ω  | x | + | y | , | p | , | x − y |  , ∀ t ∈ [0 , T ] , x, y , p ∈ l R n , | H ( t, x, p ) − H ( t, x, q ) | ≤ K 0 k X i =1 h x i λ i  | p | ∨ | q |  ν i | p − q | , ∀ t ∈ [0 , T ] , x, p, q ∈ l R n . (4.5) and | h ( x ) − h ( y ) | ≤ K 0  h x i ∨ h y i  µ − 1 | x − y | , ∀ x, y ∈ l R n . (4.6) Our main result of this se c tion is the following. Theorem 4.3. Let (HJ) hold. Supp ose V ( · , · ) and b V ( · , · ) are the vis c osity sub- and sup er-solutio n of (4 . 1) and (4 . 2) , r esp ectively . Moreov er, let | V ( t, x ) − V ( t, y ) | , | b V ( t, x ) − b V ( t, y ) | ≤ K  h x i ∨ h y i  µ − 1 | x − y | , ∀ t ∈ [0 , T ] , x, y ∈ l R n , (4.7) for so me K > 0 . Then V ( t, x ) ≤ b V ( t, x ) , ∀ ( t, x ) ∈ [0 , T ] × l R n . (4.8) A s imilar result as a bove was prov ed in [7], with most technical details o mitted. Our conditions are a little diﬀerent fro m those assumed in [7]. F o r re aders’ convenience, we provide a detailed pro of here. Pr o of. Suppos e ( ¯ t, ¯ x ) ∈ [0 , T ) × l R n such tha t V ( ¯ t, ¯ x ) − b V ( ¯ t, ¯ x ) > 0 . Let C 0 , β > 0 be undetermined. Deﬁne Q ≡ Q ( C 0 , β ) = n ( t, x ) ∈ [0 , T ] × l R n   h x i ≤ h ¯ x i e C 0 ( t − ¯ t )+ β o , 13 and G ≡ G ( C 0 , β ) = n ( t, x, y ) ∈ [0 , T ] × l R n × l R n   ( t, x ) , ( t, y ) ∈ Q o . Now, fo r δ > 0 small, deﬁne ψ ( t, x ) ≡ ψ C 0 ,δ ( t, x ) = h h x i h ¯ x i e C 0 ( ¯ t − t ) i 1 δ ≡ e 1 δ  log h x i h ¯ x i + C 0 ( ¯ t − t )  . Then ψ ( ¯ t, ¯ x ) = 1 , ψ ( T , x ) = h h x i h ¯ x i e C 0 ( ¯ t − T ) i 1 δ , and ψ t ( t, x ) = − C 0 ψ ( t, x ) δ , ψ x ( t, x ) = xψ ( t, x ) δ h x i 2 . F or an y ( t, x ) ∈ ¯ Q , w e ha ve h x i ≤ h ¯ x i e β + C 0 ( t − ¯ t ) ≤ h ¯ x i e β + C 0 ( T − ¯ t ) . Thu s, Q is b ounded and ¯ G is compact. W e in tro duce Ψ( t, x, y ) = V ( t, x ) − b V ( t, y ) − | x − y | 2 ε − σ ψ ( t, x ) − σ ( T − t ) T − ¯ t , ( t, x, y ) ∈ ¯ G, where ε > 0 small a nd 0 < σ ≤ V ( ¯ t, ¯ x ) − b V ( ¯ t, ¯ x ) 3 . Clearly , Ψ( ¯ t, ¯ x, ¯ x ) = V ( ¯ t, ¯ x ) − b V ( ¯ t, ¯ x ) − σ − σ ≥ 3 σ − 2 σ = σ > 0 . (4.9) Since Ψ( · , · , · ) is c ontin uous on the compact set ¯ G , we ma y let ( t 0 , x 0 , y 0 ) ∈ ¯ G b e a maximum of Ψ( · , · , · ) ov er ¯ G . By the optimality o f ( t 0 , x 0 , y 0 ), we hav e V ( t 0 , x 0 ) − b V ( t 0 , x 0 ) − σ ψ ( t 0 , x 0 ) − σ ( T − t 0 ) T − ¯ t = Ψ( t 0 , x 0 , x 0 ) ≤ Ψ( t 0 , x 0 , y 0 ) = V ( t 0 , x 0 ) − b V ( t 0 , y 0 ) − | x 0 − y 0 | 2 ε − σ ψ ( t 0 , x 0 ) − σ ( T − t 0 ) T − ¯ t , which implies | x 0 − y 0 | 2 ε ≤ b V ( t 0 , x 0 ) − b V ( t 0 , y 0 ) ≤ K  h x 0 i ∨ h y 0 i  µ − 1 | x 0 − y 0 | . Thu s, | x 0 − y 0 | ≤ K  h x 0 i ∨ h y 0 i  µ − 1 ε. Now, if t 0 = T , then h x 0 i , h y 0 i ≤ h ¯ x i e β + C 0 ( T − ¯ t ) . Hence, Ψ( T , x 0 , y 0 ) = h ( x 0 ) − h ( y 0 ) − | x 0 − y 0 | 2 ε − σ ψ ( T , x 0 ) ≤ K 0  h x 0 i ∨ h y 0 i  µ − 1 | x 0 − y 0 | ≤ K 0 K  h x 0 i ∨ h y 0 i  2( µ − 1) ε ≤ K 0 K h ¯ x i 2( µ − 1) e 2( µ − 1)[ β + C 0 ( T − ¯ t )] ε. Thu s, for ε > 0 small enough, the following holds: Ψ( T , x 0 , y 0 ) < σ ≤ Ψ( ¯ t, ¯ x, ¯ x ) ≤ Ψ( t 0 , x 0 , y 0 ) , 14 which means that t 0 ∈ [0 , T ). Next, we note that for ( t, x ) ∈  ∂ Q  ∩  (0 , T ) × l R n  , o ne has log h x i h ¯ x i + C 0 ( ¯ t − t ) = β , and 0 < t < T , which implies ψ ( t, x ) = e β δ → ∞ , δ → 0 , unifor mly in ( t, x ) ∈  ∂ Q  ∩  (0 , T ) × l R n  . (4.10) This implies that for δ > 0 sma ll (only depending on β ), ( t 0 , x 0 , y 0 ) ∈ G ∪ h { 0 } × l R n × l R n i . By Lemma 4.2, we have 0 ≤ σ ψ t ( t 0 , x 0 ) − σ T − ¯ t + H  t 0 , x 0 , 2( x 0 − y 0 ) ε + σ ψ x ( t 0 , x 0 )  − H  t 0 , y 0 , − 2( y 0 − x 0 ) ε  = σψ t ( t 0 , x 0 ) − σ T − ¯ t + H  t 0 , x 0 , 2( x 0 − y 0 ) ε + σ ψ x ( t 0 , x 0 )  − H  t 0 , x 0 , 2( x 0 − y 0 ) ε  + H  t 0 , x 0 , 2( x 0 − y 0 ) ε  − H  t 0 , y 0 , 2( x 0 − y 0 ) ε  ≤ σ ψ t ( t 0 , x 0 ) − σ T − ¯ t + K 0 k X i =1 h x 0 i λ i  2 | x 0 − y 0 | ε + σ | ψ x ( t 0 , x 0 ) |  ν i σ | ψ x ( t 0 , x 0 ) | + ω  | x 0 | + | y 0 | , 2 | x 0 − y 0 | ε , | x 0 − y 0 |  ≤ − σ C 0 δ ψ ( t 0 , x 0 ) − σ T − ¯ t + σ K 0 k X i =1 h x 0 i λ i  2 K  h x 0 i ∨ h y 0 i  µ − 1 + σ ψ ( t 0 , x 0 ) δ h x 0 i  ν i i ψ ( t 0 , x 0 ) δ h x 0 i + ω  | x 0 | + | y 0 | , 2 | x 0 − y 0 | ε , | x 0 − y 0 |  . Note that ( t 0 , x 0 , y 0 ) ≡ ( t 0 ,ε , x 0 ,ε , y 0 ,ε ) ∈ ¯ G ( C 0 , β ) (a ﬁxed co mpa ct set). Let ε → 0 along a suitable sequence, we hav e | x 0 ,ε − y 0 ,ε | → 0 . F o r notational simplicit y , we deno te ( t 0 ,ε , x 0 ,ε , y 0 ,ε ) → ( t 0 , x 0 , x 0 ). In the a bove, by c a nceling σ , and then send ε → 0 and σ → 0, one o bta ins (canceling σ ) 1 T − ¯ t ≤ − C 0 ψ ( t 0 , x 0 ) δ + K 0 k X i =1 h x 0 i λ i  2 K h x 0 i µ − 1  ν i ψ ( t 0 , x 0 ) δ h x 0 i = − n C 0 − K 0 k X i =1 (2 K ) ν i h x 0 i λ i +( µ − 1) ν i − 1 o ψ ( t 0 , x 0 ) δ ≤ − n C 0 − K 0 k X i =1 (2 K ) ν i o ψ ( t 0 , x 0 ) δ ≡ −  C 0 − e K 0  ψ ( t 0 , x 0 ) δ . Thu s, by tak ing C 0 > e K 0 , we obtain a contradiction, proving our co nclusion. W e now ma ke so me comment s on the uniqueness/no n-uniqueness of viscosity solutions. Firs t of all, let us look a t the following ex a mple which is adopted from [5, 4], Example 4.4. It is known that there a re t wo diﬀerent b ounded strictly increasing con tin uous diﬀeren- tiable functions f i : l R → l R ( i = 1 , 2) such that b ( x ) ≡ f ′ 1 ( f − 1 1 ( x )) = f ′ 2 ( f − 1 2 ( x )) , x ∈ l R . F urther, if we deﬁne X i ( t ; x 0 ) = f i ( t + f − 1 i ( x 0 )) , t ∈ l R 15 then X 1 ( · ; x 0 ) a nd X 2 ( · ; x 0 ) a re tw o diﬀeren t solutions to the following initial v alue pro blem:    d dt X ( t ; x 0 ) = b  X ( t ; x 0 )  , t ∈ l R , X (0; x 0 ) = x 0 . By deﬁning V i ( t, x ) = h ( X i ( T − t ; x )) , ( t, x ) ∈ [0 , T ] × l R , we o btain t wo diﬀer e n t viscosity solutions to the follo wing HJ equation: ( V t ( t, x ) + b ( x ) V x ( t, x ) = 0 , ( t, x ) ∈ [0 , T ] × l R , V ( T , x ) = h ( x ) , x ∈ l R . (4.11) Therefore, the viscosity solution to the above HJ equation is no t unique in the set of contin uous functions. How ever, w e note that in the current c ase, H ( t, x, p ) = b ( x ) p, ( t, x, p ) ∈ [0 , T ] × l R × l R . Thu s, | H ( t, x, p ) − H ( t, x, q ) | ≤ C | p − q | , ∀ t ∈ [0 , T ] , x, p, q ∈ l R , which means that (4 .5) holds with k = 1, λ 1 = ν 1 = 0. Hence, for any µ ≥ 1, as long a s (4.6) holds, viscos it y solution to (4.11) is unique in the class of con tinuous functions s a tisfying (4.7 ). Example 4.5. Consider − x 2 − a xV x ( x ) + | V x ( x ) | 2 = 0 , x ∈ l R , with a ≥ 0 . Thus, H ( x, p ) = − x 2 − a xp + p 2 , ( x, p ) ∈ l R 2 . Then le t V ( x ) = λx 2 , x ∈ l R . W e should hav e 0 = − 1 − 2 aλ + 4 λ 2 . Hence, λ = 2 a ± √ 4 a 2 + 1 6 8 = a ± √ a 2 + 4 4 . Therefore, there are tw o solutions to the HJ equation: V ± ( x ) = a ± √ a 2 + 4 4 x 2 , x ∈ l R . Both of these solutions are a nalytic. Note that | H ( x, p ) − H ( x, q ) | ≤  a | x | + 2  | p | ∨ | q |   | p − q | , x, p, q ∈ l R , and | V ± ( x ) − V ± ( y ) | ≤ | a ± √ a 2 + 4 | 4  h x i ∨ h y i  | x − y | . Thu s, in our ter mino logy , µ = 2 , k = 2 with λ 1 = 0 , ν 1 = 1 , λ 2 = 0 , ν 2 = 1 . Consequently , λ i + ( µ − 1) ν i = 1 , i = 1 , 2 . This means tha t although (4.4) is sa tis ﬁe d, the corresp onding HJ eq uation has mor e than one viscosity solution. This example sho ws that stationar y problems a re diﬀeren t from evolution problems, as far as the uniqueness o f vis c o sity solutio n is concerned. 16 5 Upp er and Lo w er V alue F unctions In this section, we ar e going to deﬁne the upp er and low er v a lue functions via the so-called Elliott–K alton strategies. So me ba sic pro pe r ties of upper and low er v alue functions will be established carefully . 5.1 State tra jectories and Elliot t–Kalton strat egies Let us in tr o duce the follo wing h y po theses which ar e strengthened v ersio ns of (H1)–(H3). (H1) ′ Map f : [0 , T ] × l R n × U 1 × U 2 → l R n satisﬁes (H1). Moreover, for some µ 0 , µ 1 , µ 2 , | f ( t, x, u 1 , u 2 ) − f ( t, y , u 1 , u 2 ) | ≤  h x i ∨ h y i  µ 0 +  h x i ∨ h y i  µ 1 | u 1 | σ 1 +  h x i ∨ h y i  µ 2 | u 2 | σ 2  | x − y | , ∀ ( t, u 1 , u 2 ) ∈ [0 , T ] × U 1 × U 2 , x, y ∈ l R n , (5.1) and h f ( t, x, u 1 , u 2 ) − f ( t, y , u 1 , u 2 ) , x − y i ≤ L | x − y | 2 , ∀ ( t, u 1 , u 2 ) ∈ [0 , T ] × U 1 × U 2 , x, y ∈ l R n . (5.2) W e note that condition (5.1) implies the lo cal Lipschitz con tinuit y of the map x 7→ f ( t, x, u 1 , u 2 ), with the Lipsch tiz constant p ossibly dep ending on | u 1 | σ 1 and | u 2 | σ 2 . This is the ca se if we are considering A Q t wo-perso n ze r o-sum diﬀerential games (see Section 2). On the other hand, condition (5.2) will be us e d to establish the lo cal Lipschitz co n tin uit y o f the upper and lower v alue functions, with the Lipsc hitz co nstant being of p olynomial order of h x i ∨ h y i . It is impor tant that the right hand side o f (5 .2) is indepe ndent of ( u 1 , u 2 ); Otherwise, the Lipschitz constant of the upper and lower v alue functions will be some exponential function of h x i ∨ h y i , for which w e do not know if the uniqueness of viscos it y solution to the cor resp onding HJI equation holds. By the wa y , we point out that (5.2 ) do es not imply the lo cal Lipschitz contin uity of the map x 7→ f ( t, x, u 1 , u 2 ). F or example, f ( x ) = x 1 3 , with x ∈ l R. (H2) ′ Map g : [0 , T ] × l R n × U 1 × U 2 → l R satisﬁes (H2). Moreov e r , | g ( t, x, u 1 , u 2 ) − g ( t, y , u 1 , u 2 ) | ≤  h x i ∨ h y i  µ − 1 + | u 1 | ρ 1 ( µ − 1) µ + | u 2 | ρ 2 ( µ − 1) µ  | x − y | , ∀ ( t, u 1 , u 2 ) ∈ [0 , T ] × U 1 × U 2 , x, y ∈ l R n . (5.3) Also, map h : l R n → l R is contin uo us and    | h ( x ) − h ( y ) | ≤ L  h x i ∨ h y i  µ − 1 | x − y | , ∀ x, y ∈ l R n , | h (0) | ≤ L. (5.4) F urther, the compatibility h yp othesis (H3) is no w replaced by the fo llowing: (H3) ′ The constan ts σ 1 , σ 2 , ρ 1 , ρ 2 , µ app ear in (H1) ′ –(H2) ′ satisfy the following: σ i µ < ρ i , i = 1 , 2 . (5.5) Let us ﬁrst prese nt the follo wing Gronw all type inequalit y . Lemma 5. 1. Let θ, α, β : [ t, T ] → l R + and θ 0 ≥ 0 sa tisfy θ ( s ) 2 ≤ θ 2 0 + Z s t  α ( r ) θ ( r ) 2 + β ( r ) θ ( r )  dr , s ∈ [ t, T ] . (5.6) 17 Then θ ( s ) ≤ e 1 2 R T t α ( τ ) dτ θ 0 + 1 2 e R T t α ( τ ) dτ Z s t β ( r ) dr , s ∈ [ t, T ] . (5.7) Pr o of. First, b y the usual Gr onw all’s inequa lit y , we hav e θ ( s ) 2 ≤ e R s t α ( τ ) dτ θ 2 0 + Z s t e R s r α ( τ ) dτ β ( r ) θ ( r ) dr ≤ e R T t α ( τ ) dτ θ 2 0 + e R T t α ( τ ) dτ Z s t β ( r ) θ ( r ) dr ≡ Θ( s ) . Then d ds p Θ( s ) = 1 2 Θ( s ) − 1 2 ˙ Θ( s ) = 1 2 Θ( s ) − 1 2 e R T t α ( τ ) dτ β ( s ) θ ( s ) ≤ 1 2 e R T t α ( τ ) dτ β ( s ) . Consequently , θ ( s ) ≤ p Θ( s ) ≤ e 1 2 R T t α ( τ ) dτ θ 0 + 1 2 e R T t α ( τ ) dτ Z s t β ( r ) dr , s ∈ [ t, T ] , proving our co nclusion. W e no w prove the following result co nc e rning the state tra jectories. Prop ositio n 5.2. Let (H1) ′ hold. Then, for any ( t, x ) ∈ [0 , T ) × l R n , ( u 1 ( · ) , u 2 ( · )) ∈ U σ 1 1 [ t, T ] × U σ 2 2 [ t, T ] , state equation (1 . 