Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

Optimal con trol of a tumor gro wth mo del with h yp erb olic relaxation of the c hemical p oten tial Pierluigi Colli (1) e-mail: pierluigi.colli@unipv.it Elisabett a R occa (1) e-mail: elisabetta.rocca@unipv.it J ¨ ur gen Sprekels (2) e-mail: juergen.sprekels@wias-berlin.de (1) Dipartimen to di Matematica “F. Casorati” Univ ersit` a di P avia via F errata 5, I-27100 Pa via, Italy (2) W eierstrass Institute for Applied Analysis and Sto c hastics An ton-Wilhelm-Amo-Strasse 39, D-10117 Berlin, Germany Abstract In this pap er, w e study the optimal con trol of a phase ﬁeld model for a tumor gro wth mo del of Cahn–Hilliard type in which the often assumed parab olic relaxation of the c hemical potential is replaced b y a hyperb olic one. Both the cases when the double- w ell potential gov erning the phase ev olution is of either regular or logarithmic type are cov ered b y the analysis. W e sho w the F r ´ ec het diﬀeren tiability of the associated con trol-to-state op erator in suitable Banac h spaces and establish ﬁrst-order neces- sary optimality conditions in terms of a v ariational inequality in volving the adjoin t state v ariables. The necessary optimalit y conditions are then used to deriv e sparsit y results for the optimal con trols. Key w ords: T umor growth models, singular potentials, h yp erb olic relaxation, optimal control, necessary optimalit y conditions, sparsity AMS (MOS) Sub ject Classiﬁcation: 49J20, 49K20, 49K40, 35K57, 37N25 1 In tro duction Let α > 0 , τ > 0, and let Ω ⊂ R 3 denote some op en and b ounded domain ha ving a smo oth boundary Γ = ∂ Ω with out w ard normal n and corresp onding out ward normal deriv ativ e ∂ n . Moreov er, w e ﬁx some ﬁnal time T > 0 and introduce for ev ery t ∈ (0 , T ] the sets Q t := Ω × (0 , t ) and Σ t := Γ × (0 , t ), where we put, for the sak e of brevit y , Q := Q T and Σ := Σ T . W e consider in this pap er the follo wing optimal control problem: 1 2 Colli – Rocca – Sprekels ( CP ) Minimize the cost functional J (( µ, ϕ, σ ) , u ) = b 1 2 Z Q | ϕ − b ϕ Q | 2 + b 2 2 Z Ω | ϕ ( T ) − b ϕ Ω | 2 + b 3 2 Z Q | u | 2 + κ G ( u ) = : J (( µ, ϕ, σ ) , u ) + κ G ( u ) (1.1) sub ject to the state system α∂ tt µ + ∂ t ϕ − ∆ µ = P ( ϕ )( σ + χ (1 − ϕ ) − µ ) − h ( ϕ ) u 1 in Q , (1.2) τ ∂ t ϕ − ∆ ϕ + F ′ ( ϕ ) = µ + χ σ in Q , (1.3) ∂ t σ − ∆ σ = − χ ∆ ϕ − P ( ϕ )( σ + χ (1 − ϕ ) − µ ) + u 2 in Q , (1.4) ∂ n µ = ∂ n ϕ = ∂ n σ = 0 on Σ , (1.5) µ (0) = µ 0 , ∂ t µ (0) = µ ′ 0 , ϕ (0) = ϕ 0 , σ (0) = σ 0 in Ω , (1.6) and to the control constrain t u = ( u 1 , u 2 ) ∈ U ad , (1.7) where U ad is some nonempty , closed, b ounded and conv ex subset of the space L ∞ ( Q ) × L 2 ( Q ) that will b e sp eciﬁed later. Moreo ver, b 1 ≥ 0, b 2 ≥ 0, b 3 > 0 and κ ≥ 0 are prescrib ed constants, b ϕ Q ∈ L 2 ( Q ) and b ϕ Ω ∈ H 1 (Ω) are giv en target functions, and the functional G : L 2 ( Q ) × L 2 ( Q ) → R is nonnegativ e, con tinuous and con v ex. A protot ypical form for G , which enhances the occurrence of sp arsity , is giv en by G ( u ) = ∥ u ∥ L 1 ( Q ) = Z Q ( | u 1 | + | u 2 | ) . (1.8) The state system (1.2)–(1.6) constitutes a simpliﬁed and relaxed version of the four- sp ecies thermo dynamically consisten t mo del for tumor growth originally prop osed b y Ha wkins-Daruud et al. in [20] that additionally includes chemoctatic terms. Let us brieﬂy review the role of the o ccurring sym b ols. The primary v ariables ϕ, µ, σ denote the phase ﬁeld, the asso ciated c hemical p oten tial, and the nutrien t concen tration, resp ectiv ely . Moreov er, the additional term α ∂ tt µ is a hyperb olic regularization of equa- tion (1.2), while the term τ ∂ t ϕ is a viscosit y con tribution to the Cahn–Hilliard equation. In the ab o ve mo del equations, there are tw o functions acting as distributed con trols in the phase and nutrien t equations, resp ectively . The con trol v ariable u 2 can mo del the supply of either a medication or some nutrien t, while u 1 , whic h is nonlinearly coupled to the state v ariable ϕ in the phase equation (1.2) through the term h ( ϕ ) u 1 , mo dels the application of a cytoto xic drug into the system. Here, h is a truncation function in order to hav e the action restricted to the spatial region in which the tumor cells are lo cated; for instance, it can b e assumed that h ( − 1) = 0 , h (1) = 1, and 0 < h ( ϕ ) < 1 if − 1 < ϕ < 1. W e refer to [17, 19, 21, 23] for p ossible c hoices of h . The nonlinearity P denotes a proliferation function, and the p ositiv e constant χ represen ts the chemotactic sensitivity . Finally , the nonlinear function F is a double-w ell p oten tial whose deriv ativ e represen ts the thermo dynamic force driving the ev olution of the system. T ypical examples, which will b e included by our analysis, are given b y the regular and logarithmic p otentials, which Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 3 are deﬁned, in this order, by F reg ( r ) = 1 4  1 − r 2  2 for r ∈ R , (1.9) F log ( r ) =    (1 + r ) ln(1 + r ) + (1 − r ) ln(1 − r ) − k 1 r 2 for r ∈ ( − 1 , 1) 2 ln(2) − k 1 for r ∈ {− 1 , 1 } , + ∞ for r ∈ [ − 1 , 1] (1.10) where k 1 > 1 so that F log is nonconv ex. F or technical reasons, w e do not consider non- smo oth p otentials like the double obstacle p otential in this pap er. The logarithmic p o- ten tial F log is of particular relev ance in the applications, since it is closely connected with the conﬁgurational entrop y . W e observ e that the thermo dynamic force F ′ log ( r ) b ecomes un b ounded as r → ± 1, whic h forces the phase v ariable ϕ to attain its v alues in the ph ysically meaningful range ( − 1 , 1). As far as well-posedness is concerned, the ab ov e mo del has b een in vestigated in the recen t pap er [7], in which results concerning existence, uniqueness, regularit y , and con- tin uous dep endence ha ve b een established (see, in particular, the Theorems 2.2 and 2.3 stated b elo w in Section 2). In this connection, w e remark that there exists a large b o dy of literature devoted to well-posedness results for diﬀerent v arian ts of the ab ov e mo del. F or an accoun t of these con tributions, we refer the interested reader to the introduction and the references given in [7]. In this pap er, w e fo cus on the optimal control of the state system (1.2)–(1.6). The optimal con trol of tumor growth phase ﬁeld systems constitutes an imp ortan t direction of researc h, since it can b e applied directly to devise and implement strategies for the medical treatmen t of cancer patien ts. W orks on boundary and distributed con trol ap- p eared in [3, 4, 18]. Control strategies incorp orating c hemotaxis, active transport, v ariable mobilities, and Keller–Segel dynamics w ere developed in [1, 2, 5, 8, 14, 23, 34]. F urther con- tributions included adv anced optimality conditions in [15, 16], as well as a reﬁned analysis of treatmen t-time optimization and related asymptotics in [28 – 30, 35]. Singular logarith- mic and double obstacle p oten tials w ere addressed in [26, 27], while sparse controls and second-order suﬃcient optimality conditions were studied in [9, 13, 32]. W ell-p osedness, regularit y , and asymptotic b ehavior for mo dels relev an t to con trol applications and in- cluding chemotaxis were dev elop ed in [8]. The ab ov emen tioned results collectively pro vide a rigorous framework for the design and optimization of therap eutic strategies gov erned b y diﬀuse–interface tumor-gro wth mo dels. Concerning the hyperb olic relaxation of the chemical p oten tial in the viscous Cahn– Hilliard equation (uncoupled from the n utrient and without mass sources), we refer to the recent contributions [11, 12], whic h inspired the presen t w ork. In [11], well-posedness, con tinuous dep endence, and regularit y results w ere established, along with an analysis of the asymptotic b ehavior as the relaxation parameter α tends to 0. A related optimal con trol problem w as studied in [12]. The deriv ation of optimalit y conditions for the control problem ( CP ) mak es it necessary to establish diﬀeren tiabilit y prop erties of the con trol-to-state operator asso ciated with the state system (1.2)–(1.6). The pro of of suc h diﬀerentiabilit y prop erties, in turn, app ears to b e p ossible only if the state v ariable ϕ attains its v alue in a prop er compact subset of the in terior of the domain of the nonlinearity F ′ (the in terv al ( − 1 , 1), in the case 4 Colli – Rocca – Sprekels of the logarithmic potential (1.10)). In other w ords, the so-called uniform sep ar ation c ondition m ust b e v alid. A t this p oin t, a signiﬁcan t diﬀerence betw een regular potentials lik e (1.9) and irregular p otentials like (1.10) b ecomes apparent: indeed, it follows from the Theorems 2.2 and 2.3 b elow that, under suitable assumptions on the other data of the system (1.2)–(1.6), for regular p oten tials the uniform separation condition is alwa ys v alid if the control v ariable u 1 b elongs to L ∞ ( Q ), while in singular cases like (1.10) this can only b e guaran teed under the stronger condition u 1 ∈ L ∞ ( Q ) ∩ L 2 (0 , T ; V ). As a consequence of this fact, the sets U ad of admissible controls and the entire analysis of the problem ( CP ) are likely to diﬀer in the t wo cases. W e ha v e c hosen to treat the t wo cases in separate sections. The pap er is organized as follows: in the following Section 2, we form ulate the general setting of our problem and state well-posedness results for the state system (1.2)–(1.6) pro ved in [7]. In Section 3, w e then consider the case of regular p otentials. W e show the F r ´ e c het diﬀeren tiabilit y of the con trol-to-state operator b etw een suitable Banach spaces and the well-posedness of the associated adjoin t state system, and deriv e ﬁrst-order necessary optimalit y conditions for ( CP ). In Section 4, the same program is p erformed for the singular case, where many parts of the analysis of Section 3 carry ov er to this situation. The ﬁnal Section 5 then brings some results concerning the sparsit y of optimal controls. Throughout the pap er, we mak e rep eated use of H¨ older’s inequality , of the elementary Y oung’s inequalit y ab ≤ δ | a | 2 + 1 4 δ | b | 2 ∀ a, b ∈ R , ∀ δ > 0 , (1.11) as w ell as of the contin uit y of the embeddings H 1 (Ω) ⊂ L p (Ω) for 1 ≤ p ≤ 6 and H 2 (Ω) ⊂ C 0 (Ω). Notice that the latter embedding is also compact, while this holds true for the former embeddings only if p < 6. 