SECOND-ORDER SUFFICIENT CONDITIONS IN THE SPARSE OPTIMAL CONTROL OF A PHASE FIELD TUMOR GROWTH MODEL WITH LOGARITHMIC POTENTIAL

. This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn–Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the L 1 –norm in order to enhance the occurrence of sparsity effects in the optimal controls, i


Introduction
Let α > 0, β > 0, χ > 0, and let Ω ⊂ R 3 denote some open and bounded domain having a smooth boundary Γ = ∂Ω and the unit outward normal n with associated outward normal derivative ∂ n .Moreover, we fix some final time T > 0 and introduce for every t ∈ (0, T ) the sets Q t := Ω × (0, t) and Q t := Ω × (t, T ).We also set, for convenience, Q := Q T and Σ := Γ × (0, T ).We then consider the following optimal control problem: (CP) Minimize the cost functional J((µ, ϕ, σ), u) subject to the state system and to the control constraint u = (u 1 , u 2 ) ∈ U ad . ( Here, b 1 and b 2 are nonnegative constants, while b 3 and κ are positive; ϕ Q and ϕ Ω are given target functions.The term g(u) accounts for possible sparsity effects.Moreover, the set of admissible controls U ad is a nonempty, closed and convex subset of the control space U := L ∞ (Q) 2 . (1.8) The state system (1.2)-(1.6)constitutes a simplified and relaxed version of the fourspecies thermodynamically consistent model for tumor growth originally proposed by Hawkins-Daruud et al. in [38] that additionally includes the chemotaxis-like terms χ σ in (1.3) and − χ ∆ϕ in (1.4).Let us briefly review the role of the occurring symbols.The primary (state) variables ϕ, µ, σ denote the tumor fraction, the associated chemical potential, and the nutrient concentration, respectively.Furthermore, the additional term α∂ t µ corresponds to a parabolic regularization of equation (1.2), while β∂ t ϕ is the viscosity contribution to the Cahn-Hilliard equation.The nonlinearity P denotes a proliferation function, whereas the positive constant χ represents the chemotactic sensitivity and provides the system with a cross-diffusion coupling.
The evolution of the tumor fraction is mainly governed by the nonlinearities F 1 and F 2 whose derivatives occur in (1.3).Here, F 2 is smooth, typically a concave function.As far as F 1 is concerned, we admit in this paper functions of logarithmic type such as for r ∈ (−1, 1) 2 ln (2) for r ∈ {−1, 1} , +∞ for r ∈ [−1, 1] (1.9) We assume that F = F 1 + F 2 is a double-well potential.This is actually the case if F 2 (r) = k(1 − r 2 ) with a sufficiently large k > 0. Note also that F ′ 1,log (r) becomes unbounded as r ց −1 and r ր 1.
The control variable u 2 occurring in (1.4) can model either an external medication or some nutrient supply, while u 1 , which occurs in the phase equation (1.2), models the application of a cytotoxic drug to the system.Usually, u 1 is multiplied by a truncation function (ϕ) in order to have the action only in the spatial region where the tumor cells are located.Typically, one assumes that (−1) = 0, (1) = 1, and (ϕ) is in between if −1 < ϕ < 1; see [29,35,41,42] for some insights on possible choices of .Also in [15,17,54], this kind of nonlinear coupling between u 1 and ϕ has been admitted.For our following analysis, this nonlinear coupling is too strong, and, only for technical reasons, we have chosen to simplify the original system somewhat by assuming that = (x, t) is a bounded nonnegative function that does not depend on ϕ.We stress the fact that this simplification does not have any impact on the validity of the results from [17] to be used below.
As far as well-posedness is concerned, the above model was already investigated in the case χ = 0 in [6][7][8][9], and in [25] with α = β = χ = 0.There the authors also pointed out how the relaxation parameters α and β can be set to zero, by providing the proper framework in which a limit system can be identified and uniquely solved.We also note that in [13] a version has been studied in which the Laplacian in the equations (1.2)-(1.4)has been replaced by fractional powers of a more general class of selfadjoint operators having compact resolvents.A model which is similar to the one studied in this note was the subject of [15,54].