1) admits a unique solution y ( · ) ≡ y ( · ; t, x, u 1 ( · ) , u 2 ( · )) ≡ y t,x ( · ) . Moreov er , there exists a constant C 0 > 0 o nly de p ends on L, T , t s uch that h y t,x ( s ) i ≤ C 0 n h x i + Z s t  | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr o , s ∈ [ t, T ] , (5.8) | y t,x ( s ) − x | ≤ C 0 n h x i ( s − t ) + Z s t  | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr o , s ∈ [ t, T ] . (5.9) and for ( ¯ t, ¯ x ) ∈ [0 , T ] × l R n with ¯ t ∈ [ t, T ] , a nd y ¯ t, ¯ x ( · ) ≡ y ( · ; ¯ t, ¯ x, u 1 ( · ) , u 2 ( · )) | y t,x ( s ) − y ¯ t , ¯ x ( s ) | ≤ C 0 n | x − ¯ x | + h x i ( ¯ t − t ) + Z ¯ t t  | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr o , s ∈ [ t, T ] . (5.10) Pr o of. First, under (H1) ′ , for any ( t, x ) ∈ [0 , T ) × l R n , and any ( u 1 ( · ) , u 2 ( · )) ∈ U σ 1 1 [ t, T ] × U σ 2 2 [ t, T ], the map y 7→ f ( s, y , u 1 ( s ) , u 2 ( s )) is lo cally Lipschitz contin uous . Th us, state equa tion (1.1) admits a unique lo cal so lution y ( · ) = y ( · ; t, x, u 1 ( · ) , u 2 ( · )). Next, by (5.2), w e ha ve h x, f ( t, x, u 1 , u 2 ) i = h x, f ( t, x, u 1 , u 2 ) − f ( t, 0 , u 1 , u 2 ) i + h x, f ( t, 0 , u 1 , u 2 ) i ≤ L | x | 2 + L | x |  1 + | u 1 | σ 1 + | u 2 | σ 2  , ∀ ( t, x, u 1 , u 2 ) ∈ [0 , T ] × l R n × U 1 × U 2 . Thu s, h y ( s ) i 2 = h x i 2 + 2 Z s t h y ( r ) , f ( r, y ( r ) , u 1 ( r ) , u 2 ( r )) i dr ≤ h x i 2 + 2 Z s t L  h y ( r ) i 2 + h y ( r ) i  1 + | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2   dr . Then, it follows fr o m L e mma 5.1 that h y ( s ) i ≤ e L ( T − t ) h x i + Le 2 L ( T − t ) Z s t  1 + | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr . 18 This implies that the solution y ( · ) of the state eq uation (1.1 ) glo bally exists on [ t, T ] and (5.8) holds. Also, we have | y ( s ) − x | 2 = 2 Z s t h y ( r ) − x, f ( r, y ( r ) , u 1 ( r ) , u 2 ( r )) i dr ≤ 2 Z s t  L | y ( r ) − x | 2 + h y ( r ) − x, f ( r, x, u 1 ( r ) , u 2 ( r )) i  dr ≤ 2 L Z s t  | y ( r ) − x | 2 + | y ( r ) − x |  h x i + | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2   dr . Thu s, by Lemma 5.2 ag ain, we obtain (5.9). Now, for any ( t, x ) , ( ¯ t, ¯ x ) ∈ [0 , T ] × l R n , with 0 ≤ t ≤ ¯ t < T , denote y t,x ( · ) = y ( · ; t, x, u 1 ( · ) , u 2 ( · )), and y ¯ t , ¯ x ( · ) = y ( · ; ¯ t, ¯ x, u 1 ( · ) , u 2 ( · )). Then for s ∈ [ ¯ t, T ], w e hav e | y t,x ( s ) − y ¯ t, ¯ x ( s ) | 2 = | y t,x ( ¯ t ) − ¯ x | 2 +2 Z s ¯ t h y t,x ( r ) − y ¯ t, ¯ x ( r ) , f ( r , y t,x ( r ) , u 1 ( r ) , u 2 ( r )) − f ( r, y ¯ t, ¯ x ( r ) , u 1 ( r ) , u 2 ( r )) i dr ≤ | y t,x ( ¯ t ) − x | 2 + 2 L Z s ¯ t | y t,x ( r ) − y ¯ t, ¯ x ( r ) | 2 dr . Thu s, it follows from the Gronw all’s inequality that | y t,x ( s ) − y ¯ t, ¯ x ( s ) | ≤ e L ( s − ¯ t ) | y t,x ( ¯ t ) − x | ≤ e L ( s − ¯ t )  | x − ¯ x | + | y t,x ( ¯ t ) − x |  ≤ e L ( s − ¯ t ) n | x − ¯ x | + Le 2 L ( T − t )  h x i ( ¯ t − t ) + Z ¯ t t | u 1 ( r ) | σ 1 dr + Z ¯ t t | u 2 ( r ) | σ 2 dr o ≤ C n | x − ¯ x | + h x i ( ¯ t − t ) + Z ¯ t t  | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr o . This completes the proof. F rom the ab ov e prop osition, together with (H2) ′ , we see that for an y u i ( · ) ∈ U ρ i i [ t, T ] (which is smaller than U σ i i [ t, T ]), i = 1 , 2, the p erformance functional J ( t, x ; u 1 ( · ) , u 2 ( · )) is well-deﬁned. Let us now intro duce the following deﬁnitio n which is a mo diﬁcation of the notion introduced in [11]. Deﬁnition 5.3. A map α 1 : U 1 2 [ t, T ] → U ∞ 1 [ t, T ] is called an Ellio tt–Kalton (E-K , for short) s trategy for Play er 1 if it is non-ant icip ating , namely , for any u 2 ( · ) , ¯ u 2 ( · ) ∈ U 1 2 [ t, T ], and any ˆ t ∈ [ t, T ], α 1 [ u 2 ( · )]( s ) = α 1 [ ¯ u 2 ( · )]( s ) , a.e. s ∈ [ t, ˆ t ] , provided u 1 ( s ) = ¯ u 1 ( s ) , a.e. s ∈ [ t, ˆ t ] . The set of all E-K strategies for Play er 1 is denoted b y A 1 [ t, T ]. An E - K strategy α 2 : U 1 2 [ t, T ] → U ∞ 1 [ t, T ] for P lay er 2 can b e deﬁned similarly . The set of all E-K str ategies for Play e r 2 is deno ted b y A 2 [ t, T ]. Note that as far as the state equation is concerned, one could deﬁne a n E- K strategy α 1 for Play er I a s a map α 1 : U σ 2 2 [ t, T ] → U σ 1 1 [ t, T ]. Whereas, as far as the p erformanc e functiona l is concerned, o ne might hav e to restrictively deﬁne α 1 : U ρ 2 2 [ t, T ] → U ρ 1 1 [ t, T ]. W e note that the num b ers σ 1 , σ 2 , ρ 1 , ρ 2 app eared in (H1) ′ –(H2) ′ might no t be the “optimal” o nes, in s o me sense (for exa mple, σ 1 and σ 2 might b e larger than necessary , and ρ 1 and ρ 2 could b e smaller tha n they should b e , a nd so on). Our ab ov e deﬁnition is s omehow “universal”. The domain U 1 2 [ t, T ] of α 1 is la rge eno ugh to cover p ossible u 2 ( · ) in some lar ger space than U σ 2 2 [ t, T ], and the co-do main U ∞ 1 [ t, T ] is la rge enough so that the integrabilit y o f α 1 [ u 2 ( · )] is ensure d a nd the suprem um will remain the same due to the densit y of U ∞ 1 [ t, T ] in U ρ 1 1 [ t, T ]. In what follows, we simply denote U i [ t, T ] = U ∞ i [ t, T ] , i = 1 , 2 . 