2 General setting and the state system In this section, w e in tro duce the general setting of our problem and state w ell-p osedness results for the state system (1.2)–(1.6) prov ed in [7]. T o b egin with, for a Banac h space X w e denote by ∥ · ∥ X the norm in the space X or in a p ow er thereof, and b y X ∗ its dual space. The only exception to this rule applies to the norms of the L p spaces and of their p o wers, whic h we often denote by ∥ · ∥ p , for 1 ≤ p ≤ ∞ . As usual, for Banach spaces X and Y that are con tained in the same top ological v ector space, we introduce the linear space X ∩ Y , whic h b ecomes a Banach space when endow ed with its natural norm ∥ u ∥ X ∩ Y := ∥ u ∥ X + ∥ u ∥ Y , for u ∈ X ∩ Y . Moreov er, we in tro duce the spaces H := L 2 (Ω) , V := H 1 (Ω) , W := { v ∈ H 2 (Ω) : ∂ n v = 0 on Γ } . (2.1) F urthermore, we denote b y ( · , · ), ∥ · ∥ , and ⟨· , ·⟩ the standard inner pro duct and norm in H , as w ell as the dualit y pairing b et ween V and its dual V ∗ . W e then hav e the dense and compact em b eddings V ⊂ H ⊂ V ∗ , with the standard iden tiﬁcation ⟨ v , w ⟩ = ( v , w ) for v ∈ H and w ∈ V . W e also introduce for t ∈ [0 , T ] and elemen ts w ∈ L 1 (0 , T ; V ∗ ) the notation (1 ∗ w )( t ) := Z s 0 w ( s ) ds . (2.2) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 5 W e no w provide precise assumptions for the data of the system (1.2)–(1.6). (A1) α, τ and χ are p ositive constan ts. (A2) F = F 1 + F 2 satisﬁes: F 2 ∈ C 3 ( R ) has a Lipschitz contin uous deriv ativ e F ′ 2 ; F 1 : R → [0 , + ∞ ] is con v ex and low er semicontin uous with F 1 (0) = 0, and there are constan ts −∞ ≤ r − < 0 < r + ≤ + ∞ such that the restriction of F 1 to ( r − , r + ) b elongs to C 3 ( r − , r + ) and satisﬁes lim r ↘ r − F ′ 1 ( r ) = −∞ and lim r ↗ r + F ′ 1 ( r ) = + ∞ . (2.3) (A3) P ∈ C 2 ( R ) is nonnegativ e, b ounded, and P and P ′ are Lipsc hitz con tinuous. (A4) h ∈ C 2 ( R ) is nonnegativ e, b ounded, and h and h ′ are Lipsc hitz con tinuous. Let us note that b oth the p oten tials (1.9) and (1.10) fulﬁll the condition ( A2 ), where in the latter case we ha v e ( r − , r + ) = ( − 1 , 1). Observ e also that (A2) implies that the sub diﬀeren tial ∂ F 1 of F 1 is a maximal monotone graph in R × R . Moreo ver, by virtue of the gro wth conditions (2.3), its eﬀective domain D ( ∂ F 1 ) is the op en in terv al ( r − , r + ). Since the restriction of F 1 to ( r − , r + ) b elongs to C 3 ( r − , r + ) by (A2) , we ha ve for every r ∈ D ( ∂ F 1 ) that ∂ F 1 ( r ) = { F ′ 1 ( r ) } . Also, since F 1 attains its minimum v alue 0 at 0, it turns out that 0 ∈ D ( ∂ F 1 ) and 0 ∈ ∂ F 1 (0). Finally , w e observe that the assumptions on F 2 imply that F 2 gro ws at most quadratically . Next, w e in tro duce our notion of a solution to (1.2)–(1.6). Deﬁnition 2.1. A triple ( µ, ϕ, σ ) is c al le d a solution to the state system (1.2) – (1.6) if µ ∈ H 2 (0 , T ; V ∗ ) ∩ W 1 , ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; V ) , (2.4) ϕ ∈ W 1 , ∞ (0 , T ; H ) ∩ H 1 (0 , T ; V ) ∩ L ∞ (0 , T ; W ) ∩ C 0 ( Q ) , (2.5) ϕ ∈ ( r − , r + ) a.e. in Q, (2.6) σ ∈ H 1 (0 , T ; H ) ∩ C 0 ([0 , T ]; V ) ∩ L 2 (0 , T ; W ) , (2.7) F ′ 1 ( ϕ ) ∈ L ∞ (0 , T ; H ) , (2.8) and if ( µ, ϕ, σ ) satisﬁes ⟨ α∂ tt µ, v ⟩ + Z Ω ∂ t ϕ v + Z Ω ∇ µ · ∇ v = Z Ω P ( ϕ )( σ + χ (1 − ϕ ) − µ ) v − Z Ω h ( ϕ ) u 1 v for every v ∈ V and a.e. in (0 , T ) , (2.9) τ ∂ t ϕ − ∆ ϕ + F ′ ( ϕ ) = µ + χ σ, a.e. in Q , (2.10) ∂ t σ − ∆ σ = − χ ∆ ϕ − P ( ϕ )( σ + χ (1 − ϕ ) − µ ) + u 2 a.e. in Q , (2.11) as wel l as µ (0) = µ 0 , ∂ t µ (0) = µ ′ 0 , ϕ (0) = ϕ 0 , σ (0) = σ 0 , a.e. in Ω . (2.12) 6 Colli – Rocca – Sprekels It is worth noting that the homogeneous Neumann b oundary conditions (1.5) are enco ded in the conditions (2.5) and (2.7) for ϕ and σ (cf. the deﬁnition of the space W ), and in the v ariational equality (2.9) for µ . Notice also that the initial conditions (2.12) are meaningful, b ecause (2.5) and (2.7) imply that ϕ, σ ∈ C 0 ([0 , T ]; V ), while, owing to (2.4), it turns out that µ ∈ C 1 ([0 , T ]; V ∗ ) ∩ C 0 ([0 , T ]; H ) and, consequen tly , ∂ t µ is at least w eakly contin uous from [0 , T ] to H . W e also observe that our notion of solution is a special case of that giv en in [7, Def. 2.1]. Indeed, there the solutions were quadruples ( µ, ϕ, ξ , σ ) where ξ ∈ L ∞ (0 , T ; H ) satisﬁed in place of the identit y (2.10) the inclusion condition τ ∂ t ϕ − ∆ ϕ + ξ + F ′ 2 ( ϕ ) = µ + χ σ, ξ ∈ ∂ F 1 ( ϕ ) , a.e. in Q. But, as men tioned abov e, then it follows that ξ = F ′ 1 ( ϕ ) a.e. in Q , whence w e obtain our equation (2.10). Concerning the w ell-p osedness of the state system, w e hav e the following result. Theorem 2.2. Supp ose that (A1) – (A4) ar e fulﬁl le d, and let the initial data satisfy µ 0 ∈ V , µ ′ 0 ∈ H, σ 0 ∈ V , ϕ 0 ∈ W , with r − < ϕ 0 ( x ) < r + for al l x ∈ Ω . (2.13) Then the state system (1.2) – (1.6) has for every u = ( u 1 , u 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) a unique solution ( µ, ϕ, σ ) in the sense of Deﬁnition 2.1, and ther e exists a c onstant K 1 > 0 , which dep ends only on the norm ∥ u ∥ L ∞ ( Q ) × L 2 ( Q ) and the data of the system, such that ∥ µ ∥ H 2 (0 ,T ; V ∗ ) ∩ W 1 , ∞ (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ ϕ ∥ W 1 , ∞ (0 ,T ; H ) ∩ H 1 (0 ,T ; V ) ∩ L ∞ (0 ,T ; W ) ∩ C 0 ( Q ) + ∥ ξ ∥ L ∞ (0 ,T ; H ) + ∥ σ ∥ H 1 (0 ,T ; H ) ∩ C 0 ([0 ,T ]; V ) ∩ L 2 (0 ,T ; W ) ≤ K 1 . (2.14) Mor e over, whenever ( µ i , ϕ i , σ i ) , i = 1 , 2 , ar e two solutions to (1.2) – (1.6) asso ciate d with the data u i = ( u i 1 , u i 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) , i = 1 , 2 , then we have ∥ µ 1 − µ 2 ∥ L ∞ (0 ,T ; H ) + ∥ 1 ∗ ( µ 1 − µ 2 ) ∥ L ∞ (0 ,T ; V ) + ∥ ϕ 1 − ϕ 2 ∥ L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V ) + ∥ σ 1 − σ 2 ∥ L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V ) ≤ K 2  ∥ u 1 1 − u 2 1 ∥ L 2 (0 ,T ; H ) + ∥ u 1 2 − u 2 2 ∥ L 2 (0 ,T ; H )  , (2.15) with a c onstant K 2 > 0 which only dep ends on the data of the system and the norms ∥ u i ∥ L ∞ ( Q ) × L 2 ( Q ) , i = 1 , 2 . Pr o of. Observ e that, in view of the contin uit y of the embedding W ⊂ C 0 (Ω), the initial datum ϕ 0 is contin uous in Ω. The last condition in (2.13) therefore implies that ϕ 0 attains its v alues in a compact subset of ( r − , r + ). Hence, by virtue of (A2) , w e hav e F 1 ( ϕ 0 ) ∈ L 1 (Ω) and F ′ 1 ( ϕ 0 ) ∈ H , so that all of the conditions for the application of Theorem 2.2 in [7] are met. The result is then a direct consequence of [7, Thm. 2.2]. The next result provides further regularit y for more regular initial data and controls u = ( u 1 , u 2 ). In particular, it yields a uniform sep ar ation c ondition , which will pro ve to b e fundamen tal for the con trol theory in the case of singular p otentials like (1.10) when −∞ < r − < r + < + ∞ . Observe that in the regular case ( r − , r + ) = ( −∞ , + ∞ ) such a separation condition is automatically satisﬁed: indeed, the estimate (2.14) yields, in particular, that ∥ ϕ ∥ C 0 ( Q ) ≤ K 1 . Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 7 Theorem 2.3. Assume that (A1) – (A4) hold true, and let the initial data fulﬁl l (2.13) as wel l as the additional assumptions µ 0 ∈ W , µ ′ 0 ∈ V , σ 0 ∈ L ∞ (Ω) . (2.16) In addition, supp ose that u = ( u 1 , u 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) satisﬁes u = ( u 1 , u 2 ) ∈ L 2 (0 , T ; V ) × L ∞ (0 , T ; H ) . (2.17) Then the solution ( µ, ϕ, σ ) to (1.2) – (1.6) in the sense of Deﬁnition 2.1 satisﬁes σ ∈ L ∞ ( Q ) , µ ∈ H 1 (0 , T ; V ) ∩ L ∞ (0 , T ; W ) , F ′ 1 ( ϕ ) ∈ L ∞ ( Q ) . (2.18) Mor e over, ther e ar e c onstants K 3 > 0 and r − < r ∗ < r ∗ < r + , which dep end only on the data of the system and the norm ∥ u ∥ ( L ∞ ( Q ) ∩ L 2 (0 ,T ; V )) × L ∞ (0 ,T ; H ) , such that ∥ σ ∥ L ∞ ( Q ) + ∥ µ ∥ H 1 (0 ,T ; V ) ∩ L ∞ (0 ,T ; W ) + ∥ F ′ 1 ( ϕ ) ∥ L ∞ ( Q ) ≤ K 3 , (2.19) r ∗ ≤ ϕ ( x.t ) ≤ r ∗ for al l ( x, t ) ∈ Q . (2.20) Pr o of. A t ﬁrst, recall that the range of the con tin uous function ϕ 0 lies in a compact subset of ( r − , r + ) so that, by (A2) , F ′ 1 ( ϕ 0 ) ∈ L ∞ (Ω). Therefore, all of the conditions for the application of Theorem 3.1 in [7] are met, whence it follo ws that the conditions (2.18) and (2.19) are v alid. The existence of suitable constants r − < r ∗ < r ∗ < r + satisfying (2.20) then follo ws from the asymptotic growth assumptions made in (2.3). 3 The case of regular p oten tials In this section, we consider the case of regular p oten tials deﬁned on the whole real line. Throughout, we work under the assumptions of Theorem 2.2. In this setting, assumption (A2) implies the following regularit y prop ert y: F 1 and F 2 b elong to C 3 ( R ) . (3.1) Then, b y Theorem 2.2, the control-to-state op erator S : L ∞ ( Q ) × L 2 ( Q ) ∋ u = ( u 1 , u 2 ) 7→ S ( u ) := ( µ, ϕ, σ ) (3.2) is w ell deﬁned as a mapping into the space X =  H 2 (0 , T ; V ∗ ) ∩ W 1 , ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; V )  ×  W 1 , ∞ (0 , T ; H ) ∩ H 1 (0 , T ; V ) ∩ L ∞ (0 , T ; W ) ∩ C 0 ( Q )  ×  H 1 (0 , T ; H ) ∩ C 0 (0 , T ; V ) ∩ L 2 (0 , T ; W )  , (3.3) whic h is a Banach space when endow ed with its natural norm. Moreo v er, S is lo cally Lipsc hitz contin uous in the sense of (2.15), then in a larger space. W e now improv e this result. 