For some nonlocal variations of the above model we refer to [27,28,47].Moreover, in order to better emulate in-vivo tumor growth, it is possible to include in similar models the effects generated by the fluid flow development by postulating a Darcy law or a Stokes-Brinkman law.In this direction, we refer to [11, 21, 24, 27, 29-33, 35, 59], and we also mention [36], where elastic effects are included.For further models, discussing the case of multispecies, we refer the reader to [21,27].The investigation of associated optimal control problems also presents a wide number of results of which we mention [10, 13, 15, 22, 23, 28, 34, 37, 42, 48-52, 54, 56].Sparsity in the optimal control theory of partial differential equations is a very active field of research.The use of sparsity-enhancing functionals goes back to inverse problems and image processing.Soon after the seminal paper [57], many results were published.We mention only very few of them with closer relation to our paper, in particular [1,39,40], on directional sparsity, and [5] on a general theorem for second-order conditions; moreover, we refer to some new trends in the investigation of sparsity, namely, infinite horizon sparse optimal control (see, e.g., [43,44]), and fractional order optimal control (cf.[46], [45]).These papers concentrated on first-order optimality conditions for sparse optimal controls of single elliptic and parabolic equations.In [3,4], first-and second-order optimality conditions have been discussed in the context of sparsity for the (semilinear) system of FitzHugh-Nagumo equations.Moreover, we refer to the measure control of the Navier-Stokes system studied in [2].
The optimal control problem (CP) has recently been investigated in [17] for the case of logarithmic potentials F 1 and without sparsity terms, where second-order sufficient op-timality conditions have been derived using the τ -critical cone and the splitting technique as described in the textbook [58].In [54] and [18], sparsity terms have been incorporated, where in the latter paper not only logarithmic nonlinearities but also nondifferentiable double obstacle potentials have been admitted.However, second-order sufficient optimality conditions have not been derived.
The derivation of meaningful second-order conditions for locally optimal controls of (CP) in the logarithmic case with sparsity term is the main object of this paper.In particular, we aim at constructing suitable critical cones which are as small as possible.In our approach, we follow the recent work [55] on the sparse optimal control of Allen-Cahn systems, which was based on ideas developed in [4].
The paper is organized as follows.In the next section, we list and discuss our assumptions, and we collect known results from [18] concerning the properties of the state system (1.2)-(1.6)and of the control-to-state operator.In Section 3, we study the optimal control problem.We derive first-order necessary optimality conditions and results concerning the full sparsity of local minimizers, and we establish second-order sufficient optimality conditions for the optimal control problem (CP).In an appendix, we prove auxiliary results that are needed for the main theorem on second-order sufficient conditions.
Prior to this, let us fix some notation.For any Banach space X, we denote by • X the norm in the space X, by X * its dual space, and by • , • X the duality pairing between X * and X.For any 1 ≤ p ≤ ∞ and k ≥ 0, we denote the standard Lebesgue and Sobolev spaces on Ω by L p (Ω) and W k,p (Ω), and the corresponding norms by • L p (Ω) = • p and • W k,p (Ω) , respectively.For p = 2, they become Hilbert spaces, and we employ the standard notation H k (Ω) := W k,2 (Ω).As usual, for Banach spaces X and Y that are both continuously embedded in some topological vector space Z, we introduce the linear space X ∩ Y which becomes a Banach space when equipped with its natural norm v X∩Y := v X + v Y , for v ∈ X ∩ Y .Moreover, we recall the definition (1.8) of the control space U and introduce the spaces (1.10) Furthermore, by ( • , • ) and • we denote the standard inner product and related norm in H, and, for simplicity, we also set Throughout the paper, we make repeated use of Hölder's inequality, of the elementary Young inequality as well as the continuity of the embeddings We close this section by introducing a convention concerning the constants used in estimates within this paper: we denote by C any positive constant that depends only on the given data occurring in the state system and in the cost functional, as well as on a constant that bounds the (L ∞ (Q) × L ∞ (Q))-norms of the elements of U ad .The actual value of such generic constants C may change from formula to formula or even within formulas.Finally, the notation C δ indicates a positive constant that additionally depends on the quantity δ.
2 General setting and properties of the control-tostate operator In this section, we introduce the general setting of our control problem and state some results on the state system (1.2)-(1.6)and the control-to-state operator that in its present form have been established in [17,18].
We make the following assumptions on the data of the system.