19 Recall that 0 ∈ U i ( i = 1 , 2). F or later conv enience, we herea fter let u 0 1 ( · ) ∈ U 1 [ t, T ] and u 0 2 ( · ) ∈ U 2 [ t, T ] b e deﬁned by u 0 1 ( s ) = 0 , u 0 2 ( s ) = 0 , ∀ s ∈ [ t, T ] , and let α 0 1 ∈ A 1 [ t, T ] be the E -K strateg y that α 0 1 [ u 2 ( · )]( s ) = 0 , ∀ s ∈ [ t, T ] , u 2 ( · ) ∈ U 1 2 [ t, T ] . W e call suc h an α 0 1 the zer o E-K s t r ate gy for Pla yer 1. Similarly , w e deﬁne zero E-K s trategy α 0 2 ∈ A 2 [ t, T ] for P lay er 2. Now, we deﬁne        V + ( t, x ) = sup α 2 ∈A 2 [ t,T ] inf u 1 ( · ) ∈U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) , V − ( t, x ) = inf α 1 ∈A 1 [ t,T ] sup u 2 ( · ) ∈U 2 [ t,T ] J ( t, x ; α 1 [ u 2 ( · )] , u 2 ( · )) . ( t, x ) ∈ [0 , T ] × l R n , (5.11) which are called upp er and lower value functions of our tw o -p erson zer o -sum diﬀerential game. 5.2 Upp er and lo wer v alue functions, and principle of optimalit y W e no w in tro duce the following notations: F or r > 0, U i [ t, T ; r ] = n u i ∈ U i [ t, T ]    Z T t | u i ( s ) | ρ i ds ≤ r o , i = 1 , 2 , and      A 1 [ t, T ; r ] = n α 1 : U 1 2 [ t, T ] → U 1 [ t, T ; r ]   α 1 ∈ A 1 [ t, T ] o , A 2 [ t, T ; r ] = n α 2 : U 1 1 [ t, T ] → U 2 [ t, T ; r ]   α 2 ∈ A 2 [ t, T ] o . W e point out that althoug h the upper and low e r v alue functions are for mally deﬁned in (5.1 1), there seems to b e no guarantee that they are well-deﬁned. The following result sta tes that under suitable conditions, V ± ( · , · ) a re indeed w ell-deﬁned. Theorem 5.4. L et (H1) ′ –(H3) ′ hold. Then the upper and lower v alue functions V ± ( · , · ) are well-deﬁned and there exists a constant C > 0 such that | V ± ( t, x ) | ≤ C h x i µ , ( t, x ) ∈ [0 , T ] × l R n . (5.12) Moreov er,        V + ( t, x ) = sup α 2 ∈A 2 [ t,T ; N ( | x | )] inf u 1 ( · ) ∈U 1 [ t,T ; N ( | x | )] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) , V − ( t, x ) = inf α 1 ∈A 1 [ t,T ; N ( | x | )] sup u 2 ( · ) ∈U 2 [ t,T ; N ( | x | )] J ( t, x ; α 1 [ u 2 ( · )] , u 2 ( · )) , (5.13) where N ( | x | ) = C h x i µ , fo r so me constant C > 0. Pr o of. First of all, for any ( t, x ) ∈ [0 , T ] × l R n and u 1 ( · ) ∈ U 1 [ t, T ], b y Prop osition 5.2 , we hav e h y ( s ) i ≤ C 0 n h x i + Z s t | u 1 ( r ) | σ 1 dr o ≤ C 0 n h x i + k u 1 ( · ) k σ 1 L σ 1 ( t,T ) o . 20 Then J ( t, x ; u 1 ( · ) , 0) = Z T t g ( s, y ( s ) , u 1 ( s ) , 0) ds + h ( y ( T )) ≥ Z T t h c | u 1 ( s ) | ρ 1 − L h y ( s ) i µ i ds − L h y ( T ) i µ ≥ Z T t h c | u 1 ( s ) | ρ 1 − LC µ 0  h x i + Z s t | u 1 ( r ) | σ 1 dr  µ i ds − LC µ 0  h x i + k u 1 ( · ) k σ 1 L σ 1 ( t,T )  µ ≥ − C h x i µ − C k u 1 ( · ) k σ 1 µ L σ 1 ( t,T ) + Z T t c | u 1 ( s ) | ρ 1 ds. Since (note µ ≥ 1) k u 1 ( · ) k σ 1 µ L σ 1 ( t,T ) =  Z T t | u 1 ( r ) | σ 1 dr  µ ≤ ( T − t ) µ − 1 Z T t | u 1 ( r ) | σ 1 µ dr , we o btain (taking into account σ 1 µ < ρ 1 ) J ( t, x ; u 1 ( · ) , 0) ≥ − C h x i µ + Z T t h c | u 1 ( s ) | ρ 1 − C | u 1 ( s ) | σ 1 µ i ds ≥ − C h x i µ + c 2 Z T t | u 1 ( s ) | ρ 1 ds ≥ − C h x i µ . (5.14) Consequently , V + ( t, x ) = sup α 2 ∈A 2 [ t,T ] inf u 1 ( · ) ∈U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) ≥ inf u 1 ( · ) ∈U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α 0 2 [ u 1 ( · )]) ≥ − C h x i µ . Likewise, for a n y u 2 ( · ) ∈ U 2 [ t, T ], w e hav e J ( t, x ; 0 , u 2 ( · )) = Z T t g ( s, y ( s ) , 0 , u 2 ( s )) ds + h ( y ( T )) ≤ C h x i µ . (5.15) Thu s, V + ( t, x ) = sup α 2 ∈A 2 [0 ,T ] inf u 1 ( · ) ∈U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) ≤ sup α 2 ∈A 2 [ t,T ] J ( t, x ; u 0 1 ( · ) , α 2 [ u 0 1 ( · )]) ≤ C h x i µ . Similar results also hold for the low er v alue function V − ( · , · ). There fo re, w e obtain that V ± ( t, x ) are well- deﬁned fo r all ( t, x ) ∈ [0 , T ] × l R n and (5.1 2 ) holds. Next, for the co nstant C > 0 appear ing in (5.12), we set N ( r ) = 4 C c h r i µ . Then fo r any u 1 ( · ) ∈ U 1 [ t, T ] \ U 1 [ t, T ; N ( | x | )], from (5.14), we see that J ( t, x ; u 1 ( · ) , α 0 2 [ u 1 ( · )]) ≥ − C h x i µ + c 2 Z T t | u 1 ( s ) | ρ 1 ds > C h x i µ ≥ V + ( t, x ) = sup α 2 ∈A 2 [ t,T ] inf u 1 ( · ) ∈U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) . Thu s, V + ( t, x ) = sup α 2 ∈A 2 [ t,T ] inf u 1 ( · ) ∈U 1 [ t,T ; N ( | x | )] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) . (5.16) 21 Consequently , from (5.15), for an y u 1 ( · ) ∈ U 1 [ t, T ; N ( | x | )], we hav e − C h x i µ ≤ V + ( t, x ) ≤ sup α 2 ∈A 2 [ t,T ] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) ≤ C h x i µ + C Z T t | u 1 ( s ) | ρ 1 ds − c 2 Z T t | α 2 [ u 1 ( · )]( s ) | ρ 2 ds ≤ C h x i µ + 2 C 2 h x i µ − c 2 Z T t | α 2 [ u 1 ( · )]( s ) | ρ 2 ds. This implies that c 2 Z T t | α 2 [ u 1 ( · )]( s ) | ρ 2 ds ≤ e C h x i µ , ∀ u 1 ( · ) ∈ U 1 [ t, T ; N ( | x | )] , (5.17) with e C = 2 C ( C + 1) > 0 b eing another a bsolute constant. Hence, if we repla ce the origina l N ( r ) b y the following: N ( r ) = 4 e C c h r i µ , and let A 2 [ t, T ; r ] = n α 2 ∈ A 2 [ t, T ]    Z T t | α 2 [ u 1 ( · )]( s ) | ρ 2 ds ≤ N ( | x | ) o , then the ﬁrst re la tion in (5.13) holds . The seco nd relation in (5.13 ) can be prov ed similarly . Next, we want to establish a mo diﬁed Bellman’s principle of optimality . T o this end, for any ( t, x ) ∈ [0 , T ) × l R n and ¯ t ∈ ( t, T ], let U i [ t, ¯ t ; r ] = n u i ( · ) ∈ U i [ t, T ]   Z ¯ t t | u i ( s ) | ρ i ds ≤ r o , i = 1 , 2 , and      A 1 [ t, ¯ t ; r ] = n α 1 : U 1 2 [ t, T ] → U 1 [ t, ¯ t ; r ]   α 1 ∈ A 1 [ t, T ] o , A 2 [ t, ¯ t ; r ] = n α 2 : U 1 1 [ t, T ] → U 2 [ t, ¯ t ; r ]   α 2 ∈ A 2 [ t, T ] o . It is clear that ( U i [ t, T ; r ] ⊆ U i [ t, ¯ t ; r ] ⊆ U i [ t, T ] , A i [ t, T ; r ] ⊆ A i [ t, ¯ t ; r ] ⊆ A i [ t, T ] , i = 1 , 2 . Thu s, fro m the pro of of Theorem 5.4, we see that for a suitable choice of N ( · ), say , N ( r ) = C (1 + r µ ) for some la rge C > 0, the following holds:        V + ( t, x ) = sup α 2 ∈A 2 [ t, ¯ t ; N ( | x | )] inf u 1 ( · ) ∈U 1 [ t,T ; N ( | x | )] J ( t, x ; u 1 ( · ) , α 2 [ u 1 ( · )]) , V − ( t, x ) = inf α 1 ∈A 1 [ t, ¯ t ; N ( | x | )] sup u 2 ( · ) ∈U 2 [ t, ¯ t ; N ( | x | )] J ( t, x ; α 1 [ u 2 ( · )] , u 2 ( · )) . (5.18) W e no w state the fo llowing mo diﬁed Bellman’s principle of optimalit y . Theorem 5.5. L e t (H1) ′ –(H3) ′ hold. Let ( t, x ) ∈ [0 , T ) × l R n and ¯ t ∈ ( t, T ] . Let N : [0 , ∞ ) → [0 , ∞ ) b e a nondecr easing co ntin uo us function suc h that (5 . 