8 Colli – Rocca – Sprekels Theorem 3.1. Supp ose that the assumptions of The or em 2.2 and (3.1) ar e satisﬁe d, and let u i = ( u i 1 , u i 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) ar e given with the asso ciate d solutions S ( u i ) = ( µ i , ϕ i , σ i ) , for i = 1 , 2 with the same initial data. Then ther e is a c onstant K 4 > 0 , which only dep ends on the data of the system and on the ( L ∞ ( Q ) × L 2 ( Q )) − norms of u 1 and u 2 , such that ∥ S ( u 1 ) − S ( u 2 ) ∥ X ≤ K 4  ∥ u 1 1 − u 2 1 ∥ L 2 (0 ,T ; H ) + ∥ u 1 2 − u 2 2 ∥ L 2 (0 ,T ; H )  . (3.4) Pr o of. W e in tro duce the quan tities µ := µ 1 − µ 2 , ϕ := ϕ 1 − ϕ 2 , σ := σ 1 − σ 2 , u 1 := u 1 1 − u 2 1 , u 2 := u 1 2 − u 2 2 , whic h then satisfy the iden tities ⟨ α∂ tt µ, v ⟩ + Z Ω ∇ µ · ∇ v = − Z Ω ∂ t ϕv + Z Ω ( P ( ϕ 1 ) − P ( ϕ 2 ))( σ 1 + χ (1 − ϕ 1 ) − µ 1 ) v + Z Ω P ( ϕ 2 )( σ − χ ϕ − µ ) v − Z Ω ( h ( ϕ 1 ) − h ( ϕ 2 )) u 1 1 v − Z Ω h ( ϕ 2 ) u 1 v for ev ery v ∈ V and a.e. in (0 , T ), (3.5) τ ∂ t ϕ − ∆ ϕ = − F ′ ( ϕ 1 ) + F ′ ( ϕ 2 ) + µ + χ σ a.e. in Q , (3.6) ∂ t σ − ∆ σ = − χ ∆ ϕ − ( P ( ϕ 1 ) − P ( ϕ 2 ))( σ 1 + χ (1 − ϕ 1 ) − µ 1 ) − P ( ϕ 2 )( σ − χ ϕ − µ ) + u 2 a.e. in Q , (3.7) µ (0) = 0 , ∂ t µ (0) = 0 , ϕ (0) = 0 , σ (0) = 0 , a.e. in Ω . (3.8) No w recall that ϕ i ∈ C 0 ( Q ), i = 1 , 2. Hence, there is some constant L > 0 such that | P ( ϕ 1 ) − P ( ϕ 2 ) | + | h ( ϕ 1 ) − h ( ϕ 2 ) | + | F ′ ( ϕ 1 ) − F ′ ( ϕ 2 ) | ≤ L | ϕ | a.e. in Q. (3.9) Note that the constan t L depends only on the quan tit y R := max {∥ ϕ 1 ∥ C 0 ( Q ) , ∥ ϕ 2 ∥ C 0 ( Q ) } , and thus, according to (2.14), only on the data of the state system (1.2)–(1.6) and the norms ∥ u i ∥ L ∞ ( Q ) × L 2 ( Q ) , i = 1 , 2. In the follo wing, we denote by C suc h p ositive constan ts. W e b egin b y p ointing out that the righ t-hand side of (3.6) is at least in L 2 (0 , T ; H ) and that its norm can b e estimated with the help of (3.9) and (2.15). Moreo ver, the initial datum for ϕ in (3.8) is null and therefore b elongs to V . Then, by applying a w ell-known parab olic regularity estimate (see, e.g., [25, Chapter 3]), w e ﬁnd out that ∥ ϕ ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ≤ C ∥ L | ϕ | + | µ | + χ | σ |∥ L 2 (0 ,T ; H ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . (3.10) Next, the same procedure can b e applied to the parab olic equation (3.7). W e hav e to verify that the L 2 (0 , T ; H )-norm of the righ t-hand side of (3.7) is suitably b ounded in L 2 (0 , T ; H ). In view of (A3) , (2.15), and (3.10), note that only the second term on the righ t-hand side of (3.7) requires a sp ecial attention. Thanks to the b oundedness of σ 1 , ϕ 1 , µ 1 in L ∞ (0 , T ; L 4 (Ω)), the H¨ older inequality , and the con tinuous em b edding Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 9 V ⊂ L 4 (Ω), w e ha ve that ∥ ( P ( ϕ 1 ) − P ( ϕ 2 ))  σ 1 + χ (1 − ϕ 1 ) − µ 1  ∥ 2 L 2 (0 ,T ; H ) ≤ Z Q L 2 | ϕ | 2  χ + | σ 1 | + χ | ϕ 1 | + | µ 1 |  2 ≤ C Z T 0  ∥ ϕ ( s ) ∥ 2 + ∥ ϕ ( s ) ∥ 2 4  ∥ σ 1 ( s ) ∥ 2 4 + ∥ ϕ 1 ( s ) ∥ 2 4 + ∥ µ 1 ( s ) ∥ 2 4   ds ≤ C ∥ ϕ ∥ 2 L 2 (0 ,T ; V ) ≤ C  ∥ u 1 ∥ 2 L 2 (0 ,T ; H ) + ∥ u 2 ∥ 2 L 2 (0 ,T ; H )  . But then it follows from (3.10) that ∥ σ ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . (3.11) W e no w turn our atten tion to the h yp erb olic equation (3.5). Recalling the theory in [25, Chapter 3, Sections 8–9], the solution µ satisﬁes µ ∈ C 1 (0 , T ; H ) ∩ C 0 (0 , T ; V ) and ob eys the asso ciated energy iden tity , obtained b y formally choosing v = ∂ t µ in (3.5) and in tegrating ov er (0 , t ). This identit y holds for ev ery t ∈ [0 , T ]. T aking (3.8) into accoun t and applying Y oung’s inequality repeatedly , we deduce that α 2 ∥ ∂ t µ ( t ) ∥ 2 + 1 2 ∥∇ µ ( t ) ∥ 2 ≤ 1 2 Z Q t  | ∂ t ϕ | 2 + | ∂ t µ | 2  + L Z Q t | ϕ |  χ + | σ 1 | + χ | ϕ 1 | + | µ 1 |  | ∂ t µ | + C Z Q t  | σ | 2 + | ϕ | 2 + | µ | 2 + | u 1 | 2 + | ∂ t µ | 2  . (3.12) Note that, in order to obtain the last term on the right-hand side, w e ha v e in particular exploited the b ound ∥ u 1 1 ∥ L ∞ ( Q ) , which is absorb ed in the constan t C , together with the assumptions (A3) and (A4) . As for the second term on the right-hand side of (3.12), whic h w e denote by I 1 , w e apply H¨ older’s and Y oung’s inequalities, as w ell as the fact that σ 1 , ϕ 1 , and µ 1 are b ounded in L ∞ (0 , T ; L 4 (Ω)). It then follows, up on also inv oking (2.15), that | I 1 | ≤ C Z t 0 ∥ ϕ ( s ) ∥ 4  1 + ∥ σ 1 ( s ) ∥ 4 + ∥ ϕ 1 ( s ) ∥ 4 + ∥ µ 1 ( s ) ∥ 4  ∥ ∂ t µ ( s ) ∥ ds ≤ C Z Q t | ∂ t µ | 2 + C ∥ ϕ ∥ 2 L 2 (0 ,T ; V ) ≤ C Z Q t | ∂ t µ | 2 + C  ∥ u 1 ∥ 2 L 2 (0 ,T ; H ) + ∥ u 2 ∥ 2 L 2 (0 ,T ; H )  . (3.13) Com bining (3.12) and (3.13) with (2.15) and (3.10), w e th us obtain that α 2 ∥ ∂ t µ ( t ) ∥ 2 + 1 2 ∥∇ µ ( t ) ∥ 2 ≤ C Z Q t | ∂ t µ | 2 + C  ∥ u 1 ∥ 2 L 2 (0 ,T ; H ) + ∥ u 2 ∥ 2 L 2 (0 ,T ; H )  . (3.14) 10 Colli – Rocca – Sprekels Then, an application of the Gronw all lemma leads us to the estimate ∥ µ ∥ W 1 , ∞ (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . (3.15) In addition, from the previous estimates and a comparison in (3.5) it also follows that ∥ µ ∥ H 2 (0 ,T ; V ∗ ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . (3.16) A t this p oint, w e may observ e that the right-hand side of (3.6) is in H 1 (0 , T ; H ) and its time deriv ativ e is given b y − F ′′ ( ϕ 1 ) ∂ t ϕ 1 + F ′′ ( ϕ 2 ) ∂ t ϕ 2 + ∂ t µ + χ ∂ t σ = − ( F ′′ ( ϕ 1 ) − F ′′ ( ϕ 2 )) ∂ t ϕ 1 − F ′′ ( ϕ 2 ) ∂ t ϕ + ∂ t µ + χ ∂ t σ, whic h b elongs to L 2 (0 , T ; H ), since | F ′′ ( ϕ 1 ) − F ′′ ( ϕ 2 ) | ≤ C | ϕ | b y (A2) and (2.14), ∂ t ϕ 1 is b ounded in L 2 (0 , T ; V ), and F ′′ ( ϕ 2 ) is ob viously b ounded. Moreov er, the initial v alue for ∂ t ϕ resulting from (3.6) and (3.8) is still zero on account of ϕ 1 (0) = ϕ 2 (0) = ϕ 0 (cf. (2.12)). Then, applying again the parab olic regularity [25] at the level of ∂ t ϕ , it turns out that ϕ ∈ H 2 (0 , T ; H ) ∩ W 1 , ∞ (0 , T ; V ) ∩ H 1 (0 , T ; W ) solves τ ∂ tt ϕ − ∆( ∂ t ϕ ) = − ( F ′′ ( ϕ 1 ) − F ′′ ( ϕ 2 )) ∂ t ϕ 1 − F ′′ ( ϕ 2 ) ∂ t ϕ + ∂ t µ + χ ∂ t σ a.e. in Q , (3.17) ∂ t ϕ (0) = 0 a.e. in Ω . (3.18) No w, multiplying (3.17) b y ∂ t ϕ , in tegration ov er Q t and Y oung’s inequality lead to the estimate τ 2 ∥ ∂ t ϕ ( t ) ∥ 2 + Z Q t |∇ ∂ t ϕ | 2 ≤ C Z Q t | ϕ | | ∂ t ϕ 1 | | ∂ t ϕ | + C Z Q t | ∂ t ϕ | 2 + Z Q t | ( ∂ t µ + χ ∂ t σ ) ∂ t ϕ | ≤ C Z t 0 ∥ ϕ ( s ) ∥ 4 ∥ ∂ t ϕ 1 ( s ) ∥ 4 ∥ ∂ t ϕ ( s ) ∥ ds + C Z Q t  | ∂ t µ | 2 + | ∂ t σ | 2 + | ∂ t ϕ | 2  , (3.19) and for the ﬁrst term on the righ t-hand side we ha ve that C Z t 0 ∥ ϕ ( s ) ∥ 4 ∥ ∂ t ϕ 1 ( s ) ∥ 4 ∥ ∂ t ϕ ( s ) ∥ ds ≤ C Z Q t | ∂ t ϕ | 2 + C ∥ ϕ ∥ 2 L ∞ (0 ,T ; V ) ∥ ∂ t ϕ 1 ∥ 2 L 2 (0 ,T ; V ) , (3.20) where w e used the con tinuous embedding V  → L 4 (Ω) and the fact that ∂ t ϕ 1 ∈ L 2 (0 , T ; V ). Collecting (3.19) and (3.20), and exploiting the previously derived estimates (3.10), (3.11), (3.15), w e conclude that ∥ ϕ ∥ W 1 , ∞ (0 ,T ; H ) ∩ H 1 (0 ,T ; V ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . (3.21) Next, w e infer from (3.6) and (3.9) that, a.e. in Q , | ∆ ϕ | = | − τ ∂ t ϕ − F ′ ( ϕ 1 ) + F ′ ( ϕ 2 ) + µ + χ σ | ≤ C  | ∂ t ϕ | + | ϕ | + | µ | + | σ |  . Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 11 Therefore, w e can conclude from (3.21) and standard elliptic estimates that ∥ ϕ ∥ L ∞ (0 ,T ; W ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  , and it follo ws from [31, Sect. 8, Cor. 4] and the compactness of the em b edding W ⊂ C 0 (Ω) that also ∥ ϕ ∥ C 0 ( Q ) ≤ C  ∥ u 1 ∥ L 2 (0 ,T ; H ) + ∥ u 2 ∥ L 2 (0 ,T ; H )  . With this, the pro of of the assertion is complete. 3.1 The linearized system In the following, w e study the diﬀeren tiability prop erties of the con trol-to-state op erator S . W e b egin our analysis with the linearization of the system (1.2)–(1.6). T o this end, let u = ( u 1 , u 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) b e ﬁxed and ( µ, ϕ, σ ) = S ( u ) denote the associated solution according to Theorem 2.2. W e then consider for giv en ( h 1 , h 2 ) ∈ L 2 ( Q ) × L 2 ( Q ) the initial-b oundary v alue problem ⟨ α∂ tt η , v ⟩ + Z Ω ∂ t ψ v + Z Ω ∇ η · ∇ v = Z Ω P ( ϕ )  ξ − χ ψ − η  v + Z Ω P ′ ( ϕ )  σ + χ (1 − ϕ ) − µ  ψ v − Z Ω h ( ϕ ) h 1 v − Z Ω h ′ ( ϕ ) u 1 ψ v for all v ∈ V and a.e. in (0 , T ) , (3.22) τ ∂ t ψ − ∆ ψ + F ′′ ( ϕ ) ψ = χ ξ + η , a.e. in Q, (3.23) ∂ t ξ − ∆ ξ = − χ ∆ ψ − P ( ϕ )  ξ − χ ψ − η  − P ′ ( ϕ )  σ + χ (1 − ϕ ) − µ  ψ + h 2 , a.e. in Q, (3.24) ∂ n ψ = ∂ n ξ = 0 a.e. on Σ , (3.25) η (0) = ∂ t η (0) = ψ (0) = ξ (0) = 0 a.e. in Ω . (3.26) Ob viously , (3.22) is just the v ariational form of the equation α∂ tt η + ∂ t ψ − ∆ η = P ( ϕ )  ξ − χ ψ − η  + P ′ ( ϕ )  σ + χ (1 − ϕ ) − µ  ψ − h ( ϕ ) h 1 − h ′ ( ϕ ) u 1 ψ in Q, together with the b oundary condition ∂ n η = 0 on Σ. W e ha v e the follo wing result. Theorem 3.2. Supp ose that the assumptions of The or em 2.2 and (3.1) ar e fulﬁl le d, and let u = ( u 1 , u 2 ) ∈ L ∞ ( Q ) × L 2 ( Q ) b e given with asso ciate d ( µ, ϕ, σ ) = S ( u ) ∈ X . Then the line arize d system (3.22) – (3.26) has for every incr ement h = ( h 1 , h 2 ) ∈ L 2 ( Q ) × L 2 ( Q ) a unique solution ( η h , ψ h , ξ h ) ∈ X , and ther e exists a c onstant K 5 > 0 , which dep ends only on the data of the system and the ( L ∞ ( Q ) × L 2 ( Q )) − norm of u , such that ∥ ( η h , ψ h , ξ h ) ∥ X ≤ K 5  ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H )  . (3.27) In other wor ds, the line ar mapping h 7→ ( η h , ψ h , ξ h ) is c ontinuous fr om L 2 (0 , T ; H ) × L 2 (0 , T ; H ) into X . 12 Colli – Rocca – Sprekels Pr o of. The existence result is pro ved by means of a F aedo–Galerkin appro ximation using the eigenfunctions { e j } j ∈ N of the eigenv alue problem − ∆ e j = λ j e j in Ω, ∂ n e j = 0 on Γ, as basis functions. F or the sak e of brevity , we av oid here writing the ﬁnite-dimensional system explicitly and only perform the relev an t a priori estimates formally on the con tin- uous system (3.22)–(3.26). W e note, how ev er, that these formal estimates are in any case fully justiﬁed on the level of the F aedo–Galerkin approximations. W e also observ e that ϕ attains its v alues in a compact subset of R , so that all of the functions P ( ϕ ), P ′ ( ϕ ), h ( ϕ ), h ′ ( ϕ ), and F ′′ ( ϕ ), are b ounded in Q b y a constant that only dep ends on the data of the system and the L ∞ ( Q ) × L 2 ( Q )–norm of u . In the remainder of this proof, w e denote constan ts having this prop ert y b y C . Let t ∈ (0 , T ] b e ﬁxed. W e ﬁrst add R Ω η v to b oth sides of (3.22) and then c ho ose ∂ t η as test function, a.e. in (0 , T ). Integrating with resp ect to time o ver [0 , t ], in voking (A3) and (A4) , and employing Y oung’s inequality appropriately , we then obtain that α 2 ∥ ∂ t η ( t ) ∥ 2 + 1 2 ∥ η ( t ) ∥ 2 V ≤ C Z Q t  | η | 2 + (1 + ∥ u 1 ∥ 2 L ∞ ( Q ) ) | ψ | 2 + | ξ | 2 + | h 1 | 2 + | ∂ t η | 2  + τ 4 Z Q t | ∂ t ψ | 2 + C Z Q t | ψ | | ∂ t η | + C Z t 0  ∥ µ ( s ) ∥ 4 + ∥ ϕ ( s ) ∥ 4 + ∥ σ ( s ) ∥ 4  ∥ ψ ( s ) ∥ 4 ∥ ∂ t η ( s ) ∥ ds ≤ C Z Q t  | η | 2 + | ψ | 2 + | ξ | 2 + | h 1 | 2 + | ∂ t η | 2  + τ 4 Z Q t | ∂ t ψ | 2 + C Z t 0 ∥ ψ ( s ) ∥ 2 V ds , (3.28) where in the last estimate we hav e also used the con tinuit y of the em b edding V ⊂ L 4 (Ω) and the fact that µ, ϕ, σ are b ounded in L ∞ (0 , T ; L 4 (Ω)). Next, w e add ψ to both sides of (3.23), m ultiply b y ∂ t ψ , and integrate ov er Q t . Using Y oung’s inequalit y , we see immediately that τ Z Q t | ∂ t ψ | 2 + 1 2 ∥ ψ ( t ) ∥ 2 V ≤ C Z Q t  | η | 2 + | ψ | 2 + | ξ | 2  + τ 4 Z Q t | ∂ t ψ | 2 . (3.29) Finally , we m ultiply (3.24) b y ξ and in tegrate o ver Q t . Inv oking (A3) , emplo ying Y oung’s inequalit y , and arguing as ab ov e in the deriv ation of (3.28), w e arriv e at the estimate 1 2 ∥ ξ ( t ) ∥ 2 + Z Q t |∇ ξ | 2 ≤ χ Z Q t ∇ ψ · ∇ ξ + C Z Q t  | η | 2 + | ψ | 2 + | h 2 | 2 + | ξ | 2  + C Z t 0 ∥ ψ ( s ) ∥ 2 V ds ≤ 1 2 Z Q t |∇ ξ | 2 + C Z t 0 ∥ ψ ( s ) ∥ 2 V ds + C Z Q t  | η | 2 + | ψ | 2 + | h 2 | 2 + | ξ | 2  . (3.30) A t this p oint, we add the term R Q t | ξ | 2 to b oth sides of the inequality (3.30). Next, w e sum estimates (3.28)–(3.30) and rearrange the resulting terms. W e can then apply Gron wall’s lemma, whic h yields the estimate ∥ η ∥ W 1 , ∞ (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ ψ ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ ξ ∥ L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V ) ≤ C  ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H )  . (3.31) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 13 Ha ving established the estimate (3.31), w e can easily conclude. Indeed, w e ﬁrst observ e that the L 2 (0 , T ; H ) − norm of the expression χ ξ + η − F ′′ ( ϕ ) ψ − τ ∂ t ψ is b ounded b y an expression of the form C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ), whence, inv oking standard elliptic estimates, w e immediately obtain that ∥ ψ ∥ L 2 (0 ,T ; W ) ≤ C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ) . (3.32) Next, w e observe that, thanks to the fact that µ, ϕ, σ are b ounded in L ∞ (0 , T ; L 4 (Ω)), and o wing to the estimate (3.31), it is easily v eriﬁed that also the L 2 ( Q ) − norm of the righ t-hand side of (3.24) is b ounded by C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ). Therefore, we can conclude from standard linear parab olic theory that ∥ ξ ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ≤ C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ) . (3.33) Moreo ver, a comparison in (3.22) rev eals that ∥ η ∥ H 2 (0 ,T ; V ∗ ) ≤ C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ) . (3.34) Finally , w e diﬀerentiate (3.23) with resp ect to t , obtaining the iden tity τ ∂ tt ψ − ∆ ∂ t ψ = χ ∂ t ξ + ∂ t η − F ′′ ( ϕ ) ∂ t ψ − F ′′′ ( ϕ ) ∂ t ϕ ψ . (3.35) The L 2 ( Q ) − norm of the right-hand side is b ounded by C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ): this is obviuosly true for the ﬁrst three summands, while, o wing to (3.1), the last term can b e estimated as follows: Z Q | F ′′′ ( ϕ ) | 2 | ∂ t ϕ | 2 | ψ | 2 ≤ C Z T 0 ∥ ∂ t ϕ ( s ) ∥ 2 4 ∥ ψ ( s ) ∥ 2 4 ds ≤ C ∥ ∂ t ϕ ∥ 2 L 2 (0 ,T ; V ) ∥ ψ ∥ 2 L ∞ (0 ,T ; V ) ≤ C ( ∥ h 1 ∥ 2 L 2 (0 ,T ; H ) + ∥ h 2 ∥ 2 L 2 (0 ,T ; H ) ) . Therefore, testing (3.35) ﬁrst by ∂ t ψ and then b y − ∆ ψ , we easily obtain from Y oung’s inequalit y and standard elliptic estimates that ∥ ψ ∥ W 1 , ∞ (0 ,T ; H ) ∩ H 1 (0 ,T ; V ) ∩ L ∞ (0 ,T ; W ) ≤ C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ) . (3.36) Finally , since the em b edding W ⊂ C 0 (Ω) is compact, we can infer from [31, Sect. 8, Cor. 4] that also ∥ ψ ∥ C 0 ( Q ) ≤ C ( ∥ h 1 ∥ L 2 (0 ,T ; H ) + ∥ h 2 ∥ L 2 (0 ,T ; H ) ) . (3.37) Com bining the ab ov e estimates, we ha v e th us shown that the system (3.22)–(3.26) has a solution ( η , ψ , ξ ) ∈ X for whic h the inequalit y (3.27) is v alid. It is readily seen that the solution is unique. Indeed, if ( η i , ψ i , ξ i ) ∈ X , i = 1 , 2, are t wo solutions, then their diﬀerence ( η , ψ , ξ ) = ( η 1 − η 2 , ψ 1 − ψ 2 , ξ 1 − ξ 2 ) satisﬁes the system (3.22)–(3.26) with h 1 = h 2 = 0. It then follo ws from the estimates sho wn ab ov e that η = ψ = ξ = 0, whence the uniqueness follows. With this, the assertion is completely pro ved. 14 Colli – Rocca – Sprekels 3.2 Existence of optimal controls In the remainder of Section 3, w e study the optimal control problem ( CP ) for the case of regular potentials. In addition to the assumptions (A1) – (A4) and (3.1), w e generally assume for the quantities occurring in the cost functional (1.1): (A5) b 1 ≥ 0, b 2 ≥ 0, b 3 > 0 and κ ≥ 0 are giv en constan ts. (A6) b ϕ Q ∈ L 2 ( Q ) and b ϕ Ω ∈ V are prescrib ed target functions. (A7) G : L 2 ( Q ) × L 2 ( Q ) → R is nonnegativ e, con tin uous and con vex. W e no w sp ecify the set U ad of admissible con trols: we in tro duce the control space U := L ∞ ( Q ) × L 2 ( Q ) (3.38) and set U ad := { u = ( u 1 , u 2 ) ∈ U : u i ≤ u i ≤ b u i a.e. in Q , for i = 1 , 2 } , (3.39) where the threshold functions satisfy u i , b u i ∈ L ∞ ( Q ) , i = 1 , 2 , and u i ≤ b u i , a.e. in Q , for i = 1 , 2 . (3.40) A t this p oin t, w e ﬁx once and for all a constan t R > 0 such that U ad ⊂ U R := { u = ( u 1 , u 2 ) ∈ U : ∥ u ∥ U < R } . (3.41) W e then observe the follo wing fact: the constan ts K 1 , ..., K 5 constructed in the pro ofs of the Theorems 2.2, 2.3, 3.1 and 3.2 can b e chosen indep enden tly of the con trols, as long as the latter b elong to U R ; the constan ts then dep end only on the data of the state system and R , and no longer on the sp ecial control in U R . In particular, it follo ws from (2.14) that the second solution comp onent ϕ is uniformly b ounded in Q by the same constan t K 1 pro vided the associated con trol u b elongs to U R . Consequen tly , there exists a constan t K 6 > 0, whic h dep ends only on R and the data of the state system, such that max i =0 , 1 , 2  ∥ P ( i ) ( ϕ ) ∥ C 0 ( Q ) + ∥ h ( i ) ( ϕ ) ∥ C 0 ( Q )  + max j =1 , 2 max i =0 , 1 , 2 , 3 ∥ F ( i ) j ( ϕ ) ∥ C 0 ( Q ) ≤ K 6 , (3.42) whenev er ϕ is the second solution comp onent corresponding to some u ∈ U R . W e ha ve the follo wing existence result. Theorem 3.3. Supp ose that the c onditions of The or em 2.2, along with (A5) – (A7) , (3.1) and (3.39) – (3.41) , ar e fulﬁl le d. Then the optimal c ontr ol pr oblem ( CP ) has at le ast one solution. Pr o of. A t ﬁrst, we observe that the cost functional is nonnegativ e and therefore bounded from b elow. Since U ad  = ∅ , we can pick a minimizing sequence { u n } n ∈ N ⊂ U ad , that is, w e hav e, with ( µ n , ϕ n , σ n ) = S ( u n ), n ∈ N , lim n →∞ J ( S ( u n ) , u n ) = inf v ∈ U ad J ( S ( v ) , v ) ≥ 0 . Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 15 Since U ad is a closed, bounded, and con v ex subset of U , we ma y without loss of generality assume that u n → u weakly star in U for some u ∈ U ad . Moreo ver, since the b ound (2.14) derived in the pro of of Theorem 2.2 is v alid with a constan t K 1 that do es not dep end on the sp ecial con trol in U R , w e can claim that { ( µ n , ϕ n , σ n ) } n ∈ N is b ounded in X . W e may therefore assume that there is a triple ( µ, ϕ, σ ) ∈ X such that µ n → µ w eakly star in H 2 (0 , T ; V ∗ ) ∩ W 1 , ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; V ) , ϕ n → ϕ w eakly star in W 1 , ∞ (0 , T ; H ) ∩ H 1 (0 , T ; V ) ∩ L ∞ (0 , T ; W ) and strongly in C 0 ( Q ) , σ n → σ weakly star in H 1 (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L 2 (0 , T ; W ) . But then also P ( ϕ n ) → P ( ϕ ) , h ( ϕ n ) → h ( ϕ ) , F ′ ( ϕ n ) → F ′ ( ϕ ) , all strongly in C 0 ( Q ) . Therefore, taking the limit as n → ∞ in the state system (2.9)–(2.12), written for the pairs (( µ n , ϕ n , σ n ) , u n ), n ∈ N , we easily v erify that the triple ( µ, ϕ, σ ) satisﬁes the state system for the control u . By uniqueness, w e m ust hav e ( µ, ϕ, σ ) = S ( u ), whic h means that (( µ, ϕ, σ ) , u ) is an admissible pair for ( CP ). A t this p oint, w e notice that the conv ex and con tinuous functional G is weakly se- quen tially low er semicontin uous in L 2 ( Q ) × L 2 ( Q ), which then also applies to the func- tional v 7→ J ( S ( v ) , v ). Hence, J (( S ( u ) , u ) ≤ lim inf n →∞ J ( S ( u n ) , u n ) , whic h means that u ∈ U ad is an optimal control. 3.3 Diﬀeren tiabilit y of the con trol-to-state op erator In this subsection, w e are going to at pro ve a diﬀeren tiabilit y result for the control-to-state op erator S in suitable Banach spaces. T o this end, we in tro duce the space Y :=  W 1 , ∞ (0 , T ; H ) ∩ L ∞ (0 , T ; V )  ×  H 1 (0 , T ; H ) ∩ C 0 ([0 , T ]; V ) ∩ L 2 (0 , T ; W )  ×  H 1 (0 , T ; H ) ∩ C 0 ([0 , T ]; V ) ∩ L 2 (0 , T ; W )  , (3.43) whic h is a Banac h space when equipp ed with its natural norm ∥ · ∥ Y . W e ha ve the follo wing result. Theorem 3.4. Supp ose that the assumptions of The or em 2.2, (3.1) , and (3.41) ar e ful- ﬁl le d. Then the c ontr ol-to-state op er ator S is F r´ echet diﬀer entiable in U R as a map- ping fr om U into Y . Mor e over, for every u = ( u 1 , u 2 ) ∈ U R the F r ´ echet derivative D S ( u ) ∈ L ( U , Y ) is given as fol lows: for every dir e ction h = ( h 1 , h 2 ) ∈ U , it holds that D S ( u )[ h ] = ( η h , ψ h , ξ h ) is the unique solution to the line arize d system (3.22) – (3.26) . Pr o of. Let u = ( u 1 , u 2 ) ∈ U R b e ﬁxed. By Theorem 3.2, the linear mapping h 7→ ( η h , ψ h , ξ h ) is b ounded and therefore con tinuous from L 2 ( Q ) × L 2 ( Q ) in to X , hence, a fortiori, also as a mapping from U in to the space Y . W e hav e to show that lim ∥ h ∥ U → 0 ∥ S ( u + h ) − S ( u ) − ( η h , ψ h , ξ h ) ∥ Y ∥ h ∥ U = 0 . (3.44) 16 Colli – Rocca – Sprekels Let ( µ, ϕ, σ ) = S ( u ). Then there is some ρ > 0 suc h that u + h ∈ U R for all h ∈ U with ∥ h ∥ U < ρ . In the follo wing, w e just consider suc h increments h = ( h 1 , h 2 ) ∈ U with ∥ h ∥ U < ρ , and we denote b y C > 0 constants that dep end only on R , ρ and the data of the state system, but not on the sp ecial c hoice of suc h incremen ts. F or an y suc h h , we deﬁne the quan tities ( µ h , ϕ h , σ h ) := S ( u + h ) , y h := µ h − µ − η h , z h := ϕ h − ϕ − ψ h , w h := σ h − σ − ξ h , where ( η h , ψ h , ξ h ) ∈ X is the unique solution to the linearized system (3.22)–(3.26) as- so ciated with h = ( h 1 , h 2 ). Clearly , we hav e ( y h , z h , w h ) ∈ X , and we see that (3.44) certainly holds true if there is some C > 0 suc h that ∥ ( y h , z h , w h ) ∥ Y ≤ C ∥ h ∥ 2 U , (3.45) whic h we are going to prov e in the follo wing. T o this end, w e ﬁrst observ e that all of the constan ts C > 0 app earing in the statemen ts of Theorem 2.2 and Theorem 3.1 dep end only on the data of the system (1.2)–(1.6) and the U − norm of the controls. Therefore, there is some constant C > 0 such that ∥ ( µ h , ϕ h , ξ h ) ∥ X ≤ C and ∥ ( µ h − µ, ϕ h − ϕ, σ h − σ ) ∥ X ≤ C ∥ h ∥ U , for all h ∈ U with ∥ h ∥ U ≤ ρ. (3.46) In particular, the estimate (3.42) is v alid for ϕ and ϕ h , for all h ∈ U with ∥ h ∥ U ≤ ρ . A t this p oint, w e subtract from the equations (1.2)–(1.6) satisﬁed by ( µ h , ϕ h , σ h ) the sum of the corresp onding equations for ( µ, ϕ, σ ) and the equations (3.22)–(3.26) satisﬁed b y ( η h , ψ h , ξ h ). A little algebra then sho ws