(A4) The initial data satisfy µ 0 , σ 0 ∈ H 1 (Ω) ∩ L ∞ (Ω), ϕ 0 ∈ W 0 , as well as (A5) With fixed given constants u i , u i satisfying u i < u i , i = 1, 2, we have Remark 2.1.Observe that (A3) implies that the functions P, P ′ , P ′′ are Lipschitz continuous on R. Let us also note that F 1 = F ) Moreover, there is a constant K 1 > 0, which depends on Ω, T, R, α, β and the data of the system, but not on the choice of u ∈ U R , such that Furthermore, there are constants r * , r * , which depend on Ω, T, R, α, β and the data of the system, but not on the choice of u ∈ U R , such that Also, there is some constant K 2 > 0 having the same dependencies as K 1 such that max i=0,1,2,3 (2.9) Finally, if u i ∈ U R are given controls and (µ i , ϕ i , σ i ) the corresponding solutions to (1.2)-(1.6),for i = 1, 2, then, with a constant K 3 > 0 having the same dependencies as K 1 , (2.10) Remark 2.3.Condition (2.8), known as the separation property, is especially important for the case of singular potentials such as F 1 = F 1,log , since it guarantees that the phase variable ϕ always stays away from the critical values −1, +1.The singularity of F ′ 1 is therefore no longer an obstacle for the analysis, as the values of ϕ range in some interval in which F ′ 1 is smooth.
Owing to Theorem 2.2, the control-to-state operator is well defined as a mapping between U = L ∞ (Q) 2 and the Banach space specified by the regularity results (2.4)-(2.6).We now discuss its differentiability properties.For this purpose, some functional analytic preparations are in order.We first define the linear spaces which are Banach spaces when endowed with their natural norms.Next, we introduce the linear space which becomes a Banach space when endowed with the norm Finally, we put (2.15) For fixed (ϕ * , µ * , σ * ), we first discuss an auxiliary result for the linear initial-boundary value problem ) which for λ 1 = λ 2 = 1 and λ 3 = λ 4 = 0 coincides with the linearization of the state equation at ((µ * , ϕ * , σ * ), (u * 1 , u * 2 )).We have the following result.Lemma 2.4.Suppose that λ 1 , λ 2 , λ 3 , λ 4 ∈ {0, 1} are given and that the assumptions Moreover, the linear mapping Proof.The existence result and the continuity of the mapping (2.21) between the spaces  [17] we can conclude that the mapping (2.21) is also continuous between the spaces Then, according to [17,Thm. 4.4], the control-to-state operator S is twice continuously Fréchet differentiable at u * as a mapping from U into Y.Moreover, for every h where (η h , ξ h , θ h ) ∈ Y is the unique solution to the linearization of the state system given by the initial-boundary value problem (2.16)-(2.20)with λ 1 = λ 2 = 1 and λ 3 = λ 4 = 0.
Remark 2.5.Observe that, in view of the continuity of the embedding Y ⊂ Z × X × Z, the operator S ′ (u * ) ∈ L(U, Y) also belongs to the space L(U, Z × X ×Z) and, owing to the density of U in L 2 (Q) 2 , can be extended continuously to an element of L(L 2 (Q) 2 , Z × X × Z) without changing its operator norm.Denoting the extended operator still by S ′ (u * ), we see that the identity In addition, it also follows from the proof of [17,Lem. 4.1] that there is a constant K 4 > 0, which depends only on R and the data, such that (2.22) Next, we show a Lipschitz property for the extended operator S ′ .
Lemma 2.6.The mapping , is Lipschitz continuous in the following sense: there is a constant K 5 > 0, which depends only on R and the data, such that, for all controls (2.23) Then it follows from (2.10) in Theorem 2.2 that Moreover, (η, ξ, θ) solves the problem which is of the form (2.16)-(2.20)with λ 1 = λ 3 = 1 and λ 2 = λ 4 = 0, and where (2.31) We therefore conclude from Lemma 2.4 that Hence, the proof will be finished once we can show that To this end, we first use the mean value theorem, (2.9), Hölder's inequality, the continuity of the embedding V ⊂ L 4 (Ω), as well as (2.10) and (2.24), to find that (2.33) Here, we have for convenience omitted the argument s in the third integral.We will do this repeatedly in the following.For the three summands on the right-hand side of (2.30), which we denote by A 1 , A 2 , A 3 , in this order, we obtain by similar reasoning the estimates where in the last estimate we also used (2.7) and (2.33).With this, the assertion is proved.
