On the Nonuniqueness and Instability of Solutions of Tracking-Type Optimal Control Problems
aa r X i v : . [ m a t h . O C ] J u l Manuscript submitted to doi:10.3934/xx.xx.xx.xxAIMS’ JournalsVolume X , Number , XX pp. X–XX
ON THE NONUNIQUENESS AND INSTABILITY OF SOLUTIONSOF TRACKING-TYPE OPTIMAL CONTROL PROBLEMS
Constantin Christof ∗ Department of Mathematics, Technische Universität München,Boltzmannstr. 3, 85748 Garching b. München, Germany
Dominik Hafemeyer
Department of Mathematics, Technische Universität München,Boltzmannstr. 3, 85748 Garching b. München, Germany (Communicated by the associate editor name)
Abstract.
We study tracking-type optimal control problems that involve anon-affine, weak-to-weak continuous control-to-state mapping, a desired state y d , and a desired control u d . It is proved that such problems are alwaysnonuniquely solvable for certain choices of the tuple ( y d , u d ) and instable inthe sense that the set of solutions (interpreted as a multivalued function of ( y d , u d ) ) does not admit a continuous selection. Introduction.
This paper is concerned with the uniqueness and the stabilityof solutions of tracking-type optimal control problems of the form min ( y,u ) ∈ Y × U k y − y d k pY + k u − u d k pU s.t. y = S ( u ) . (P)Our standing assumptions on the quantities in (P) are as follows: Assumption 1.1. (i) ( Y, k·k Y ) and ( U, k·k U ) are uniformly convex, uniformly smooth Banach spaces,(ii) p ∈ (1 , ∞ ) is arbitrary but fixed,(iii) y d ∈ Y and u d ∈ U are given,(iv) S : U → Y is a function that is not affine-linear and satisfies u n n →∞ −−−− ⇀ u in U = ⇒ S ( u n ) n →∞ −−−− ⇀ S ( u ) in Y. Here, the symbol “ ⇀ ” denotes weak convergence. Due to their simple structure and since they allow to easily construct situationswith known analytic solutions (just choose u d := ¯ u and y d := S (¯ u ) for some given ¯ u ∈ U ), tracking-type optimal control problems of the form (P) are considered veryfrequently in the literature - in particular in the case where the exponent p is equalto two and the spaces Y and U are Hilbert. Compare, for instance, with [1, 4, 8, 11,14, 15, 16] and the tangible examples in Section 2 in this context. Very recently, it Mathematics Subject Classification.
Key words and phrases. optimal control, nonuniqueness, global solutions, nonlinear operators.This research was conducted within the International Research Training Group IGDK 1754,funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF) underproject number 188264188/GRK1754. ∗ Corresponding author: Constantin Christof. was demonstrated in [24] by means of an explicit construction for a boundary controlproblem with u d = 0 governed by a semilinear elliptic partial differential equationthat problems of the type (P) can possess multiple global solutions. The aim of thisbrief note is to point out that tracking-type optimal control problems which involvea desired state y d , a desired control u d , and a non-affine, weak-to-weak continuouscontrol-to-state map S : u y are indeed always nonuniquely solvable for certainchoices of the tuple ( y d , u d ) - regardless of whether the control-to-state operatorarises from a partial differential equation, a variational inequality, a differentialinclusion or something else. We further demonstrate that the same effects that areresponsible for this nonuniqueness of solutions also cause the problem (P) to beinstable in the sense that the set of solutions of (P) (interpreted as a multivaluedmap of ( y d , u d ) ) does not admit a continuous selection. For the main results of thisnote, we refer the reader to Theorems 2.2 and 2.3.Although, at the end of the day, just consequences of classical results from non-linear approximation theory and a simple identification with a metric projection, webelieve that the observations made in this paper are of sufficient interest to justifypointing them out and making them available in a tangible format - in particulardue to their very general nature and their potential consequences for, e.g., the studyof turnpike properties, cf. the discussion in [24] and the references therein. We re-mark that, for the special case of trajectory control problems, arguments analogousto those in this note have already been used in [10, 21, 27].The organization of the remainder of this paper (i.e., Section 2) is as follows:In Proposition 2.1, we briefly check that the problem (P) possesses at least oneglobal solution for every choice of the tuple ( y d , u d ) ∈ Y × U . The subsequentTheorem 2.2 then establishes our first main result - the nonuniqueness of solutionsof (P) for certain choices of the desired state and the desired control. Afterwards,in Theorem 2.3, we demonstrate that the solution set of (P) indeed does not admita selection which depends continuously on the problem data ( y d , u d ) . The paperconcludes with some additional comments and tangible examples in Remark 2.4and Examples 2.5 to 2.8.2. Nonuniqueness and Instability of Solutions.