that the quan tit y ( y h , z h , w h ) is a solution to the system ⟨ α∂ tt y h , v ⟩ + Z Ω ∇ y h · ∇ v = − Z Ω ∂ t z h v + Z Ω Λ h 1 v + Z Ω Λ h 2 v for all v ∈ V and a.e. t ∈ (0 , T ) , (3.47) τ ∂ t z h − ∆ z h = χ w h + y h + Λ h 3 a.e. in Q, (3.48) ∂ t w h − ∆ w h = − χ ∆ z h − Λ h 1 a.e. in Q, (3.49) ∂ n y h = ∂ n w h = 0 a.e. on Σ , (3.50) y h (0) = ∂ t y h (0) = z h (0) = w h (0) = 0 a.e. in Ω , (3.51) with the quan tities Λ h 1 = P ( ϕ )  w h − χ z h − y h  +  P ( ϕ h ) − P ( ϕ ) − P ′ ( ϕ ) ψ h  σ + χ (1 − ϕ ) − µ  +  ( P ( ϕ h ) − P ( ϕ )  ( σ h − σ ) − χ ( ϕ h − ϕ ) − ( µ h − µ )  , (3.52) Λ h 2 = −  h ( ϕ h ) − h ( ϕ ) − h ′ ( ϕ ) ψ h  u 1 −  h ( ϕ h ) − h ( ϕ )  h 1 , (3.53) Λ h 3 = −  F ′ ( ϕ h ) − F ′ ( ϕ ) − F ′′ ( ϕ ) ψ h  . (3.54) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 17 By virtue of T aylor’s theorem with in tegral remainder, it holds that P ( ϕ h ) − P ( ϕ ) − P ′ ( ϕ ) ψ h = P ′ ( ϕ ) z h + A h ( ϕ h − ϕ ) 2 , h ( ϕ h ) − h ( ϕ ) − h ′ ( ϕ ) ψ h = h ′ ( ϕ ) z h + B h ( ϕ h − ϕ ) 2 , F ′ ( ϕ h ) − F ′ ( ϕ ) − F ′′ ( ϕ ) ψ h = F ′′ ( ϕ ) z h + C h ( ϕ h − ϕ ) 2 , (3.55) with the remainders A h := Z 1 0 (1 − s ) P ′′ ( sϕ h + (1 − s ) ϕ )) ds, B h := Z 1 0 (1 − s ) h ′′ ( sϕ h + (1 − s ) ϕ )) ds, C h := Z 1 0 (1 − s ) F ′′′ ( sϕ h + (1 − s ) ϕ )) ds . Noting that all conv ex combinations of ϕ h and ϕ satisfy the condition (3.42), w e see that ∥ A h ∥ L ∞ ( Q ) + ∥ B h ∥ L ∞ ( Q ) + ∥ C h ∥ L ∞ ( Q ) ≤ C . (3.56) No w let t ∈ (0 , T ] b e ﬁxed. W e are going to estimate the L 2 ( Q t ) − norms of Λ h i , for i = 1 , 2 , 3. As far as Λ h 1 is concerned, we estimate the three summands in the three lines in (3.52) separately . At ﬁrst, w e obviously ha ve Z Q t | P ( ϕ ) | 2 | w h − χ z h − y h | 2 ≤ C Z Q t  | w h | 2 + | y h | 2 + | z h | 2  . (3.57) Next, in voking (3.42), (3.46), (3.55), (3.56), the con tin uity of the em b eddings V ⊂ L 4 (Ω) and V ⊂ L 6 (Ω), as w ell as the fact that µ, ϕ, σ are bounded in L ∞ (0 , T ; V ), w e can argue as follo ws: Z Q t | P ( ϕ h ) − P ( ϕ ) − P ′ ( ϕ ) ψ h | 2 | σ + χ (1 − ϕ ) − µ | 2 ≤ C Z Q t  | z h | 2 + | ϕ h − ϕ | 4  1 + | µ | 2 + | ϕ | 2 + | σ | 2  ≤ C Z t 0 ∥ z h ( s ) ∥ 2 ds + C Z t 0 ∥ ϕ h ( s ) − ϕ ( s ) ∥ 4 4 ds + C Z t 0 ∥ z h ( s ) ∥ 2 4  ∥ µ ( s ) ∥ 2 4 + ∥ ϕ ( s ) ∥ 2 4 + ∥ σ ( s ) ∥ 2 4  ds + C Z t 0 ∥ ϕ h ( s ) − ϕ ( s ) ∥ 4 6  ∥ µ ( s ) ∥ 2 6 + ∥ ϕ ( s ) ∥ 2 6 + ∥ σ ( s ) ∥ 2 6  ds ≤ C Z t 0 ∥ z h ( s ) ∥ 2 V ds + C ∥ h ∥ 4 U . (3.58) In addition, using the contin uous em b edding V ⊂ L 4 (Ω) and (3.46) once more, we ﬁnd Z Q t | P ( ϕ h ) − P ( ϕ ) | 2 | ( σ h − σ ) − χ ( ϕ h − ϕ ) − ( µ h − µ ) | 2 ≤ C Z t 0  ∥ ϕ h − ϕ ∥ 2 4  ∥ σ h − σ ∥ 2 4 + ∥ ϕ h − ϕ ∥ 2 4 + ∥ µ h − µ ∥ 2 4   ( s ) ds ≤ C ∥ h ∥ 4 U . (3.59) 18 Colli – Rocca – Sprekels Hence, com bining the estimates (3.57)–(3.59), we hav e shown that Z Q t | Λ h 1 | 2 ≤ C Z Q t  | w h | 2 + | y h | 2  + C Z t 0 ∥ z h ( s ) ∥ 2 V ds + C ∥ h ∥ 4 U . (3.60) In addition, we ha v e, using (3.46), (3.53), (3.55) and (3.56), and arguing similarly as ab o v e, Z Q t | Λ h 2 | 2 ≤ C Z Q t  | h ( ϕ h ) − h ( ϕ ) − h ′ ( ϕ ) ψ h ) | 2 | u 1 | 2 + | h ( ϕ h ) − h ( ϕ ) | 2 | h 1 | 2  ≤ C Z Q t  | z h | 2 + | ϕ h − ϕ | 4  + Z t 0 ∥ ( ϕ h − ϕ )( s ) ∥ 2 ∞ ∥ h 1 ( s ) ∥ 2 ds ≤ C Z Q t | z h | 2 + C ∥ h ∥ 4 U . (3.61) Similar reasoning also yields that Z Q t | Λ h 3 | 2 ≤ C Z Q t  | z h | 2 + | ϕ h − ϕ | 4  ≤ C Z Q t | z h | 2 + C ∥ h ∥ 4 U . (3.62) Ab out the h yp erb olic equation (3.47), w e recall that the solution y h is in C 1 (0 , T ; H ) ∩ C 0 (0 , T ; V ) and satisﬁes the energy identit y (cf. [25, Chapter 3, Sections 8–9]) α 2 ∥ ∂ t y h ( t ) ∥ 2 + 1 2 Z Ω |∇ y h ( t ) | 2 = Z Q t  − ∂ t z h + Λ h 1 + Λ h 2  ∂ t y h for all t ∈ [0 , T ] . This identit y is formally obtained by taking v = ∂ t y h in (3.47) and in tegrating ov er (0 , t ). Then, w e can add the term 1 2 ∥ y h ( t ) ∥ 2 = R Q t y h ∂ t y h to b oth sides and, b y virtue of Y oung’s inequalit y , (3.60), and (3.61), deduce that α 2 ∥ ∂ t y h ( t ) ∥ 2 + 1 2 ∥ y h ( t ) ∥ 2 V ≤ τ 4 Z Q t | ∂ t z h | 2 + C Z Q t  | w h | 2 + | y h | 2 + | ∂ t y h | 2  + C Z t 0 ∥ z h ( s ) ∥ 2 V ds + C ∥ h ∥ 4 U . (3.63) Next, w e add z h to b oth sides of (3.48), multiply b y ∂ t z h , and integrate o v er Q t . It then follo ws from Y oung’s inequalit y , thanks to (3.62), that 1 2 ∥ z h ( t ) ∥ 2 V + τ Z Q t | ∂ t z h | 2 ≤ τ 4 Z Q t | ∂ t z h | 2 + C Z Q t  | w h | 2 + | y h | 2 + | z h | 2  + C ∥ h ∥ 4 U . (3.64) Finally , we add w h to b oth sides of (3.49), multiply b y w h , and in tegrate ov er Q t . It then follo ws, in voking Y oung’s inequality and (3.60) again, that 1 2 ∥ w h ( t ) ∥ 2 + Z Q t |∇ w h | 2 ≤ 1 2 Z Q t |∇ w h | 2 + χ 2 2 Z Q t |∇ z h | 2 + C Z Q t  | w h | 2 + | y h | 2  + C Z t 0 ∥ z h ( s ) ∥ 2 V ds + C ∥ h ∥ 4 U . (3.65) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 19 A t this p oint, we add the inequalities (3.63), (3.64) and (3.65). Rearranging terms, w e ﬁnd that Gron wall’s lemma can b e applied, which then yields the estimate ∥ y h ∥ W 1 , ∞ (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ z h ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ w h ∥ L ∞ (0 ,T ; H ) ∩ L 2 (0 ,T ; V ) ≤ C ∥ h ∥ 2 U . (3.66) But then, in view of (3.62) and (3.66), also ∥ ∆ z h ∥ L 2 (0 ,T ; H ) = ∥ τ ∂ t z h − χ w h − y h − Λ h 3 ∥ L 2 (0 ,T ; H ) ≤ C ∥ h ∥ 2 U , and standard elliptic estimates yield that ∥ z h ∥ L 2 (0 ,T ; W ) ≤ C ∥ h ∥ 2 U . (3.67) Next, we m ultiply (3.49) b y − ∆ w h and integrate ov er Q t . Then, by Y oung’s inequal- it y , we deduce that 1 2 ∥∇ w h ( t ) ∥ 2 + Z Q t | ∆ w h | 2 ≤ 1 2 Z Q t | ∆ w h | 2 + C Z Q t  | ∆ z h | 2 + | Λ h 1 | 2  , (3.68) and w e can infer from the previous estimates and standard elliptic estimates that ∥ w h ∥ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ≤ C ∥ h ∥ 2 U . (3.69) Comparison in (3.49) then yields that also ∥ ∂ t w h ∥ L 2 (0 ,T ; H ) ≤ C ∥ h ∥ 2 U . (3.70) Finally , w e recall the con tinuit y of the embedding H 1 (0 , T ; H ) ∩ L 2 (0 , T ; W ) ⊂ C 0 ([0 , T ]; V ) , that, along with the estimates (3.66)–(3.70), completes the pro of of (3.45). Hence, The- orem 3.4 is fully prov ed. Remark 3.5. It is worth men tioning that in the ab o v e pro of the actual v alue of the constan t R did not matter (as long as it is large enough to guarantee that U ad ⊂ U R ). W e can therefore claim that the con trol-to-solution operator S is F r´ ec het diﬀeren tiable in the sense of Theorem 3.4 on the en tire control space U . 3.4 First-order necessary optimalit y conditions In this subsection, w e derive ﬁrst-order necessary optimality conditions for ( CP ) in the case of regular p otentials. Note that the same optimal con trol problem has b een studied rep eatedly for the case when the hyperb olic relaxation term α∂ tt µ is replaced by a parab olic relaxation term of the form α∂ t µ . In particular, references [8, 9] address the case without sparsity , i.e., for κ = 0, while sparsity terms are included in [10, 32, 33], where the latter pap er is concerned with a sligh tly simpliﬁed state system. 20 Colli – Rocca – Sprekels In the follo wing, w e will see that at least the ﬁrst-order necessary optimalit y conditions with sparsity term established in [8, 10, 32] for the parab olic relaxation remain v alid, with the appropriate mo diﬁcations, in the case of h yp erb olic relaxation. In order to keep the pap er at a reasonable length, how ev er, w e do not address the deriv ation of second- order suﬃcient optimalit y conditions as carried out in [9, 33], since this would require considerably more in v olved analytical argumen ts. A t this p oin t, w e in tro duce the r e duc e d c ost functionals e J and e J b y putting e J ( u ) := J ( S ( u ) , u ) , e J ( u ) := J ( S ( u ) , u ) , for u ∈ U . (3.71) Since, thanks to Theorem 3.4, the con trol-to-state mapping is F r´ ec het diﬀeren tiable from U into Y , the functional e J is a F r ´ echet diﬀeren tiable mapping from U into R . Therefore, the chain rule sho ws that, for every u = ( u 1 , u 2 ) ∈ U and h = ( h 1 , h 2 ) ∈ U , the F r´ ec het deriv ativ e D e J ( u ) satisﬁes the iden tity D e J ( u )[ h ] = b 1 Z Q ( ϕ − b ϕ Q ) ψ h + b 2 Z Ω ( ϕ ( T ) − b ϕ Ω ) ψ h ( T ) + b 3 Z Q u · h , (3.72) where ( µ, ϕ, σ ) = S ( u ) and ( η h , ψ h , ξ h ) is the solution to the linearized system (3.22)– (3.26). Notice that Theorem 3.2 implies that the expression on the right-hand side of (3.72) is meaningful also for every h ∈ L 2 ( Q ) × L 2 ( Q ), and we therefore ma y extend the F r´ ec het deriv ativ e D e J ( u ) ∈ U ∗ to an elemen t of ( L 2 ( Q ) × L 2 ( Q )) ∗ , whic h is still denoted b y D e J ( u ), by p ostulating the iden tit y (3.72) also for general h ∈ L 2 ( Q ) × L 2 ( Q ). In this w ay , expressions of the form D e J ( u )[ h ] hav e a well-deﬁned meaning also for suc h argumen ts h . W e now deriv e the announced necessary optimality conditions. W e recall at this p oin t that a con trol u ∈ U ad is called lo c al ly optimal for ( CP ) in the sense of L p ( Q ) × L p ( Q ) for some p ∈ [1 , ∞ ] if and only if there is some ε > 0 suc h that e J ( u ) ≤ e J ( u ) for all u ∈ U ad suc h that ∥ u − u ∥ L p ( Q ) × L p ( Q ) ≤ ε . It is easily seen that every lo cally optimal control in the sense of L p ( Q ) × L p ( Q ) for some p ∈ [1 , ∞ ) is also lo cally optimal in the sense of L ∞ ( Q ) × L ∞ ( Q ). No w assume that u = ( u 1 , u 2 ) ∈ U ad is a lo cally optimal control for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ). Then it is easily seen that the v ariational inequalit y D e J ( u )[ u − u ] + κ ( G ( u ) − G ( u )) ≥ 0 ∀ u ∈ U ad (3.73) m ust b e satisﬁed. Indeed, if u ∈ U ad is given and t ∈ (0 , 1) is suﬃciently small, then we ha ve that u + t ( u − u ) ∈ U ad and ∥ u + t ( u − u ) − u ∥ L ∞ ( Q ) × L ∞ ( Q ) ≤ ε . Hence, we can infer from the con v exity of G that 0 ≤ e J ( u + t ( u − u )) + κ G ( u + t ( u − u )) − e J ( u ) − κ G ( u ) ≤ e J ( u + t ( u − u )) − e J ( u ) + κ t ( G ( u ) − G ( u )) . A t this p oint w e note that S , b eing F r´ ec het diﬀeren tiable from U into Y , is also F r´ ec het diﬀeren tiable from the small er space L ∞ ( Q ) × L ∞ ( Q ) in to Y . Therefore, dividing b y t > 0, Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 21 and then taking the limit as t ↘ 0, (3.73) follows. But (3.73) implies that u solves the unconstrained con vex minimization problem min u ∈ ( L 2 ( Q ) × L 2 ( Q ))  Φ( u ) + κ G ( u ) + I U ad ( u )  , where Φ( u ) = D e J ( u )[ u ] with the extension of