Next, we turn our interest to the second Fréchet derivative is the unique solution to the bilinearization of the state system at ((µ * , ϕ * , σ * ), (u * 1 , u * 2 )), which is given by the linear initial-boundary value problem and which is again of the form (2.16)-(2.20)with λ 1 = λ 3 = 1 and λ 2 = λ 4 = 0, where are well-defined elements of the space Z × X × Z, where S ′ (u * ) now denotes the extension of the Fréchet derivative introduced in Remark 2.5.We now claim that there is a constant C > 0 that depends only on R and the data, such that (2.44) Indeed, arguing as in the derivation of the estimates (2.33)-(2.36),we obtain which proves the claim.At this point, we can conclude from Lemma 2.4 that the system (2.37)-(2.41)has for every h, k Moreover, we have, with a constant K 6 > 0 that depends only on R and the data, (2.45) Remark 2.7.Similarly as in Remark 2.5, the operator without changing its operator norm.Denoting the extended operator still by S ′′ (u * ), we see that the identity S ′′ (u * )[h, k] = (ν, ψ, ρ) is also valid for every h, k ∈ L 2 (Q) 2 , only that (ν, ψ, ρ) ∈ Z × X × Z, in general.In addition, we have We conclude our preparatory work by showing a Lipschitz property for the extended operator S ′′ that resembles (2.23).
At this point, we infer from the proof of [17,Lem. 4.1] that the assertion follows once we can show that We only show the corresponding estimate for the terms B 1 , B 4 , B 11 and leave the others to the interested reader.In the following, we make use of the mean value theorem, Hölder's inequality, the continuity of the embeddings V ⊂ L 6 (Ω) ⊂ L 4 (Ω), and the global estimates (2.7), (2.8), (2.22), and (2.46).We have 2 as well as The assertion of the lemma is thus proved.

The optimal control problem
We now begin to investigate the control problem (CP).In addition to (A1)-(A6), we make the following assumptions: Observe that (C3) implies that g is weakly sequentially lower semicontinuous on L 2 (Q) 2 .Moreover, denoting in the following by ∂ the subdifferential mapping in L 2 (Q) 2 , it follows from standard convex analysis that ∂g is defined on the entire space L 2 (Q) 2 and is a maximal monotone operator.In addition, the mapping ((µ, ϕ, σ), u) → J((µ, ϕ, σ), u) defined by the cost functional (1.1) is obviously continuous and convex (and thus weakly sequentially lower semicontinuous) on the space 2 .From a standard argument (which needs no repetition here) it then follows that the problem (CP) has a solution.
In the following, we often denote by u * = (u * 1 , u * 2 ) ∈ U ad a local minimizer in the sense of U and by (µ * , ϕ * , σ * ) = S(u * ) the associated state.The corresponding adjoint state variables solve the adjoint system, which is given by the backward-in-time parabolic system According to [17, Thm.5.2], the adjoint system has a unique weak solution ( p, q, r ) satisfying as well as − for every v ∈ V and almost every t ∈ (0, T ), and Moreover, it follows from the proof of [17, Thm.5.2] that there exists a constant K 8 > 0, which depends only on R and the data (but not on the special choice of u * ∈ U ad ), such that

First-order necessary optimality conditions
In this section, we aim at deriving associated first-order necessary optimality conditions for local minima of the optimal control problem (CP).We assume that (A1)-(A6) and (C1)-(C3) are fulfilled and define the reduced cost functionals associated with the functionals J and J 1 introduced in (1.1) by ).It thus follows from the chain rule that the smooth part J 1 of J is a twice continuously Fréchet differentiable mapping from U into R, where, for every where (η h , ξ h , θ h ) = S ′ (u * )[h] is the unique solution to the linearized system (2.16)-(2.20),with λ 1 = λ 2 = 1 and λ 3 = λ 4 = 0, associated with h.
Remark 3.1.Observe that the right-hand side of (3.16) is meaningful also for arguments h , where in this case (η h , ξ h , θ h ) = S ′ (u * )[h] with the extension of the operator S ′ (u * ) to L 2 (Q) 2 introduced in Remark 2.5.Hence, by means of the identity (3.16) we can extend the operator The extended operator, which we again denote by J ′ 1 (u * ), then becomes an element of (L 2 (Q) 2 ) * .In this way, expressions of the form J ′ 1 (u * )[h] have a proper meaning also for h ∈ L 2 (Q) 2 .