Before we turn our attentionto our main observations, we note the following:
Proposition 2.1 (Existence of Global Minimizers).
In the situation of Assumption 1.1,the minimization problem (P) admits at least one global solution (¯ y, ¯ u ) ∈ Y × U forevery choice of the tuple ( y d , u d ) ∈ Y × U .Proof. The claim follows straightforwardly from the direct method of calculus ofvariations. Indeed, if we consider a minimizing sequence { ( y n , u n ) } n ∈ N ⊂ Y × U of(P), then the sequences { y n } n ∈ N ⊂ Y and { u n } n ∈ N ⊂ U are trivially bounded bythe structure of the objective function of (P), and it follows from our assumption ofuniform convexity and the theorems of Milman-Pettis and Banach-Alaoglu, see [23]and [26, Section V-2], that the spaces Y and U are reflexive and that we may extracta subsequence of { ( y n , u n ) } n ∈ N (for simplicity denoted by the same symbol) suchthat { y n } n ∈ N converges weakly in Y to some ¯ y ∈ Y and { u n } n ∈ N converges weakly in U to some ¯ u ∈ U . Note that the weak-to-weak continuity of S implies that ¯ y = S (¯ u ) has to hold. In combination with the weak lower semicontinuity of continuous and N THE NONUNIQUENESS AND INSTABILITY OF SOLUTIONS 3 convex functions, see [5, Corollary 4.1.14], it now follows immediately that inf ( y,u ) ∈ Y × U, y = S ( u ) k y − y d k pY + k u − u d k pU = lim n →∞ k y n − y d k pY + k u n − u d k pU ≥ k ¯ y − y d k pY + k ¯ u − u d k pU . This shows that (¯ y, ¯ u ) is a global solution of (P) and completes the proof.We are now in the position to prove our first main result: Theorem 2.2 (Nonuniqueness of Global Minimizers).
In the situation ofAssumption 1.1, there always exists a tuple ( y d , u d ) ∈ Y × U such that the problem (P) possesses more than one global solution.Proof. The main idea of the proof is to identify (P) with a metric projection problemonto the graph of the control-to-state mapping S , i.e., the set M := { ( S ( u ) , u ) | u ∈ U } ⊂ Y × U (1)and to subsequently invoke classical results on the convexity of Chebyshev sets. Topursue this approach, we argue by contradiction.Assume that the minimization problem (P) possesses for each tuple ( y d , u d ) ∈ Y × U precisely one global solution (¯ y, ¯ u ) ∈ Y × U . Then, the monotonicity of thefunction [0 , ∞ ) ∋ x x /p ∈ [0 , ∞ ) implies that, for every ( y d , u d ) , the uniqueglobal minimizer of (P) is also the sole solution of the problem min ( y,u ) ∈ M k ( y, u ) − ( y d , u d ) k Y × U , (2)where M is the set in (1) and where k·k Y × U is the norm on Y × U defined by k ( y, u ) k Y × U := ( k y k pY + k u k pU ) /p ∀ ( y, u ) ∈ Y × U. (3)Note that the space Y × U endowed with the norm k·k Y × U is trivially Banach, andthat [7, Theorem 1] and our assumptions on U and Y imply that ( Y × U, k·k Y × U ) is uniformly convex. Further, the space ( Y × U, k·k Y × U ) is also uniformly smooth.Indeed, from [20, Theorem 5.5.12], we obtain that the uniform smoothness of thespaces ( Y, k·k Y ) and ( U, k·k U ) is equivalent to the uniform convexity of the duals ( Y ∗ , k·k Y ∗ ) and ( U ∗ , k·k U ∗ ) , and, using a standard calculation, it is easy to checkthat