18) holds. Then V + ( t, x ) = sup α 2 ∈A 2 [ t, ¯ t ; N ( | x | )] inf u 1 ( · ) ∈U 1 [ t, ¯ t ; N ( | x | )] n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α 2 [ u 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) o , (5.19) 22 and V − ( t, x ) = inf α 1 ∈A 1 [ t, ¯ t ; N ( | x | )] sup u 2 ( · ) ∈U 2 [ t, ¯ t ; N ( | x | )] n Z ¯ t t g ( s, y ( s ) , α 1 [ u 2 ( · )]( s ) , u 2 ( s )) ds + V − ( ¯ t , y ( ¯ t )) o . (5.20) W e note that if in (5.1 9) and (5 .20), A i [ t, ¯ t ; N ( | x | )] and U i [ t, ¯ t ; N ( | x | )] ar e r eplaced b y A i [ t, T ] and U i [ t, T ], resp ectively , the result is s tandard and the pr o of is r outine. Howev er, in the ab ov e case, some careful mo diﬁca tion is necessary . F or rea ders’ conv enie nc e , we pro v ide a pro of in the a ppendix. W e p oint out that our mo diﬁed principle of optimality w ill pla y an essential role in the next subsection. 5.3 Con t in uity of upper and low er v alue functions In this subsection, w e are go ing to establish the contin uity of the upp er and low er v alue functions. Let us state the main r esults now. Theorem 5.6 . Let (H1) ′ –(H3) ′ hold. Then V ± ( · , · ) are contin uous. Moreov e r, there exists a constant C > 0 and a nondecre asing contin uous function N : [0 , ∞ ) → [0 , ∞ ) suc h that the follo wing estimates hold: | V ± ( t, x ) − V ± ( t, ¯ x ) | ≤ C  h x i ∨ h x i  µ − 1 | x − ¯ x | , t ∈ [0 , T ] , x, ¯ x ∈ l R n , (5.21) and | V ± ( t, x ) − V ± ( ¯ t, x ) | ≤ N ( | x | ) | t − ¯ t | ρ 1 − σ 1 ρ 1 ∧ ρ 2 − σ 2 ρ 2 , ∀ t, ¯ t ∈ [0 , T ] , x ∈ l R n . (5.22) Pr o of. W e will o nly prove the conclus io ns for V + ( · , · ). The co nclusions for V − ( · , · ) ca n be prov ed similarly . First, let 0 ≤ t ≤ T , x, ¯ x ∈ l R n , a nd let N ( r ) = C h r i µ for some C > 0, such that (5.13) holds. T ake u 1 ( · ) ∈ U ρ 1 1 [ t, T ; N ( | x | ∨ | ¯ x | )] , α 2 ∈ e A ρ 2 2 [ t, T ; N ( | x | ∨ | ¯ x | )] . (5.23) Denote u 2 ( · ) = α 2 [ u 1 ( · )]. Then Z T t | u i ( r ) | σ i dr ≤ C  Z T t | u i ( r ) | ρ i dr  σ i ρ i ≤ C  h x i ∨ h x i  σ i µ ρ i ≤ C h x i ∨ h ¯ x i , i = 1 , 2 . Making use of P rop osition 5 .1 , we hav e | y t,x ( s ) | , | y t, ¯ x ( s ) | ≤ C 0 h h x i ∨ h ¯ x i + Z T t  | u 1 ( r ) | σ 1 + | u 2 ( r ) | σ 2  dr i ≤ C  h x i ∨ h ¯ x i  , s ∈ [ t, T ] , and | y t,x ( s ) − y t, ¯ x ( s ) | ≤ C 0 | x − ¯ x | , s ∈ [ t, T ] . Consequently , | J ( t, x ; u 1 ( · ) , u 2 ( · )) − J ( t, ¯ x ; u 1 ( · ) , u 2 ( · )) | ≤ Z T t | g ( s, y t,x ( s ) , u 1 ( s ) , u 2 ( s )) − g ( s, y t, ¯ x ( s ) , u ( s )) | ds + | h ( y t,x ( T )) − h ( y t, ¯ x ( T )) | ≤ Z T t L   h y t,x ( s ) i ∨ h y t, ¯ x ( s ) i  µ − 1 + | u 1 ( s ) | ρ 1 ( µ − 1) µ + | u 2 ( s ) | ρ 2 ( µ − 1) µ  | y t,x ( s ) − y t, ¯ x ( s ) | ds + L  h y t,x ( T ) i ∨ h y t, ¯ x ( T ) i  µ − 1 | y t,x ( T ) − y t, ¯ x ( T ) | ≤ C n  h x i ∨ h ¯ x i  µ − 1 +  Z T t | u 1 ( s ) | ρ 1 ds  µ − 1 µ +  Z T t | u 2 ( s ) | ρ 2 ds  µ − 1 µ o | x − ¯ x | ≤ C  h x i ∨ h ¯ x i  µ − 1 | x − ¯ x | . 23 Since the ab ov e estimate is unifor m in ( u 1 ( · ) , α 2 ) sa tisfying (5.2 3), we obtain (5.21) for V + ( · , · ). W e now prov e the co nt inuit y in t . F rom the modiﬁed principle of optimality , we s ee that for any ε > 0 , there exis ts a n α ε 2 ∈ A 2 [ t, ¯ t ; N ( | x | )] such tha t V + ( t, x ) − ε ≤ inf u 1 ( · ) ∈U 1 [ t, ¯ t ; N ( | x | )] n Z ¯ t t g ( s, y ( s ) , u 1 ( · ) , α ε 2 [ u 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) o ≤ Z ¯ t t g ( s, y ( s ) , 0 , α ε 2 [ u 0 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) ≤ Z ¯ t t L  h y ( s ) i µ − c | α 2 [ u 0 1 ( · )]( s )) | ρ 2  ds + V + ( ¯ t, x ) + | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | ≤ Z ¯ t t L h y ( s ) i µ ds + V + ( ¯ t, x ) + | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | . By Pr op osition 5.2 , w e ha ve (denote u ε 2 ( · ) = α ε 2 [ u 0 1 ( · )]) | y ( ¯ t ) − x | ≤ C h h x i ( ¯ t − t ) + Z ¯ t t | u ε 2 ( s ) | σ 2 ds i ≤ C h h x i ( ¯ t − t ) +  Z ¯ t t | u ε 2 ( s ) | ρ 2 ds  σ 2 ρ 2 ( ¯ t − t ) ρ 2 − σ 2 ρ 2 i ≤ C h h x i ( ¯ t − t ) + N ( | x | )( ¯ t − t ) ρ 2 − σ 2 ρ 2 i . Also, | y ( s ) | ≤ C 0 h h x i + Z ¯ t t | u ε 2 ( s ) | σ 2 ds i ≤ N ( | x | ) , s ∈ [ t, ¯ t ] . Hence, by the prov ed (5.21 ), we obtain | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | ≤ N ( | x | ∨ | y ( ¯ t )) | y ( ¯ t ) − x | ≤ N ( | x | )( ¯ t − t ) ρ 2 − σ 2 ρ 2 . Consequently , V + ( t, x ) − V + ( ¯ t, x ) ≤ N ( | x | )( ¯ t − t ) ρ 2 − σ 2 ρ 2 + ε , which yields V + ( t, x ) − V + ( ¯ t, x ) ≤ N ( | x | )( ¯ t − t ) ρ 2 − σ 2 ρ 2 . On the other hand, V + ( t, x ) ≥ inf u 1 ( · ) ∈U 1 [ t,T ; N ( | x | )] n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , 0) ds + V + ( ¯ t, y ( ¯ t )) o . Hence, fo r any ε > 0, ther e e xists a u ε 1 ( · ) ∈ U 1 [ t, T ; N ( | x | )] such that V + ( t, x ) + ε ≥ Z ¯ t t g ( s, y ( s ) , u ε 1 ( s ) , 0) ds + V + ( ¯ t, y ( ¯ t )) ≥ − Z ¯ t t L h y ( s ) i µ ds + c Z ¯ t t | u ε 1 ( s ) | ρ 1 ds + V + ( ¯ t, x ) − | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | ≥ − Z ¯ t t L h y ( s ) i µ ds + V + ( ¯ t, x ) − | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | . Now, in the curren t case, we hav e | y ( ¯ t ) − x | ≤ C h h x i ( ¯ t − t ) + Z ¯ t t | u ε 1 ( s ) | σ 1 ds i ≤ C h h x i ( ¯ t − t ) +  Z ¯ t t | u ε 1 ( s ) | ρ 1 ds  σ 1 ρ 1 ( ¯ t − t ) ρ 1 − σ 1 ρ 1 i ≤ C h h x i ( ¯ t − t ) + N ( | x | )( ¯ t − t ) ρ 1 − σ 1 ρ 1 i . 