In the following, we assume that u * = (u * 1 , u * 2 ) ∈ U ad is a given locally optimal control for (CP) in the sense of U, that is, there is some ε > 0 such that Notice that any locally optimal control in the sense of L p (Q) 2 with 1 ≤ p < ∞ is also locally optimal in the sense of U. Therefore, a result proved for locally optimal controls in the sense of U is also valid for locally optimal controls in the sense of L p (Q) 2 .It is of course also valid for (globally) optimal controls.Now, in the same way as in [55], we infer that then the variational inequality is satisfied.Moreover, denoting by the symbol ∂ the subdifferential mapping in L 2 (Q) 2 (recall that g is a convex continuous functional on L 2 (Q) 2 ), we conclude from [55, Thm.4.5] that there is some As usual, we simplify the expression J ′ 2 ) ∈ U ad be a locally optimal control of (CP) in the sense of U with associated state (µ * , ϕ * , σ * ) = S(u * ) and adjoint state (p * , q * , r * ).Then there exists some Remark 3.3.We underline again that (3.20) is also necessary for all globally optimal controls and all controls which are even locally optimal in the sense of L p (Q) ×L p (Σ) with p ≥ 1. Observe also that the variational inequality (3.20) is equivalent to two independent variational inequalities for u * 1 and u * 2 that have to hold simultaneously, namely, where

Sparsity of controls
The convex function g in the objective functional accounts for the sparsity of optimal controls, i.e., any locally optimal control can vanish in some region of the space-time cylinder Q.The form of this region depends on the particular choice of the functional g which can differ in different situations.The sparsity properties can be deduced from the variational inequalities (3.21) and (3.22) and the form of the subdifferential ∂g.In this paper, we restrict our analysis to the case of full sparsity which is characterized by the functional (recall (1.1)) Other important choices leading to the directional sparsity with respect to time and the directional sparsity with respect to space are not considered here.It is well known (see, e.g., [54]) that the subdifferential of g is given by The following sparsity result can be proved in exactly the same way as [55, Thm.4.9].

Second-order sufficient optimality conditions
In this section, we establish the main results of this paper, using auxiliary results collected in the Appendix.We provide conditions that ensure local optimality of pairs obeying the first-order necessary optimality conditions of Theorem 3.2.Second-order sufficient optimality conditions are based on a condition of coercivity that is required to hold for the smooth part J 1 of J in a certain critical cone.The nonsmooth part g contributes to sufficiency by its convexity.In the following, we generally assume that (A1)-(A6), (C1)-(C3), and the conditions u 1 < 0 < u 1 and u 2 < 0 < u 2 are fulfilled.Our analysis will follow closely the lines of [55], which in turn follows [4], where a secondorder analysis was performed for sparse control of the FitzHugh-Nagumo system.In particular, we adapt the proof of [4,Thm. 3.4] to our setting of less regularity.
To this end, we fix a pair of controls u * = (u * 1 , u * 2 ) that satisfies the first-order necessary optimality conditions, and we set (µ * , ϕ * , σ * ) = S(u * ).Then the cone 2 satisfying the sign conditions (3.30) a.e. in Q}, where is called the cone of feasible directions, which is a convex and closed subset of We also need the directional derivative of , which is given by Following the definition of the critical cone in [4, Sect.3.1], we define which is also a closed and convex subset of Remark 3.5.Let us compare the first condition in (3.33) with the situation in the differentiable control problem without sparsity terms obtained for κ = 0. Then this condition leads to the requirement that v An analogous condition results for v 2 .
One might be tempted to define the critical cone using (3.35) and its counterpart for v 2 also in the case κ > 0. This, however, is not a good idea, because it leads to a critical cone that is larger than needed, in general.As an example, we mention the particular case when the control u * = 0 satisfies the first-order necessary optimality conditions and when | − p * | < κ and |r * | < κ hold a.e. in Q.Then the upper relation of (3.33), and its counterpart for v 2 , lead to C u * = {0}, the smallest possible critical cone.
However, thanks to u * 1 = 0, the variational inequality (3.21) implies that − p * +κλ * 1 + b 3 u * 1 = 0 a.e. in Q, i.e., the condition | − (x, t)p * (x, t) + κλ * 1 (x, t) + b 3 u * 1 (x, t)| > 0 can only be satisfied on a set of measure zero.Moreover, also the sign conditions (3.30) do not restrict the critical cone.Hence, the largest possible critical cone C u * = L 2 (Q) 2 would be obtained, provided that analogous conditions hold for u * 2 and r * in Q.In this example, the quadratic growth condition (3.43) below is valid for the choice (3.32) as critical cone even without assuming the coercivity condition (3.42) below (here the so-called first-order sufficient conditions apply), while the use of a cone based on (3.35) leads to postulating (3.42) on the whole space L 2 (Q) 2 for the quadratic growth condition to be valid.This shows that the choice of (3.32) as critical cone is essentially better than of one based on (3.35).