the dual of ( Y × U, k·k Y × U ) is isometrically isomorphic to the product space Y ∗ × U ∗ endowed with the norm k ( y ∗ , u ∗ ) k Y ∗ × U ∗ := (cid:16) k y ∗ k p/ ( p − Y ∗ + k u ∗ k p/ ( p − U ∗ (cid:17) ( p − /p ∀ ( y ∗ , u ∗ ) ∈ Y ∗ × U ∗ . In combination with [7, Theorem 1], the above implies that ( Y ∗ × U ∗ , k·k Y ∗ × U ∗ ) is uniformly convex, and, by [20, Proposition 5.2.7 and Theorem 5.5.12], that thespace ( Y × U, k·k Y × U ) is uniformly smooth as claimed.Taking into account all of the above and the structure of the problem (2), wemay conclude that, in the considered situation and under the assumption that theproblem (P) is uniquely solvable for all ( y d , u d ) ∈ Y × U , the metric projection inthe uniformly convex and uniformly smooth Banach space ( Y × U, k·k Y × U ) onto theset M defined in (1) is well-defined and single-valued everywhere. In other words, M is a Chebyshev subset of ( Y × U, k·k Y × U ) in the sense of [18, Section 0]. Fromthe weak-to-weak continuity of the control-to-state mapping S , we further obtainthat every sequence { ( y n , u n ) } n ∈ N ⊂ M that converges weakly in Y × U to some (˜ y, ˜ u ) has to satisfy ˜ y n →∞ ↼ −−−− y n = S ( u n ) n →∞ −−−− ⇀ S (˜ u ) . C. CHRISTOF AND D. HAFEMEYER
The set M is thus not only Chebyshev but also weakly closed and we may invoke[18, Corollary 4.2] to deduce that M has to be convex, i.e., we have λ ( y , u ) + (1 − λ )( y , u ) = ( λS ( u ) + (1 − λ ) S ( u ) , λu + (1 − λ ) u ) ∈ M (4)for all λ ∈ [0 , and all ( y , u ) , ( y , u ) ∈ M . Due to the definition of M , (4) canonly be true if S ( λu + (1 − λ ) u ) = λS ( u ) + (1 − λ ) S ( u ) (5)holds for all λ ∈ [0 , and all u , u ∈ U . This property, however, implies incombination with our assumptions on S that the map L ( · ) := S ( · ) − S (0) is linearand continuous as a function from U to Y . Indeed, for every arbitrary but fixed u ∈ U , (5) yields L ( αu ) = S ( αu + (1 − α )0) − S (0) = αS ( u ) − αS (0) = αL ( u ) ∀ α ∈ [0 , and αL ( u ) = αL (cid:18) α αu (cid:19) = L ( αu ) ∀ α ∈ (1 , ∞ ) . From these equations, it readily follows that L ( u + u ) = S (cid:18)
12 (2 u ) + 12 (2 u ) (cid:19) − S (0) = 12 S (2 u ) + 12 S (2 u ) − S (0)= 12 L (2 u ) + 12 L (2 u ) = L ( u ) + L ( u ) ∀ u , u ∈ U. In particular, L ( − u ) = − L ( u ) for all u ∈ U , and we may conclude that L ( αu + u ) = L ( αu ) + L ( u ) = αL ( u ) + L ( u ) ∀ u , u ∈ U ∀ α ∈ R . The function L : U → Y is thus linear as claimed and, since the weak closednessof the set M immediately yields the closedness of the graph of L in Y × U , alsocontinuous by the closed graph theorem, see, e.g., [26, Section II-6].In summary, we now arrive at the conclusion that the map S has to be an affine-linear function. This contradicts our standing assumptions and establishes that (P)cannot possess precisely one solution for all ( y d , u d ) ∈ Y × U . As we already knowthat (P) possesses at least one solution for each ( y d , u d ) by Proposition 2.1, theassertion of the theorem now follows immediately.Next, we address the issue of instability: Theorem 2.3 (Nonexistence of a Continuous Selection of Minimizers).