the mapping D e J ( u ) to L 2 ( Q ) × L 2 ( Q ), and where I U ad denotes the indicator function of U ad . Hence, denoting by the symbol ∂ the sub diﬀeren tial mapping in L 2 ( Q ) × L 2 ( Q ), w e ha ve that 0 ∈ ∂  Φ + κ G + I U ad  ( u ) . Then we may infer from the w ell-known rules for sub diﬀeren tials of con vex functionals that 0 ∈ { D e J ( u ) } + κ∂ G ( u ) + ∂ I U ad ( u ) . In other w ords, there are λ ∈ ∂ G ( u ) and b λ ∈ ∂ I U ad ( u ) suc h that 0 = D e J ( u ) + κ λ + b λ . (3.74) But, by the deﬁnition of ∂ I U ad ( u ), we ha v e b λ [ u − u ] ≤ 0 for ev ery u ∈ U ad . Hence, thanks to (3.74), 0 ≤ D e J ( u )[ u − u ] + κ λ [ u − u ] ∀ u ∈ U ad . Th us, inv oking (3.72), and iden tifying λ with the corresponding element of L 2 ( Q ) × L 2 ( Q ) according to the Riesz isomorphism, we ha ve sho wn the following result. Lemma 3.6. Assume that the hyp otheses of The or em 2.2, (A5) – (A7) , (3.1) , (3.39) and (3.40) ar e fulﬁl le d. If u ∈ U ad is a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ) , then ther e is some λ ∈ ∂ G ( u ) ⊂ ( L 2 ( Q ) × L 2 ( Q )) such that D e J ( u )[ u − u ] + κ Z Q λ · ( u − u ) = b 1 Z Q ( ϕ − b ϕ Q ) ψ h + b 2 Z Ω ( ϕ ( T ) − b ϕ Ω ) ψ h ( T ) + Z Q ( b 3 u + κ λ ) · ( u − u ) ≥ 0 for al l u ∈ U ad , (3.75) wher e ( η h , ψ h , ξ h ) is the solution to the line arize d system (3.22) – (3.26) for h = u − u . Next, we aim at simplifying the expression D e J ( u )[ u − u ] in (3.75) by in tro ducing an adjoin t state. T o this end, we consider the follo wing adjoint system: ⟨ α∂ tt p, v ⟩ + Z Ω ∇ p · ∇ v = Z Ω q v − Z Ω P ( ϕ )( p − r ) v for all v ∈ V and a.e. t ∈ (0 , T ) , (3.76) − τ ∂ t q − ∂ t p − ∆ q = − χ ∆ r + P ′ ( ϕ )  σ + χ (1 − ϕ ) − µ  ( p − r ) − F ′′ ( ϕ ) q − χ P ( ϕ )( p − r ) − h ′ ( ϕ ) u 1 p + b 1 ( ϕ − b ϕ Q ) a.e. in Q , (3.77) − ∂ t r − ∆ r = χ q + P ( ϕ )( p − r ) a.e. in Q , (3.78) ∂ n q = ∂ n r = 0 a.e. on Σ , (3.79) p ( T ) = 0 , ∂ t p ( T ) = 0 , q ( T ) = b 2 τ ( ϕ ( T ) − b ϕ Ω ) , r ( T ) = 0 , a.e. in Ω . (3.80) 22 Colli – Rocca – Sprekels W e ha v e the following w ell-p osedness result. W e emphasize the regularit y of the solution comp onent p stated in (3.82), which ensures that p is a strong solution of (3.76). In particular, (3.76) can b e equiv alen tly rewritten as α∂ tt p − ∆ p = q − P ( ϕ )( p − r ) a.e. in Q, (3.81) supplemen ted with the b oundary condition ∂ n p = 0 a.e. on Σ. Theorem 3.7. Assume that the c onditions of The or em 2.2, along with (A5) – (A7) , (3.1) , (3.39) – (3.41) , ar e fulﬁl le d, and let u ∈ U R b e given with asso ciate d state ( µ, ϕ, σ ) = S ( u ) . Then the adjoint system (3.76) – (3.80) has a unique solution ( p, q , r ) with the r e gularity p ∈ W 2 , ∞ (0 , T ; H ) ∩ W 1 , ∞ (0 , T ; V ) ∩ L ∞ (0 , T ; W ) , (3.82) q ∈ H 1 (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L 2 (0 , T ; W ) , (3.83) r ∈ H 1 (0 , T ; H ) ∩ L ∞ (0 , T ; V ) ∩ L 2 (0 , T ; W ) ∩ L ∞ ( Q ) . (3.84) Pr o of. The existence of a solution with the requested regularit y prop erties can again b e sho wn by means of a F aedo–Galerkin approximation using the eigenfunctions { e j } j ∈ N of the elliptic eigenv alue problem − ∆ e j = λ j e j in Ω, ∂ n e j = 0 on Γ, as basis functions. Once more, w e av oid writing the F aedo–Galerkin system explicitly and deriv e the relev an t a priori estimates only formally by arguing directly on the con tinuous system (3.76)–(3.80). The subsequen t estimates, while b eing only formal, are fully justiﬁed on the level of the F aedo–Galerkin appro ximations. In the follo wing w e denote Q t := Ω × ( t, T ) for ev ery t ∈ [0 , T ). T o b egin with, let t ∈ [0 , T ) b e ﬁxed, set g 1 := b 1 ( ϕ − b ϕ Q ) , g 2 := b 2 ( ϕ ( T ) − b ϕ Ω ) , and observ e that g 1 ∈ L 2 (0 , T ; H ), while g 2 ∈ V . W e test (3.76) b y − ∂ t p , integrate ov er ( t, T ), and add 1 2 ∥ p ( t ) ∥ 2 = − R Q t p∂ t p to b oth sides of the resulting iden tity . Then, thanks to the terminal conditions for p , and o wing to Y oung’s inequalit y , w e hav e that α 2 ∥ ∂ t p ( t ) ∥ 2 + 1 2 ∥ p ( t ) ∥ 2 V ≤ C Z Q t  | p | 2 + | q | 2 + | r | 2 + | ∂ t p | 2  . (3.85) Lik ewise, adding r to b oth sides of (3.78), multiplying the resultan t b y − ∂ t r , and in te- grating o ver Q t , w e obtain the estimate 1 2 ∥ r ( t ) ∥ 2 V + Z Q t | ∂ t r | 2 ≤ Z Q t  | p | 2 + | q | 2 + | r | 2  + 1 2 Z Q t | ∂ t r | 2 . (3.86) Finally , w e multiply (3.77) by − q and inte grate o v er Q t , obtaining τ 2 ∥ q ( t ) ∥ 2 + Z Q t |∇ q | 2 ≤ 1 2 τ ∥ g 2 ∥ 2 − χ Z Q t ∇ q · ∇ r + C Z Q t  | p | 2 + | q | 2 + | r | 2 + | ∂ t p | 2 + | g 1 | 2  + C Z Q t  | σ | + | ϕ | + | µ |  | p | + | r |  | q | . (3.87) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 23 No w, we observ e that − χ Z Q t ∇ q · ∇ r ≤ 1 2 Z Q t |∇ q | 2 + χ 2 Z Q t |∇ r | 2 . (3.88) Besides, the last term on the right-hand side of (3.87), which denote by I , can b e estimated as follo ws: | I | ≤ C Z T t  ∥ σ ( s ) ∥ 4 + ∥ ϕ ( s ) ∥ 4 + ∥ µ ( s ) ∥ 4  ∥ p ( s ) ∥ 4 + ∥ r ( s ) ∥ 4  ∥ q ( s ) ∥ ds ≤ Z Q t | q | 2 + C Z T t  ∥ p ( s ) ∥ 2 V + ∥ r ( s ) ∥ 2 V  ds . (3.89) A t this p oin t, we apply (3.88) and (3.89) to the right-hand side of (3.87), then we sum the result to the inequalities (3.85) and (3.86). Rearranging the terms, w e then ﬁnd that Gron wall’s lemma (tak en bac kward in time) can be applied, and we conclude the estimate ∥ p ∥ W 1 , ∞ (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) + ∥ q ∥ L ∞ (0 ,T : H ) ∩ L 2 (0 ,T ; V ) + ∥ r ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ≤ C  ∥ g 1 ∥ L 2 (0 ,T ; H ) + ∥ g 2 ∥  . (3.90) Next, we observe that, thanks to (3.90), the righ t-hand side of (3.78) is b ounded in L ∞ (0 , T ; H ). Since also r ( T ) = 0 ∈ L ∞ (Ω), it follows from classical parab olic regularit y theory (see, e.g., [24, Thm. 7.1, p. 181]) that ∥ r ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ∩ L ∞ ( Q ) ≤ C . (3.91) Similarly , it is now easy to verify that q solv es a parab olic problem in whic h the right- hand side is already kno wn to b e b ounded in L 2 (0 , T ; H ). Therefore, and since g 2 ∈ V , it follo ws that ∥ q ∥ H 1 (0 ,T ; H ) ∩ L ∞ (0 ,T ; V ) ∩ L 2 (0 ,T ; W ) ≤ C . (3.92) Finally , we test (3.76) by ∆ ∂ t p and integrate with resp ect to time o v er ( t, T ) where t ∈ [0 , T ). Integrating b y parts, and using the zero terminal conditions for p and ∂ t p , we obtain from Y oung’s inequality and the abov e estimates that α 2 ∥∇ ∂ t p ( t ) ∥ 2 + 1 2 ∥ ∆ p ( t ) ∥ 2 = Z Q t  q − P ( ϕ )( p − r )  ∆ ∂ t p = − Z Ω  q − P ( ϕ )( p − r )  ( t )∆ p ( t ) − Z Q t  ∂ t q − P ( ϕ )( ∂ t p − ∂ t r ) − P ′ ( ϕ ) ∂ t ϕ ( p − r )  ∆ p ≤ C + 1 4 ∥ ∆ p ( t ) ∥ 2 + Z Q t | ∆ p | 2 + C Z T t ∥ ∂ t ϕ ( s ) ∥ 4  ∥ p ( s ) ∥ 4 + ∥ r ( s ) ∥ 4  ∥ ∆ p ( s ) ∥ ds ≤ C + 1 4 ∥ ∆ p ( t ) ∥ 2 + C Z Q t | ∆ p | 2 , 24 Colli – Rocca – Sprekels b y noting that ∂ t ϕ is b ounded in L 2 (0 , T ; V ) b y (2.14). Then, from Gronw all’s lemma it follo ws that ∥ p ∥ W 1 , ∞ (0 ,T ; V ) ∩ L ∞ (0 ,T ; W ) ≤ C . (3.93) Consequen tly , by in tegrating b y parts in (3.76) and comparing the terms, w e arrive at (3.81) with the regularity L ∞ (0 , T ; H ) for ∂ tt p and the prop erty ∥ p ∥ W 2 , ∞ (0 ,T ; H ) ≤ C . With the ab ov e estimates, w e ha ve shown that the adjoin t system (3.76)–(3.80) has a solution ( p, q, r ) with the required regularit y . It is easily seen that it is unique: indeed, if ( p i , q i , r i ), i = 1 , 2, are t wo such solutions, then ( p, q , r ) = ( p 1 − p 2 , q 1 − q 2 , r 1 − r 2 ) solv es the system (3.76)–(3.80) with the terms g 1 and g 2 replaced b y zero. W e then infer from (3.90) that p = q = r = 0, whence the uniqueness follo ws. This concludes the pro of of the assertion. W e are no w in a p osition to improv e the impracticable result of Lemma 3.6. Theorem 3.8. Supp ose that the c onditions of The or em 2.2, (A5) – (A7) , (3.1) , (3.39) , and (3.40) ar e fulﬁl le d, and let u ∈ U ad b e a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ) with the asso ciate d state ( µ, ϕ, σ ) = S ( u ) and the adjoint state ( p, q , r ) . Then ther e exists some λ ∈ ∂ G ( u ) such that Z Q  d 0 + κ λ + b 3 u  · ( u − u ) ≥ 0 for al l u ∈ U ad , (3.94) wher e d 0 := ( − h ( ϕ ) p, r ) . Pr o of. W e set h = ( h 1 , h 2 ) = ( u 1 − u 1 , u 2 − u 2 ) = u − u and let ( η , ψ , ξ ) denote the solution to the linearized system (3.22)–(3.26) corresp onding to this increment h . Then w e test (3.22) by p , (3.23) b y q , and (3.24) by r , and add the resulting three equations. In tegrating by parts with resp ect to time and space, and using the b oundary conditions (3.25) and (3.79), the initial conditions (3.26), as w ell as the terminal conditions (3.80), w e obtain: 0 = Z T 0 ⟨ α∂ tt p ( t ) , η ( t ) ⟩ dt − Z Q ψ ∂ t p + Z Q ∇ η · ∇ p + Z Q p  − P ( ϕ )( ξ − χ ψ − η ) − P ′ ( ϕ )( σ + χ (1 − ϕ ) − µ ) ψ + h ( ϕ ) h 1 + h ′ ( ϕ ) u 1 ψ  + τ Z Ω ψ ( T ) q ( T ) − τ Z Q ψ ∂ t q − Z Q q ∆ ψ + Z Q q  F ′′ ( ϕ ) ψ − χ ξ − η  − Z Q ξ ∂ t r + Z Q ( − ∆ ξ + χ ∆ ψ ) r + Z Q r  P ( ϕ )( ξ − χ ψ − η ) + P ′ ( ϕ )( σ + χ (1 − ϕ ) − µ ) ψ − h 2  = Z Q  h ( ϕ ) p h 1 − r h 2 ) + b 2 Z Ω ψ ( T )( ϕ ( T ) − b ϕ Ω ) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 25 + Z T 0 ⟨ α∂ tt p ( t ) , η ( t ) ⟩ + Z Q ∇ p · ∇ η + Z Q  P ( ϕ )( p − r ) − q  η + Z Q  − ∂ t r − ∆ r − P ( ϕ )( p − r ) − χ q  ξ + Z Q  − ∂ t p − τ ∂ t q − ∆ q + χ ∆ r + F ′′ ( ϕ ) q + χ P ( ϕ )( p − r ) − P ′ ( ϕ )( σ + χ (1 − ϕ ) − µ )( p − r ) + h ′ ( ϕ ) u 1 p  ψ = − Z Q d 0 · h + b 1 Z Q ψ ( ϕ − b ϕ Q ) + b 2 Z Ω ψ ( T )( ϕ ( T ) − b ϕ Ω ) , (3.95) since ( p, q , r ) solv es the adjoin t system. Hence, in view of (3.75), the assertion follows. Remark 3.9. If the sparsit y functional G has the sp ecial form G ( u ) = G 1 ( u 1 ) + G 2 ( u 2 ) = G 1 ( I 1 ( u )) + G 2 ( I 2 ( u )) for u = ( u 1 , u 2 ) ∈ L 2 ( Q ) × L 2 ( Q ) , with nonnegativ e, con v ex and contin uous functionals G i : L 2 ( Q ) → R and the linear pro jection mappings I i ( u ) = u i , for i = 1 , 2, then it follo ws from the sum and c hain rules for sub diﬀerentials (cf., e.g., [22, Sect. 4.2.2, Thm. 1 and Thm. 2]) that ∂ G ( u ) = { λ = ( λ 1 , λ 2 ) : λ i ∈ ∂ G i ( u i ) , i = 1 , 2 } . In this case, w e hav e λ = ( λ 1 , λ 2 ) with λ i ∈ ∂ G i ( u i ) for i = 1 , 2, and (3.94) is equiv alen t to t wo independent v ariational inequalities that ha ve to be satisﬁed sim ultaneously , namely Z Q  − h ( ϕ ) p + κλ 1 + b 3 u 1  u 1 − u 1  ≥ 0 if u 1 ∈ L ∞ ( Q ) and u 1 ≤ u 1 ≤ b u 1 , (3.96) Z Q  r + κλ 2 + b 3 u 2  u 2 − u 2  ≥ 0 if u 2 ∈ L ∞ ( Q ) and u 2 ≤ u 2 ≤ b u 2 . (3.97) Then a standard argument leads to the p oin twise pro jection formulas u 1 = max n u 1 , min n b u 1 , − b − 1 3 ( − h ( ϕ ) p + κλ 1 ) oo a.e. in Q , (3.98) u 2 = max n u 2 , min n b u 2 , − b − 1 3 ( r + κλ 2 ) oo a.e. in Q . (3.99) 4 The case of singular p oten tials W e no w turn our in terest to nonregular potentials. T o this end, w e assume the condition: it holds that − ∞ < r − < 0 < r + < + ∞ . (4.1) In this case, the conditions of Theorem 2.2 do not suﬃce to carry out an analysis as in the previous section. Indeed, although the b oundedness condition (2.14) is v alid also in this situation, it fails to guarantee the uniform b oundedness of ∥ F ′ 1 ( ϕ ) ∥ L ∞ ( Q ) , since it cannot b e excluded that the v alues of the phase v ariable ϕ get arbitrarily close to the critical v alues r − , r + at which | F ′ 1 | blows up. Therefore, the estimate (3.42), which was 26 Colli – Rocca – Sprekels an indispensable prerequisite for the analysis of Section 3 to work, is not at our disp osal. In this connection, notice that also the uniform Lipschitz inequality (3.9), which was fundamen tal in the pro of of Theorem 3.1, is a consequence of (3.42). T o ov ercome this inheren t diﬃcult y , w e assume for the remainder of this section that the stronger conditions of Theorem 2.3 are fulﬁlled, since then the uniform separation (2.20) is satisﬁed (which implies the v alidit y of (3.42)), pro vided that the ( L ∞ ( Q ) ∩ L 2 (0 , T ; V )) × L ∞ (0 , T ; H ))– norms of the admissible controls are uniformly b ounded. It therefore follows that under the assumptions of Theorem 2.3 the F r´ ec het diﬀeren tiability result of Theorem 3.4 is also v alid in the singular case when only (4.1) is satisﬁed. It is not adv an tageous to p ostulate that a con trol v ariable ( u 1 , in this case) b elongs to a b ounded subset of L 2 (0 , T ; V ), since it imp oses a severe restriction on the av ailable con trol spaces in practical situations. In particular, the usual technique s to deriv e p oint- wise pro jection formulas from v ariational inequalities (cf. Remark 3.9) fail. W e therefore restrict ourselv es to the follo wing sp ecial cases. Sc enario 1: W e consider u 1 ∈ L ∞ ( Q ) ∩ L 2 (0 , T ; V ) as a ﬁxed given datum and apply the control action only to the second v ariable u 2 . F or con v enience, we write u 1 in place of u 1 , in the follo wing. This b ecomes formally a special case of the one discussed in Section 3 if we make the follo wing choice (cf. (3.40)) in the deﬁnition (3.39) of U ad : u 1 = u 1 = b u 1 ∈ L ∞ ( Q ) ∩ L 2 (0 , T ; V ) . (4.2) Clearly , then ( CP ) b ecomes in realit y a con trol problem only in the v ariable u 2 , but w e can take adv antage of the machi nery dev elop ed in the previous section. Indeed, the uniform separation condition (2.20) is v alid for all second solution comp onents asso ciated with controls u ∈ U R , and the fundamental estimate (3.42) holds true. F rom this p oin t on, it is easily seen that the whole analysis carried out in Section 3 can b e rep eated accordingly . Since in this case w e actually ha ve to deal only with the con trol v ariable u 2 , the analysis ev en simpliﬁes, yielding the existence of optimal controls and a ﬁrst-order necessary optimalit y condition that resem bles (3.94) in Theorem 3.8. Notice, in particular, that the linearized system (3.22)–(3.26) changes sligh tly: in fact, in the second line of Eq. (3.22), the term − R Ω h ( ϕ ) h 1 v do es not o ccur. Since the form of the adjoin t state system (3.76)–(3.80) remains unchanged (recall that w e write u 1 in place of u 1 ), this implies that also in the calculation carried out in (3.95) the corresp onding term R Q h ( ϕ ) p h 1 do es not o ccur, which clearly corresponds to the fact that the ﬁrst con trol component is ﬁxed. Accordingly , the ﬁrst comp onent of the vector function d 0 in tro duced in Theorem 3.8 v anishes, and only the part of the v ariational inequality (3.94), whic h acts on the second con trol comp onent, yields some useful information. T o write this condition more explicitly , w e assume that the sparsity functional G is of the sp ecial form G ( u ) = G (( u 1 , u 2 )) = G 2 ( u 2 ) for every u = ( u 1 , u 2 ) ∈ L 2 ( Q ) × L 2 ( Q ) , (4.3) where G 2 : L 2 ( Q ) → R is nonnegativ e, con tin uous and con vex. W e also in tro duce the set U 1 ad := { u ∈ L ∞ ( Q ) : u 2 ≤ u ≤ b u 2 a.e. in Q } . (4.4) A lo cally optimal control for ( CP ) in the sense of L ∞ ( Q ) is then any u ∈ U 1 ad suc h that there is some ε > 0 suc h that, with the ﬁxed giv en function u 1 , J ( S (( u 1 , u ) , ( u 1 , u ))) ≤ J ( S (( u 1 , v ) , ( u 1 , v ))) for all v ∈ U 1 ad with ∥ v − u ∥ L ∞ ( Q ) ≤ ε . Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 27 W e then conclude from Theorem 3.8 the following optimalit y condition. Theorem 4.1. Supp ose that the c onditions of The or em 2.3 ar e satisﬁe d with the exc eption of (2.17) . In addition, let (A5) – (A7) , (3.40) , and (4.1) – (4.4) b e fulﬁl le d. Mor e over, let u 2 ∈ U 1 ad , with asso ciate d state ( µ, ϕ, σ ) = S (( u 1 , u 2 )) and adjoint state ( p, q , r ) , b e a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ ( Q ) . Then ther e is some λ 2 ∈ ∂ G 2 ( u 2 ) such that the variational ine quality (3.97) and the p ointwise pr oje ction formula (3.99) ar e satisﬁe d. Sc enario 2: Assume that the assumptions Theorem 2.3 are satisﬁed, except the con- dition (2.17). W e then ﬁx an element b z ∈ L ∞ (Ω) ∩ V and consider con trols u 1 of the pro duct type u 1 ( x, t ) = b z ( x ) u ( t ) for a.e. ( x, t ) ∈ Q , (4.5) that is, the real control v ariable u dep ends only on time. Accordingly , w e consider the set of admissible con trols U 2 ad :=  ( u, u 2 ) ∈ L ∞ (0 , T ) × L ∞ ( Q ) : u 1 ≤ u ≤ b u 1 a.e. in (0 , T ) and u 2 ≤ u 2 ≤ b u 2 a.e. in Q  , (4.6) where u 1 , b u 1 ∈ L ∞ (0 , T ) and u 1 ≤ b u 1 a.e. in (0 , T ) , u 2 , b u 2 ∈ L ∞ ( Q ) and u 2 ≤ b u 2 a.e. in Q . (4.7) W e c hose some R > 0 suc h that U 2 ad ⊂ U R :=  ( u, u 2 ) ∈ L ∞ (0 , T ) × L ∞ ( Q ) : ∥ ( u, u 1 ) ∥ L ∞ (0 ,T ) × L ∞ ( Q ) < R  . (4.8) Then, for every ( u, u 2 ) ∈ U R , the function ( b z u, u 2 ) b elongs to the open ball of radius max  1 , ∥ b z ∥ L ∞ ( Q ) ∩ L 2 (0 ,T ; V )  R in the space ( L ∞ ( Q ) ∩ L 2 (0 , T ; V )) × L ∞ ( Q ). Therefore, the uniform separation condition (2.20) is v alid for all second solution comp onen ts asso ciated with controls ( u, u 2 ) ∈ U R , and th us also (3.42), so that the analysis carried out in Section 3 can b e rep eated analogously . In this situation, we consider the cost functional in the form J 2 (( µ, ϕ, σ ) , ( u, u 2 ) = J 2 (( µ, ϕ, σ ) , ( u, u 2 )) + κ G (( u, u 2 )) , with the diﬀeren tiable part J 2 (( µ, ϕ, σ ) , ( u, u 2 )) = b 1 2 Z Q | ϕ − b ϕ Q | 2 + b 2 2 Z Ω | ϕ ( T ) − b ϕ Ω | 2 + b 3 2  Z T 0 | u ( t ) | 2 dt + Z Q | u 2 | 2  , (4.9) and where w e assume the sparsity term in the additiv e form G (( u, u 2 )) = G 1 ( u ) + G 2 ( u 2 ) for ( u, u 2 ) ∈ L 2 (0 , T ) × L 2 ( Q ), (4.10) 28 Colli – Rocca – Sprekels with nonnegativ e, contin uous and con vex functionals G 1 : L 2 (0 , T ) → R and G 2 : L 2 ( Q ) → R . According to Remark 3.9, it then follo ws that ∂ G (( u, u 2 )) = { ( λ 1 , λ 2 ) : λ 1 ∈ ∂ G 1 ( u ) and λ 2 ∈ ∂ G 2 ( u 2 ) } . Notice also that in (4.9) we ha ve ( µ, ϕ, σ ) = S ( H [ u ] , u 2 ) with the linear op erator H : L 2 (0 , T ) ∋ u 7→ b z u =: H [ u ] ∈ L 2 ( Q ) . (4.11) Clearly H is contin uous b oth from L 2 (0 , T ) into L 2 ( Q ) and from L ∞ (0 , T ) into L ∞ ( Q ) ∩ L 2 (0 , T ; V ) and, as a linear op erator, also F r´ ec het diﬀeren tiable b et ween these spaces with the F r´ ec het deriv ativ e D H = H . It therefore follows from the chain rule that the con trol-to-state op erator, which in this situation is given b y the op erator T , where T (( u, u 2 )) := S ( H [ u ] , u 2 ) = S ( b z u, u 2 ) for ( u, u 2 ) ∈ L ∞ (0 , T ) × L ∞ ( Q ) , (4.12) is F r ´ echet diﬀerentiable from L ∞ (0 , T ) × L ∞ ( Q ) in to Y . F or ﬁxed ( u, u 2 ) ∈ L ∞ (0 , T ) × L ∞ ( Q ), its deriv ativ e D T ( u, u 2 ) is given as follows: for all directions h = ( h, h 2 ) ∈ L ∞ (0 , T ) × L ∞ ( Q ), the v alue D T ( u, u 2 )[( h, h 2 )] is giv en by the unique solution ( η h , ψ h , ξ h ) to the linearized system (3.22)–(3.26), in whic h in this case the sum of the last tw o summands in the second line of Eq. (3.22) has to b e replaced by the expression − Z Ω h ( ϕ ) b z h v − Z Ω h ′ ( ϕ ) b z u ψ v . Consequen tly , the p en ultimate term in the second line of Eq. (3.77) in the adjoint system m ust b e replaced by − h ′ ( ϕ ) b z u p . Accordingly , the sum of the corresp onding last t w o summands in the second line of the calculation (3.95), which no w is p erformed for h = ( h, h 2 ) = ( u − u, u 2 − u 2 ), b ecomes Z Q  h ( ϕ ) b z h + h ′ ( ϕ ) b z u ψ  p . It turns out that the whole analysis of Section 3.4, whic h ultimately led to Lemma 3.6 and Theorem 3.8, can b e rep eated with obvious mo diﬁcations. Here, we understand b y lo cally optimal controls ( u, u 2 ) ∈ U 2 ad for ( CP ) in the sense of L ∞ (0 , T ) × L ∞ ( Q ) pairs for which there is some ε > 0 such that J 2 ( S (( b z u, u 2 )) , ( u, u 2 )) ≤ J 2 ( S (( b z v , v 2 )) , ( v , v 2 )) for all ( v , v 2 ) in U 2 ad with ∥ ( v, v 2 ) − ( u, u 2 ) ∥ L ∞ (0 ,T ) × L ∞ ( Q ) ≤ ε. W e then infer from The- orem 3.8 and the considerations in Remark 3.9 the follo wing result. Theorem 4.2. Supp ose the c onditions of The or em 2.3 ar e satisﬁe d with the exc eption of (2.17) . Mor e over, let (A5) – (A7) , (4.1) , and (4.5) – (4.10) b e fulﬁl le d. In addition, let ( u, u 2 ) ∈ U 2 ad b e a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ (0 , T ) × L ∞ ( Q ) with asso ciate d state ( µ, ϕ, σ ) and adjoint state ( p, q , r ) . Then ther e ar e λ 1 ∈ ∂ G 1 ( u ) and λ 2 ∈ ∂ G 2 ( u 2 ) such that the variational ine quality (3.97) and the p ointwise pr oje ction formula (3.99) ar e valid. Mor e over, ther e hold the variational ine quality Z T 0  Z Ω  − h ( ϕ ( x, t )) b z ( x ) p ( x, t )  dx + κλ 1 ( t ) + b 3 u ( t )   u ( t ) − u ( t )  dt ≥ 0 for al l u ∈ L ∞ (0 , T ) such that u 1 ≤ u ≤ b u 1 a.e. in (0 , T ) , (4.13) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 29 as wel l as for almost every t ∈ (0 , T ) the p ointwise pr oje ction formula u ( t ) = max  u 1 ( t ) , min  b u 1 ( t ) , − b − 1 3  Z Ω  − h ( ϕ ( x, t )) b z ( x ) p ( x, t )  dx + κλ 1 ( t )    . (4.14) Sc enario 3: Let again the assumptions of Theorem 