At this point, we derive an explicit expression for In the following, we argue similarly as in [58,Sect. 5.7] (see also [17,Sect. 6]).At first, we readily infer that, for every ((µ, ϕ, σ), u) where the dot denotes the euclidean scalar product in R 2 .For the second-order derivative of the reduced cost functional J 1 at a fixed control u * we then find with (µ * , ϕ * , σ * ) = S(u * ) that where (η h , ξ h , θ h ), (η k , ξ k , θ k ), and (ν, ψ, ρ) stand for the unique corresponding solutions to the linearized system associated with h and k, and to the bilinearized system, respectively.From the definition of the cost functional (1.1) we readily infer that We now claim that, with the associated adjoint state (p * , q * , r * ), To prove this claim, we multiply (2.37) by p * , (2.38) by q * , (2.39) by r * , add the resulting equalities, and integrate over Q, to obtain that with the function f 1 defined in (2.42).Then, we integrate by parts and make use of the initial and terminal conditions (2.41) and (3.13) to find that whence the claim follows, since (p * , q * , r * ) solves the adjoint system (3.10)-(3.13).From this characterization, along with (3.37) and (3.38), we conclude that Observe that the expression on the right-hand side of (3.40) is meaningful also for increments h, k ∈ L 2 (Q) 2 .Indeed, in this case the expressions (η h , ξ h , θ h ) = S ′ (u * )[h], (η k , ξ k , θ k ) = S ′ (u * )[k], and (ν, ψ, ρ) = S ′′ (u * )[h, k] have an interpretation in the sense of the extended operators S ′ (u * ) and S ′′ (u * ) introduced in Remark 2.5 and Remark 2.7.Therefore, the operator J ′′ 1 (u * ) can be extended by the identity (3.40) to the space 2 .This extension, which will still be denoted by J ′′ 1 (u * ), will be frequently used in the following.We now show that it is continuous.Indeed, we claim that for all h, k ∈ L 2 (Q) 2 it holds where the constant C > 0 is independent of the choice of u * ∈ U R .Obviously, only the last integral on the right-hand side of (3.40) needs some treatment, and we estimate just its third summand, leaving the others as an exercise to the reader.We have, by virtue of Hölder's inequality, the continuity of the embedding V ⊂ L 4 (Ω), and the global bounds (2.9), (2.22), and (3.14), as asserted.
In the following, we will employ the following coercivity condition: Condition (3.42) is a direct extension of associated conditions that are standard in finitedimensional nonlinear optimization.In the optimal control of partial differential equation, it was first used in [5].As in [4,Thm 3.3] or [5], it can be shown that (3.42) is equivalent to the existence of a constant δ > 0 such that We have the following result.Theorem 3.6.(Second-order sufficient condition) Suppose that (A1)-(A6) and (C1)-(C3) are fulfilled and that u i < 0 < u i , i = 1, 2.Moreover, let u * = (u * 1 , u * 2 ) ∈ U ad , together with the associated state (µ * , ϕ * , σ * ) = S(u * ) and the adjoint state (p * , q * , r * ), fulfill the first-order necessary optimality conditions of Theorem 3.2.If, in addition, u * satisfies the coercivity condition (3.42), then there exist constants ε > 0 and τ > 0 such that the quadratic growth condition Proof.The proof follows that of [4,Thm. 3.4].We argue by contradiction, assuming that the claim of the theorem is not true.Then there exists a sequence of controls {u k } ⊂ U ad such that, for all k ∈ N, Noting that u k = u * for all k ∈ N, we define Then v k L 2 (Q) 2 = 1 and, possibly after selecting a subsequence, we can assume that for some v ∈ L 2 (Q) 2 .As in [4], the proof is split into three parts.