24 Also, | y ( s ) | ≤ C 0 h h x i + Z ¯ t t | u ε 1 ( s ) | σ 1 ds i ≤ N ( | x | ) , s ∈ [ t, ¯ t ] . Hence, by the prov ed (5.21 ), we obtain | V + ( ¯ t, y ( ¯ t )) − V + ( ¯ t, x ) | ≤ N ( | x | ∨ | y ( ¯ t )) | y ( ¯ t ) − x | ≤ N ( | x | )( ¯ t − t ) ρ 1 − σ 1 ρ 1 . Consequently , V + ( t, x ) − V + ( ¯ t, x ) ≥ − N ( | x | )( ¯ t − t ) ρ 1 − σ 1 ρ 1 − ε , which yields V + ( t, x ) − V + ( ¯ t, x ) ≥ − N ( | x | )( ¯ t − t ) ρ 1 − σ 1 ρ 1 . Hence, we obtain the estimate (5.22) for V + ( · , · ). 5.4 Characterization of the upper and low er v alue functions Having the a bove prepar ations, we ar e now at the p os itio n to characterize the upper and the low er v alue functions of our diﬀerential game. Recall that in order Theorem 4.3 applies, we need the conditions (4.4)– (4.6) (for the maps H ( · , · , · ) and h ( · ) stated in (HJ) hold, a nd the upper and low er v alue functions ha ve to be Lipschit z contin uous in a particular form (see (4.7)). It is clear that the only thing that we need is the compatibility co ndition (4.4) for the n um be r s λ i , ν i app eared in (3.24) with the pa rameter µ a ppea red in (H2) and (H2) ′ . Let us no w look at what we need here. F rom (3.24) (whic h is for the upper v alue function V + ( · , · ) o nly), a nd the similar set of conditions for low er v alue function V − ( · , · ), we should require:                                    σ 1 µ ρ 1 ≤ 1 , σ 2 µ ρ 2 ≤ 1 , ( µ − 1) σ 1 ρ 1 − σ 1 ≤ 1 , ( µ − 1) σ 2 ρ 2 − σ 2 ≤ 1 , ( µ − 1) σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) ≤ 1 , ( µ − 1) σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) ≤ 1 , σ 1 σ 2 µ ρ 1 ρ 2 + ( µ − 1) σ 1 ρ 1 ≤ 1 , σ 1 σ 2 µ ρ 1 ρ 2 + ( µ − 1) σ 2 ρ 2 ≤ 1 , σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 2 ( σ 1 + ρ 1 ) ρ 1 ρ 2 ≤ 1 , σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 1 ( σ 2 + ρ 2 ) ρ 1 ρ 2 ≤ 1 , ( µ − 1) σ 1 σ 2 ρ 1 ( ρ 2 − σ 2 ) + ( µ − 1) σ 2 ρ 2 ≤ 1 , ( µ − 1) σ 1 σ 2 ρ 2 ( ρ 1 − σ 1 ) + ( µ − 1) σ 1 ρ 1 ≤ 1 . (5.24) W e no w ha ve the following pr op osition. Prop ositio n 5.7. Let µσ i ≤ ρ i , i = 1 , 2 . (5.25) Then a ll the inequalities in (5 . 24) hold. Pr o of. First of all, w e have that µσ i ρ i ≤ 1 ⇐ ⇒ ( µ − 1) σ i ρ i − σ i ≤ 1 . Thu s, under (5.25), the la st t wo ineq ualities in the ﬁr st line of (5.24) hold. Next, b y the ab ov e equiv alence and µ ≥ 1, ( µ − 1) σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) ≤ ( µ − 1) ρ 2 µ ( ρ 2 − σ 2 ) ≤ 1 , and ( µ − 1) σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) ≤ ( µ − 1) ρ 1 µ ( ρ 1 − σ 1 ) ≤ 1 . 25 Thu s, the inequalities in the second line o f (5.2 4) hold. Now, for the third line, w e hav e σ 1 σ 2 µ ρ 1 ρ 2 + ( µ − 1) σ 1 ρ 1 ≤ σ 1 ρ 1 + ( µ − 1) σ 1 ρ 1 = µσ 1 ρ 1 ≤ 1 , and σ 1 σ 2 µ ρ 1 ρ 2 + ( µ − 1) σ 2 ρ 2 ≤ σ 2 ρ 2 + ( µ − 1) σ 2 ρ 2 = µσ 2 ρ 2 ≤ 1 . This shows that the inequalities in the third line of (5.24) hold. W e now look at the fourth line. It is seen that σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 2 ( σ 1 + ρ 1 ) ρ 1 ρ 2 ≤ σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 2 ( σ 1 + µσ 1 ) ρ 1 ρ 2 = µ 2 σ 1 σ 2 ρ 1 ρ 2 ≤ 1 , and σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 1 ( σ 2 + ρ 2 ) ρ 1 ρ 2 ≤ σ 1 σ 2 ρ 1 ρ 2 + ( µ − 1) σ 1 ( σ 2 + µσ 2 ) ρ 1 ρ 2 = µ 2 σ 1 σ 2 ρ 1 ρ 2 ≤ 1 . Finally , for the ﬁfth line, w e ha ve (making use of the inequalities in the second line of (5.24)) ( µ − 1) σ 1 σ 2 ρ 1 ( ρ 2 − σ 2 ) + ( µ − 1) σ 2 ρ 2 = σ 2 ρ 2 h ( µ − 1) σ 1 ρ 2 ρ 1 ( ρ 2 − σ 2 ) + µ − 1 i ≤ σ 2 µ ρ 2 ≤ 1 , and ( µ − 1) σ 1 σ 2 ρ 2 ( ρ 1 − σ 1 ) + ( µ − 1) σ 1 ρ 1 = σ 1 ρ 1 h ( µ − 1) σ 2 ρ 1 ρ 2 ( ρ 1 − σ 1 ) + µ − 1 i ≤ σ 1 µ ρ 1 ≤ 1 . This completes the proof. With the above r esult, we hav e the fo llowing theorem. Theorem 5.8. Le t (H1) ′ –(H3) ′ hold. Then V ± ( · , · ) are the unique viscosity solution to the upp er and low er HJI equations (1 . 3 ) , resp ectively . F urther, if the Isaacs’ co ndition holds: H + ( t, x, p ) = H − ( t, x, p ) , ∀ ( t, x, p ) ∈ [0 , T ] × l R n × l R n , (5.26) then V + ( t, x ) = V − ( t, x ) , ∀ ( t, x ) ∈ [0 , T ] × l R n . (5.27) 6 Remarks on the Existence of V iscosit y Solutions to HJ Equa- tions. W e hav e seen that under (H1)–(H3), the upper and low er Hamiltonians can be well-deﬁned and the corre- sp onding upper and low er HJI e quations ca n b e well-form ulated. Moreov er , we ha ve prov ed the unique nes s of the v iscosity solutions to the upp er and low er HJI eq ua tions within a suitable class of lo c a lly Lipschitz contin uous functions. O n the o ther hand, we hav e intro duced a little stronge r hypotheses (H1) ′ –(H3) ′ to obtain the upper and low er v alue functions V ± ( · , · ) b eing w ell-deﬁned s o that the corresp onding upp er a nd low er HJI equa tions hav e visco sit y solutions. In another word, weaker conditions ensure the uniqueness of viscosity solutio ns to the upp er and lower HJI equations , a nd stronger conditions seem to b e needed for the existence. There ar e some genera l existence results o f visco sity so lutions fo r the ﬁrst order HJ equations in the literatur e, see [1 7, 3, 23, 14, 7]. A natural question is whether the conditio ns that we assumed for the ex istence of viscos it y solutions are sharp (or close to be necessar y). In this sectio n, we present a simple situation which tells us that our co nditions ar e shar p in so me sense. W e consider the following o ne-dimensional co n tr olled linear system: ( ˙ y ( s ) = Ay ( s ) + B 1 u 1 ( s ) + B 2 u 2 ( s ) , s ∈ [ t, T ] , y ( t ) = x, (6.1) 26 with the per formance functional: J ( t, x ; u 1 ( · ) , u 2 ( · )) = Z T t h Qy ( s ) 2 + R 1 u 1 ( s ) 2 − R 2 u 2 ( s ) 2 i ds + Gy ( T ) 2 , (6.2) where A, B 1 , B 2 , A, R 1 , R 2 , G ∈ l R. W e assume that R 1 , R 2 > 0 . (6.3) Note that in the current case, σ 1 = σ 2 = 1 , µ = ρ 1 = ρ 2 = 2 . Thu s, µσ i = ρ i , i = 1 , 2 , which violates (5.5 ). In the curre nt case, w e ha ve H ± ( t, x, p ) = H ( t, x, p ) = inf u 1 sup u 2 h pf ( t, x, u 1 , u 2 ) + g ( t, x, u 1 , u 2 ) i = Apx + Qx 2 + inf u 1 h R 1 u 2 1 + p B 1 u 1 i − inf u 2 h R 2 u 2 2 − p B 2 u 2 i = Apx + Qx 2 +  B 2 2 4 R 2 − B 2 1 4 R 1  p 2 . (6.4) Consequently , the upper and low er HJI equation ha ve the same form:      V t ( t, x ) + AxV x ( t, x ) + Qx 2 +  B 2 2 4 R 2 − B 2 1 4 R 1  V x ( t, x ) 2 = 0 , ( t, x ) ∈ [0 , T ] × l R , V ( T , x ) = Gx 2 , x ∈ l R . (6.5) If the ab ov e HJI equation has a viscosity so lution, by the uniqueness, the solution has to b e of the following form: V ( t, x ) = p ( t ) x 2 , ( t, x ) ∈ [0 , T ] × l R , (6.6) where p ( · ) is the solution to the following Ricca ti equation:      ˙ p ( t ) + 2 Ap ( t ) + Q +  B 2 2 R 2 − B 2 1 R 1  p ( t ) 2 = 0 , t ∈ [0 , T ] , p ( T ) = G. (6.7) In a nother word, the solv abilit y of (6.5) is equiv alent to that of (6 .7 ). Our claim is that Ricca ti e q uation (6.7) is not always solv able for any T > 0. T o state o ur r esult in a relatively neat way , let us rewrite equation (6.7) as follo ws : ( ˙ p + αp + β p 2 + γ = 0 , p ( T ) = g , (6.8) with α = 2 A, β = B 2 2 R 2 2 − B 2 1 R 2 1 , γ = Q , g = G. Note that β could b e pos itive, neg ative, or zero. W e ha ve the follo wing result. Prop ositio n 6.1 . Riccati equation (6 . 8) admits a so lution o n [0 , T ] for any T > 0 if and only if one of the following ho lds: α 2 − 4 β γ ≥ 0 , 2 β g + α − p α 2 − 4 β γ ≤ 0; (6.9) The pro of is elementary and straightforward. F or reader’s co n venience, w e provide a pr o of in the ap- pendix . 27 It is clear that there ar e a lot of cases for which the Riccati equa tion is not solv able. F or example, α = β = γ = 1 , which violates (6.9 ). Also , the case α = 0 , β = − 1 , γ = 1 , g = − 2 , which a lso violates (6.9). F or the above tw o cases , Riccati equatio n (6.8) do es no t hav e a glo bal solution on [0 , T ] for some T > 0. Corresp onding ly we have some t wo-p erson zero-sum diﬀeren tial ga me with unbo unded controls for whic h the coerciv ity condition (5.5) fails and the upp er and low er v a lue functions could not b e deﬁned o n the whole time interv al [0 , T ], or equiv alently , the corresp onding upper/lower HJI equation have no visc o sity solutions on [0 , T ]. Ac kno wledgeme nt. The a uthors w o uld like to thank the r eferee for informing the a uthors several imp ortant references in the ﬁeld, esp ecially some most recen t pap e r s. Also, some comments made b y Pr ofessor Y. Hu (of University o f Rennes 1, F rance) on the pr evious v ersio n are rea lly appr e ciated. References [1] M. Bar di and I. Capuzzo- Do lcetta, Optimal Control a nd Viscosity Solutions of Hamilton-J acobi-Bellma n Equations, Birkh¨ auser, Boston, 1997. [2] M. Bar di and F. 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Since N ( | x | ) and ¯ t a re ﬁxed, for notational simplicity , w e denote b elow that e U 1 = U 1 [ t, ¯ t ; N ( | x | )] , e A 2 = A 2 [ t, ¯ t ; N ( | x | )] . Denote the right hand side of (5.19 ) by b V + ( t, x ). F or any ε > 0, there exists an α ε 2 ∈ e A 2 such tha t b V + ( t, x ) − ε < inf u 1 ( · ) ∈ e U 1 n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α ε 2 [ u 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) o . By the deﬁnition o f V + ( ¯ t, y ( ¯ t )), there exists an ¯ α ε 2 ∈ A 2 [ ¯ t, T ] such that V + ( ¯ t, y ( ¯ t )) − ε < inf ¯ u 1 ( · ) ∈U 1 [ ¯ t,T ] J ( ¯ t, y ( ¯ t ); ¯ u 1 ( · ) , ¯ α ε 2 [ ¯ u 1 ( · )]) . Now, we deﬁne an e xtension b α ε 2 ∈ A 2 [ t, T ] of α ε 2 ∈ A 2 [ ¯ t, T ] as follows: F or a ny u 1 ( · ) ∈ U 1 [ t, T ], b α ε 2 [ u 1 ( · )]( s ) =    α ε 2 [ u 1 ( · )]( s ) , s ∈ [ t, ¯ t ) , ¯ α ε 2 [ u 1 ( · )   [ ¯ t,T ] ]( s ) , s ∈ [ ¯ t, T ] . 29 Since α ε 2 ∈ e A 2 , we hav e Z ¯ t t | b α ε [ u 1 ( · )]( s ) | ρ 2 ds = Z ¯ t t | α ε 2 [ u 1 ( · )]( s ) | ρ 2 ds ≤ N ( | x | ) . This means that b α ε 2 ∈ e A 2 . Consequent ly , V + ( t, x ) ≥ inf u 1 ( · ) ∈ e U 1 J ( t, x ; u 1 ( · ) , b α ε 2 [ u 1 ( · )]) = inf u 1 ( · ) ∈ e U 1 n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α ε 2 [ u 1 ( · )]( s )) ds + J ( ¯ t, y ( ¯ t ); u 1 ( · )   [ ¯ t,T ] , ¯ α ε 2 [ u 1 ( · )   [ ¯ t,T ] ) o ≥ inf u 1 ( · ) ∈ e U 1 n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α ε 2 [ u 1 ( · )]( s )) ds + inf ¯ u 1 ( · ) ∈U 1 [ ¯ t,T ] J ( ¯ t, y ( ¯ t ); ¯ u 1 ( · ) , ¯ α ε 2 [ ¯ u 1 ( · )) o ≥ inf u 1 ( · ) ∈ e U 1 n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α ε 2 [ u 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) o − ε ≥ b V + ( t, x ) − 2 ε. Since ε > 0 is ar bitrary , we obtain b V + ( t, x ) ≤ V + ( t, x ) . On the other hand, for a n y ε > 0, there exists an α ε 2 ∈ e A 2 such tha t V + ( t, x ) − ε < inf u 1 ( · ) ∈ e U 1 J ( t, x ; u 1 ( · ) , α ε 2 [ u 1 ( · )]) . Also, by deﬁnition of b V + ( t, x ), b V + ( t, x ) ≥ inf u 1 ( · ) ∈ e U 1 n Z ¯ t t g ( s, y ( s ) , u 1 ( s ) , α ε 2 [ u 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) o . Thu s, there exists a u ε 1 ( · ) ∈ e U 1 such tha t b V + ( t, x ) + ε ≥ Z ¯ t t g ( s, y ( s ) , u ε 1 ( s ) , α ε 2 [ u ε 