2.3, with the exception of (2.17), and (4.1) b e satisﬁed. W e then consider con trols u 1 of the general form u 1 = e z + H [ w 1 ] , (4.15) with a ﬁxed e z ∈ L ∞ ( Q ) ∩ L 2 (0 , T ; V ), and where H : L 2 ( Q ) → L 2 ( Q ) denotes a linear mapping which is con tin uous from L 2 ( Q ) into L 2 ( Q ) and from L ∞ ( Q ) into L ∞ ( Q ) ∩ L 2 (0 , T ; V ). Notice that the Scenario 1 studied ab ov e is a sp ecial case of this one: indeed, in Scenario 1 we had e z = u 1 and H = n ull op erator. T ypical cases for linear op erators ha ving the required regularity prop erties are spatial con v olution integral op erators with suﬃcien tly smo oth k ernels or the solution op erators of a wide class of linear parab olic initial-b oundary v alue problems. In the ab ov e setting, w 1 is the true ﬁrst con trol v ariable. W e therefore consider as set of admissible con trols U 3 ad :=  ( w 1 , u 2 ) ∈ L ∞ ( Q ) × L ∞ ( Q ) : u 1 ≤ w 1 ≤ b u 1 , u 2 ≤ u 2 ≤ b u 2 a.e. in Q  (4.16) under the assumptions (3.40) on the treshold functions. Now let some R > 0 b e ﬁxed suc h that U 3 ad ⊂ U R :=  ( w 1 , u 2 ) ∈ L ∞ ( Q ) × L ∞ ( Q ) : ∥ ( w 1 , u 2 ) ∥ L ∞ ( Q ) × L ∞ ( Q ) < R  . (4.17) Then it follo ws from the con tin uity of H as a mapping from L ∞ ( Q ) in to L ∞ ( Q ) ∩ L 2 (0 , T ; V ) that there exists some b R > 0 suc h that for every ( w 1 , u 2 ) ∈ U R it holds ∥ ( e z + H [ w 1 ] , u 2 ) ∥ ( L ∞ ( Q ) ∩ L 2 (0 ,T ; V )) × L ∞ ( Q ) < b R . Therefore, the uniform separation condition (2.20) is v alid for all second solution comp onents associated with con trols ( w 1 , u 2 ) ∈ U R , and th us also (3.42), so that the analysis carried out in Section 3 can b e rep eated analo- gously . The cost functional is now assumed in the form J 3 (( µ, ϕ, σ ) , ( w 1 , u 2 ) = J 3 (( µ, ϕ, σ ) , ( w 1 , u 2 )) + κ G (( w 1 , u 2 )) , with the diﬀeren tiable part J 3 (( µ, ϕ, σ ) , ( w 1 , u 2 )) = b 1 2 Z Q | ϕ − b ϕ Q | 2 + b 2 2 Z Ω | ϕ ( T ) − b ϕ Ω | 2 + b 3 2 Z Q ( | w 1 | 2 + | u 2 | 2 ) , (4.18) where it holds ( µ, ϕ, σ ) = S (( e z + H [ w 1 ] , u 2 )). Again, w e assume the sparsity term in additiv e form, namely G (( w 1 , u 2 )) = G 1 ( w 1 ) + G 2 ( u 2 ) for ( w 1 , u 2 ) ∈ L 2 ( Q ) × L 2 ( Q ), (4.19) 30 Colli – Rocca – Sprekels with nonnegativ e, con tinuous and con vex functionals G i : L 2 ( Q ) → R , for i = 1 , 2. Accordingly , w e then ha ve ∂ G (( w 1 , u 2 )) = { ( λ 1 , λ 2 ) : λ 1 ∈ ∂ G 1 ( w 1 ) and λ 2 ∈ ∂ G 2 ( u 2 ) } . Notice that the aﬃne mapping deﬁned b y the iden tit y (4.15) is F r ´ echet diﬀerentiable b et w een the spaces L ∞ ( Q ) and L ∞ ( Q ) ∩ L 2 (0 , T ; V ) with the F r´ ec het deriv ative H . It follo ws from Theorem 3.4 and the c hain rule that the control-to-state op erator, which in this situation is given b y the op erator T , where T (( w 1 , u 2 )) := S ( e z + H [ w 1 ] , u 2 ) for ( w 1 , u 2 ) ∈ L ∞ ( Q ) × L ∞ ( Q ) , (4.20) is F r´ ec het diﬀeren tiable from L ∞ ( Q ) × L ∞ ( Q ) into Y . F or ﬁxed ( w 1 , u 2 ) ∈ L ∞ ( Q ) × L ∞ ( Q ), the deriv ative D T ( w 1 , u 2 ) is given as follows: for all directions h = ( h 1 , h 2 ) ∈ L ∞ ( Q ) × L ∞ ( Q ), the v alue D T ( w 1 , u 2 )[( h 1 , h 2 )] is giv en by the unique solution ( η h , ψ h , ξ h ) to the linearized system (3.22)–(3.26), in which in this case the sum of the last tw o summands in the second line of Eq. (3.22) has to b e replaced by the expression − Z Ω h ( ϕ ) H [ h 1 ] v − Z Ω h ′ ( ϕ )( e z + H [ w 1 ]) ψ v . Consequen tly , the p en ultimate term in the second line of Eq. (3.77) in the adjoint system m ust b e replaced b y − h ′ ( ϕ )( e z + H [ w 1 ]) p . Accordingly , the sum of the corresp onding last tw o summands in the second line of the calculation (3.95), which no w is p erformed for h = ( h 1 , h 2 ) = ( w 1 − w 1 , u 2 − u 2 ), b ecomes Z Q  h ( ϕ ) H [ h 1 ] + h ′ ( ϕ ) ( e z + H [ w 1 ]) ψ  p . A t this p oint, it is useful to in tro duce the dual op erator of H , whic h is the operator H ∗ ∈ L ( L 2 ( Q ) , L 2 ( Q )) deﬁned through the identit y Z Q H ∗ [ y ] z = Z Q y H [ z ] for all y , z ∈ L 2 ( Q ) . W e then ha ve Z Q h ( ϕ ) H [ h 1 ] p = Z Q H ∗ [ h ( ϕ ) p ] h 1 , and from the calculation (3.95) we can conclude the follo wing result. Theorem 4.3. Supp ose that the assumption of The or em 2.3 b e satisﬁe d with the exc eption of (2.17) . Mor e over, let (A5) – (A7) , (4.1) , and (4.15) – (4.19) b e fulﬁl le d. In addition, assume that ( w 1 , u 2 ) ∈ U 2 ad is a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ) . Then ther e ar e λ 1 ∈ ∂ G 1 ( w 1 ) and λ 2 ∈ ∂ G 2 ( u 2 ) such that the variational ine quality (3.97) and the p ointwise pr oje ction formula (3.99) ar e satisﬁe d. In addition, ther e hold the variational ine quality Z Q  − H ∗ [ h ( ϕ ) p ] + κλ 1 + b 3 w 1  ( w 1 − w 1 ) ≥ 0 for al l w 1 ∈ L ∞ ( Q ) with u 1 ≤ w 1 ≤ b u 1 a.e. in Q, (4.21) as wel l as almost everywher e in Q the p ointwise pr oje ction formula w 1 = max  u 1 , min  b u 1 , − b − 1 3  − H ∗ [ h ( ϕ ) p ] + κλ 1  . (4.22) Contr ol of the hyperbolic relaxa tion in a tumor gro wth model 31 5 Some remarks on sparsit y In this ﬁnal section, w e discuss the concept of sparsit y , that is, the p ossible occurrence of prop er subregions of the space-time cylinder Q in which (lo cally) optimal controls v anish. The o ccurrence of sparsity is a consequence of the choice of the functional G and the v ariational inequality (3.94) through the form of the associated subdiﬀerential ∂ G . In the follo wing, w e conﬁne ourselves to the concept of ful l sp arsity , which is connected with the functional G given in (1.8) that we consider in the following. Other concepts of sparsit y , lik e dir e ctional sp arsity with resp ect to time or to space (for these concepts see, e.g., [32]), could also b e considered. As far as the underlying double-w ell p oten tial is concerned, we conﬁne ourselves to the case of regular potentials, that is, w e assume that the assumptions of Theorem 2.2, (A5) – (A7) , (3.1), and (3.38)–(3.41) are fulﬁlled. W e notice, how ever, that results resembling that of Theorem 5.1 b elow can also b e established for singular p otentials in an y of the scenarios considered ab ov e in Section 4. No w let u = ( u 1 , u 2 ) ∈ U ad b e a ﬁxed lo cally optimal control for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ) with asso ciated state ( µ, ϕ, σ ) = S ( u ) and adjoint state ( p, q , r ). Then the statement of Theorem 3.8 is v alid. W e no w introduce the nonnegative, contin uous and con vex functional j : L 2 ( Q ) → R , j ( u ) := ∥ u ∥ L 1 ( Q ) = Z Q | u | . (5.1) It is w ell known (see, e.g., [22]) that for an y u ∈ L 2 ( Q ) the elemen ts λ of the subdiﬀerential ∂ j ( u ) are of the following form: for a.e. ( x, t ) ∈ Q it holds λ ( x, t )        = − 1 if u ( x, t ) < 0 , ∈ [ − 1 , +1] , if u ( x, t ) = 0 , = +1 if u ( x, t ) > 0 . (5.2) Next, w e observ e that w e hav e, for every u = ( u 1 , u 2 ) ∈ L 2 ( Q ) × L 2 ( Q ), G ( u ) = j ( I 1 ( u )) + j ( I 2 ( u )) , (5.3) with the pro jection op erators I i , i = 1 , 2, that were introduced in Remark 3.9. As discussed there, the v ariational inequality (3.94) then decouples into the tw o indep endent v ariational inequalities (3.96) and (3.97) with λ i ∈ ∂ j ( u i ), for i = 1 , 2. Consequen tly , the p oin t wise pro jection formulas (3.98) and (3.99) are v alid with suitable λ i ∈ ∂ j ( u i ), for i = 1 , 2. W e then obtain the following (rather standard) sparsit y result. Theorem 5.1. Assume that the c onditions of The or em 2.2, along with (A5) – (A7) , (3.1) , (3.38) – (3.41) , ar e fulﬁl le d. In addition, supp ose that the thr esholds functions u i , b u i , i = 1 , 2 , ar e c onstants satisfying the sign c ondition u i < 0 < b u i , for i = 1 . 2 . (5.4) If u = ( u 1 , u 2 ) ∈ U ad is a lo c al ly optimal c ontr ol for ( CP ) in the sense of L ∞ ( Q ) × L ∞ ( Q ) with asso ciate d state ( µ, ϕ, σ ) = S ( u ) and adjoint state ( p, q , r ) , then, for almost every 32 Colli – Rocca – Sprekels ( x, t ) ∈ Q , we have the e quivalenc e r elations u 1 ( x, t ) = 0 ⇐ ⇒ | h ( ϕ ( x, t )) p ( x, t ) | ≤ κ , (5.5) u 2 ( x, t ) = 0 ⇐ ⇒ | r ( x, t ) | ≤ κ . (5.6) Pr o of. W e only sho w (5.5); the pro of of (5.6) is analagous. First, we hav e for almost ev ery ( x, t ) ∈ Q : if u 1 ( x, t ) = 0 then, b y virtue of the p oint wise pro jection formula (3.98), 0 = − h ( ϕ ( x, t )) p ( x, t ) + κλ 1 ( x, t ), where λ 1 ( x, t ) ∈ [ − 1 , 1]. This obviously implies that | h ( ϕ ( x, t )) p ( x, t ) | ≤ κ . Con versely , we can argue for almost ev ery ( x, t ) ∈ Q as follows: supp ose that it holds | h ( ϕ ( x, t )) p ( x, t ) | ≤ κ . If u 1 ( x, t ) > 0 then λ 1 ( x, t ) = 1, and it follo ws from (3.98) that − b − 1 3 ( h ( ϕ ( x.t )) p ( x, t ) + κ ) > 0. But then h ( ϕ ( x.t ) p ( x, t ) + κ < 0, and we obtain that | h ( ϕ ( x, t ) p ( x, t ) | = − h ( ϕ ( x, t )) p ( x, t ) > κ , a contradiction. Similar reasoning shows that also the assumption u 1 ( x, t ) < 0 leads to a contradiction. W e therefore m ust ha v e u 1 ( x, t ) = 0. Remark 5.2. A t this p oin t, it mak es sense to ask whether there exists some uniform constan t b κ > 0 such that all locally optimal controls for ( CP ) v anish whenever the sparsit y parameter κ satisﬁes κ > b κ . According to the conditions (5.5) and (5.6), this will b e the case if the adjoin t v ariables p and r are b ounded by a global constant. And indeed, this is an immediate consequence of the estimates p erformed in the pro of of Theorem 3.7. In particular, it was sho wn there that there is a constant C > 0, whic h do es not dep end on the sp ecial choice of u ∈ U R , suc h that (cf. (3.91) and (3.93)) ∥ r ∥ L ∞ ( Q ) + ∥ p ∥ L ∞ ( Q ) ≤ C . Ac kno wledgmen ts PC and ER gratefully men tion their aﬃliation to the GNAMP A (Grupp o Nazionale p er l’Analisi Matematica, la Probabilit` a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and their collab oration, as Research Asso ciate, to the IMA TI – C.N.R. Pa via, Italy . PC and ER also ac knowledge the supp ort of Next Generation EU Pro ject No.P2022Z7ZAJ (A unitary mathematical framew ork for mo delling m uscular dys- trophies). References [1] M. Abatangelo, C. Ca v aterra, M. Grasselli and H. W u, Optimal distributed control for a Cahn– Hilliard–Darcy system with mass sources, unmatc hed viscosities and singular p otential, ESAIM Contr ol Optim. 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Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

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