(i) v ∈ C u * : Obviously, each v k obeys the sign conditions (3.30) and thus belongs to C(u * ).Since C(u * ) is convex and closed in L 2 (Q) 2 , it follows that v ∈ C(u * ).We now claim that Notice that by Remark 3.1 the expression J ′ 1 (u * )[v] is well defined.For every r ∈ (0, 1) and all v = (v 1 , v 2 ), u = (u 1 , u 2 ) ∈ L 2 (Q) 2 , we infer from the convexity of g that In particular, with by the variational inequality (3.20).Next, we prove the converse inequality.By (3.44), we have whence, owing to the mean value theorem, and since and thus We divide this inequality by r k and pass to the limit k → ∞.Here, we invoke Corollary 4.2 of the Appendix, and we use that g ′ (u * , v k ) → g ′ (u * , v).We then obtain the desired converse inequality J ′ 1 (u * )[v] + κg ′ (u * , v) ≤ 0 , which completes the proof of (i).
(ii) v = 0: We again invoke (3.44), now performing a second-order Taylor expansion on the left-hand side, We subtract J 1 (u * ) + κg(u * ) from both sides and use (3.46) once more to find that From the right-hand side of (3.46), and the variational inequality (3.20), it follows that thus, by (3.48), .49) Passing to the limit k → ∞, we apply Lemma 4.3 and deduce that J ′′ 1 (u * )[v, v] ≤ 0. Since we know that v ∈ C u * , the second-order condition (3.42) implies that v = 0.
(iii) Contradiction: From the previous step we know that v k → 0 weakly in L 2 (Q) 2 .Moreover, (3.40) yields that where we have set for k ∈ N. By virtue of Lemma 4.3, the sum of the last four integrals on the right-hand side converges to zero.On the other hand, v k L 2 (Q) 2 = 1 for all k ∈ N, by construction.The weak sequential semicontinuity of norms then implies that lim inf On the other hand, it is easily deduced from (3.49) and (2.47) that lim inf a contradiction.The assertion of the theorem is thus proved.
Remark 3.7.We note at this place that the formula (6.5) in [17], which resembles (3.50), contains three sign errors: indeed, the term in the second line of [17, (6.5)] involving P ′′ should carry a "plus" sign, while the two terms in the third line should carry "minus" signs.These sign errors, however, do not have an impact on the validity of the results established in [17].
Lemma 4.1.Let {u k } ⊂ U ad converge strongly in L 2 (Q) 2 to u * , and let (µ k , ϕ k , σ k ) = S(u k ) and (p k , q k , r k ), k ∈ N, denote the associated states and adjoint states.Then p k → p * weakly-star in Z and strongly in C 0 ([0, T ]; L p (Ω)) for 1 ≤ p < 6, (4.4) → 0 (at first only for a suitable subsequence, but then, owing to the uniqueness of the limit point, eventually for the entire sequence).The convergence properties (4.1)-(4.3) of the state variables are thus shown.In addition, it immediately follows from the mean value theorem and (2.9) that, as k → ∞, max i=1,2,3 Next, we conclude from the bounds (3.14) and (2.7) that there are a subsequence, which is again labeled by k ∈ N, and some triple (p, q, r) such that, as k → ∞, p k → p weakly-star in Z ∩ L ∞ (Q), (4.8) q k → q weakly-star in H 1 (0, T ; V * ) ∩ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ), (4.9) r k → r weakly-star in Z ∩ L ∞ (Q).all weakly in L 2 (Q).
At this point, we consider the time-integrated version of the adjoint system (3.10)-(3.13)with test functions in L 2 (0, T ; V ), written for ϕ k , p k , q k , r k , k ∈ N. Passage to the limit as k → ∞, using the above convergence results, immediately leads to the conclusion that (p, q, r) solves the time-integrated version of (3.10)-(3.13)with test functions in L 2 (0, T ; V ), which is equivalent to saying that (p, q, r) is a solution to (3.10)- (3.13).By the uniqueness of this solution, we must have (p, q, r) = (p * , q * , r * ).The convergence properties (4.4)-(4.6)are therefore valid for a suitable subsequence, and since the limit is uniquely determined, also for the entire sequence.Proof.We have, with u k = (u k 1 , u k 2 ) and v k = (v k 1 , v k 2 ), Owing to Lemma 4.
where (p k , q k , r k ) and (p * , q * , r * ) are the associated adjoint states.By Lemma 4.1 and its proof, the convergence properties both strongly in L 1 (Q).It remains to show that, as k → ∞, Since q k → q * weakly in L 2 (Q) by (4.9), it thus suffices to show that F (3) However, this is a simple consequence of (4.7) and (4.16).The assertion is thus proved.