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) . Now, fo r any ¯ u 1 ( · ) ∈ U 1 [ ¯ t, T ], deﬁne a particular extensio n e u 1 ( · ) ∈ U 1 [ t, T ] b y the following: e u 1 ( s ) = ( u ε 1 ( s ) , s ∈ [ t, ¯ t ) , ¯ u 1 ( s ) , s ∈ [ ¯ t, T ] . Namely , w e patc h u ε 1 ( · ) to ¯ u 1 ( · ) on [ t, ¯ t ). Since Z ¯ t t | e u 1 ( s ) | ρ 1 ds = Z ¯ t t | u ε 1 ( s ) | ρ 1 ds ≤ N ( | x | ) , we s ee that e u 1 ( · ) ∈ e U 1 . Next, we deﬁne a restr iction ¯ α ε 2 ∈ A [ ¯ t, T ] of α ε 2 ∈ e A 2 , a s follows: ¯ α ε 2 [ ¯ u 1 ( · )] = α ε 2 [ e u 1 ( · )] . F or suc h an ¯ α ε 2 , we hav e V + ( ¯ t, y ( ¯ t )) ≥ inf ¯ u 1 ( · ) ∈U 1 [ ¯ t,T ] J ( ¯ t, y ( ¯ t ) , ¯ u 1 ( · ) , ¯ α ε 2 [ ¯ u 1 ( · )]) . Hence, ther e exists a ¯ u ε 1 ( · ) ∈ U 1 [ ¯ t, T ] such that V + ( ¯ t, y ( ¯ t )) + ε > J ( ¯ t, y ( ¯ t ) , ¯ u ε 1 ( · ) , ¯ α ε 2 [ ¯ u ε 1 ( · )]) . 30 Then we further let e u ε 1 ( s ) = ( u ε 1 ( s ) , s ∈ [ t, ¯ t ) , ¯ u ε 1 ( s ) , s ∈ [ ¯ t, T ] . Again, e u ε 1 ( · ) ∈ e U 1 , a nd therefor e , b V + ( t, x ) + ε ≥ Z ¯ t t g ( s, y ( s ) , u ε 1 ( s ) , α ε 2 [ u ε 1 ( · )]( s )) ds + V + ( ¯ t, y ( ¯ t )) ≥ Z ¯ t t g ( s, y ( s ) , u ε 1 ( s ) , α ε 2 [ u ε 1 ( · )]( s )) ds + J ( ¯ t, y ( ¯ t ) , ¯ u ε 1 ( · ) , ¯ α ε 2 [ ¯ u ε 1 ( · )]) − ε = J ( t, x ; e u ε 1 ( · ) , α ε 2 [ e u ε 1 ( · )]) − ε ≥ inf u 1 ( · ) ∈ e U 1 [ t,T ] J ( t, x ; u 1 ( · ) , α ε 2 [ u 1 ( · )]) − ε ≥ V + ( t, x ) − 2 ε. Since ε > 0 is ar bitrary , we obtain b V + ( t, x ) ≥ V + ( t, x ) . This completes the proof. Pr o of of Pr op osition 6.1. Recall that we are co nsidering the following Riccati e q uation: ( ˙ p + αp + β p 2 + γ = 0 , p ( T ) = g , Case 1. β = 0. The Ricca ti equation reads ( ˙ p + αp + γ = 0 , p ( T ) = g . This is an initial v alue pr oblem for a linear equation, which admits a unique global so lution p ( · ) on [0 , T ]. Case 2. β 6 = 0. Then Riccati equation reads      ˙ p + β h p + α 2 β  2 + 4 β γ − α 2 4 β 2 i = 0 , p ( T ) = g . Let κ = p | α 2 − 4 β γ | 2 | β | ≥ 0 . There a re three subca s es. Subsc ase 1. α 2 − 4 β γ = 0. T he Riccati eq uation b ecomes      ˙ p + β  p + α 2 β  2 = 0 , p ( T ) = g . Therefore, in the ca se 2 β g + α = 0 , we have that p ( t ) ≡ − α 2 β is the (unique) global solution on [0 , T ]. Now, let 2 β g + α 6 = 0 . 31 Then we hav e dp ( p + α 2 β ) 2 = − β dt, which leads to 1 p ( t ) + α 2 β = 1 g + α 2 β − β ( T − t ) = 2 β − β (2 β g + α )( T − t ) 2 β g + α . Thu s, p ( t ) = − α 2 β + 2 β g + α 2 β − β (2 β g + α )( T − t ) , which is well-deﬁned on [0 , T ] if and only if 2 − (2 β g + α )( T − t ) 6 = 0 , t ∈ [0 , T ] . This is equiv alent to the follo wing : (2 β g + α ) T < 2 . The abov e is true fo r all T > 0 if and only if 2 β g + α ≤ 0 , Sub c ase 2. α 2 − 4 β γ < 0. T he Riccati eq uation is ˙ p + β h p + α 2 β  2 + κ 2 i = 0 . Hence, dp ( p + α 2 β ) 2 + κ 2 = − β dt, which results in 1 κ tan − 1 h 1 κ  p ( t ) + α 2 β i = − β t + C. By the terminal condition, C = β T + 1 κ tan − 1 h 1 κ  g + α 2 β i Consequently , tan − 1 h 1 κ  p ( t ) + α 2 β i = κβ ( T − t ) + tan − 1 h 1 κ  g + α 2 β i . Then p ( t ) = α 2 β + κ tan n κβ ( T − t ) + tan − 1  2 β g + α 2 κβ o . The abov e is w e ll-deﬁned for t ∈ [0 , T ] if and o nly if − π 2 < tan − 1 2 β g + α 2 κβ + κ β T < π 2 , which is true for a ll T > 0 if and only if β = 0. Sub c ase 3. α 2 − 4 β γ > 0. T he Riccati eq uation b ecomes ˙ p + β h p + α 2 β  2 − κ 2 i = 0 . If (2 β g + α − 2 κβ )(2 β g + α + 2 κβ ) ≡ 4 β 2  g + α 2 β − κ  g + α 2 β + κ  = 0 , (A 1) 32 then o ne of the following p ( t ) ≡ − α 2 β ± κ , t ∈ [0 , T ] , is the unique g lo bal so lution to the Riccati equation. W e now let (2 β g + α − 2 κβ )(2 β g + α + 2 κβ ) ≡ 4 β 2  g + α 2 β − κ  g + α 2 β + κ  6 = 0 . Then dp ( p + α 2 β ) 2 − κ 2 = − β dt. Hence, 1 2 κ ln    p ( t ) + α 2 β − κ p ( t ) + α 2 β + κ    = − β t + e C , which implies p ( t ) + α 2 β − κ p ( t ) + α 2 β + κ = C e − 2 κβ t , with C = e 2 κβ T g + α 2 β − κ g + α 2 β + κ = e 2 κβ T 2 β g + α − 2 κβ 2 β g + α + 2 κβ . Then p ( t ) + α 2 β − κ p ( t ) + α 2 β + κ = e 2 κβ ( T − t ) 2 β g + α − 2 κ β 2 β g + α + 2 κ β . Consequently , p ( t ) + α 2 β − κ = e 2 κβ ( T − t ) 2 β g + α − 2 κβ 2 β g + α + 2 κβ h p ( t ) + α 2 β + κ i . Thu s, p ( · ) globally exists on [0 , T ] if and only if e 2 κβ ( T − t ) 2 β g + α − 2 κ β 2 β g + α + 2 κ β − 1 6 = 0 , ∀ t ∈ [0 , T ] , which is equiv alent to ψ ( t ) ≡ e 2 κβ ( T − t ) (2 β g + α − 2 κβ ) − (2 β g + α + 2 κβ ) 6 = 0 , ∀ t ∈ [0 , T ] . Since ψ ′ ( t ) do es not change sig n on [0 , T ], the ab ov e is equiv alent to the follo wing: 0 < ψ (0) ψ ( T ) = h e 2 κβ T (2 β g + α − 2 κβ ) − (2 β g + α + 2 κβ ) i ( − 4 κβ ) , which is equiv alent to h e 2 κβ T (2 β g + α − 2 κβ ) − (2 β g + α + 2 κ β ) i β < 0 . Note when (A1) holds, the abov e it true. In the case β > 0 , the ab ov e rea ds e 2 κβ T (2 β g + α − 2 κβ ) < 2 β g + α + 2 κ β , which is true for a ll T > 0 if and only if 2 β g + α − 2 κ β ≤ 0 . (A2) Finally , if β < 0, then 0 < e 2 κβ T (2 β g + α − 2 κβ ) − (2 β g + α + 2 κβ ) = e − 2 κ | β | T ( − 2 | β | g + α + 2 κ | β | ) − ( − 2 | β | g + α − 2 κ | β | ) = e − 2 κ | β | T h −  2 | β | g − α − 2 κ | β |  + e 2 κ | β | T  2 | β | g − α + 2 κ | β | i , 33 which is true for a ll T > 0 if and only if 0 ≤ 2 | β | g − α + 2 κ | β | = − (2 β g + α − 2 κ | β | ) . Thu s, 2 β g + α − 2 κ | β | ≤ 0 . which has the same for m as (A2). This completes the pro of. 34

Hamilton-Jacobi Equations and Two-Person Zero-Sum Differential Games with Unbounded Controls

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