In the situation of Assumption 1.1, there always exist a tuple ( y d , u d ) ∈ Y × U ,sequences { ( y d,n , u d,n ) } n ∈ N ⊂ Y × U and { ( y ′ d,n , u ′ d,n ) } n ∈ N ⊂ Y × U , and elements (¯ y, ¯ u ) ∈ Y × U and (¯ y ′ , ¯ u ′ ) ∈ Y × U such that the following is true:(i) { ( y d,n , u d,n ) } n ∈ N and { ( y ′ d,n , u ′ d,n ) } n ∈ N converge strongly in Y × U to ( y d , u d ) ,(ii) (¯ y, ¯ u ) is the unique solution of (P) with data ( y d,n , u d,n ) for all n ∈ N , i.e., { (¯ y, ¯ u ) } = arg min ( y,u ) ∈ Y × U, y = S ( u ) k y − y d,n k pY + k u − u d,n k pU ∀ n ∈ N , (iii) (¯ y ′ , ¯ u ′ ) is the unique solution of (P) with data ( y ′ d,n , u ′ d,n ) for all n ∈ N , i.e., { (¯ y ′ , ¯ u ′ ) } = arg min ( y,u ) ∈ Y × U, y = S ( u ) (cid:13)(cid:13) y − y ′ d,n (cid:13)(cid:13) pY + (cid:13)(cid:13) u − u ′ d,n (cid:13)(cid:13) pU ∀ n ∈ N , (iv) (¯ y, ¯ u ) = (¯ y ′ , ¯ u ′ ) . N THE NONUNIQUENESS AND INSTABILITY OF SOLUTIONS 5
Proof.
In the considered situation, we obtain from exactly the same arguments asin the proof of Theorem 2.2 that (P) is equivalent to the projection problem (2)and from Theorem 2.2 itself that there exists a tuple ( y d , u d ) ∈ Y × U such that(P) (and thus also (2)) possesses two nonidentical global solutions (¯ y, ¯ u ) ∈ Y × U and (¯ y ′ , ¯ u ′ ) ∈ Y × U . Define ( y d,t , u d,t ) := t (¯ y, ¯ u ) + (1 − t )( y d , u d ) ∀ t ∈ (0 , and ( y ′ d,t , u ′ d,t ) := t (¯ y ′ , ¯ u ′ ) + (1 − t )( y d , u d ) ∀ t ∈ (0 , . Then, the uniform convexity of the space ( Y × U, k·k Y × U ) (with k·k Y × U defined asin (3), see again [7, Theorem 1]) and exactly the same calculations as in the proofof [17, Theorem 2.1] yield that { (¯ y, ¯ u ) } = arg min ( y,u ) ∈ M k ( y, u ) − ( y d,t , u d,t ) k Y × U ∀ t ∈ (0 , and { (¯ y ′ , ¯ u ′ ) } = arg min ( y,u ) ∈ M (cid:13)(cid:13) ( y, u ) − ( y ′ d,t , u ′ d,t ) (cid:13)(cid:13) Y × U ∀ t ∈ (0 , holds, where M is the set in (1). To establish the assertion of the theorem, it nowsuffices to choose an arbitrary sequence { t n } n ∈ N ⊂ (0 , with t n → , to define ( y d,n , u d,n ) := ( y d,t n , u d,t n ) and ( y ′ d,n , u ′ d,n ) := ( y ′ d,t n , u ′ d,t n ) for all n ∈ N , and toagain exploit the equivalence between the problems (P) and (2).Some remarks regarding the last two results are in order: Remark 2.4. (i) The assumption that both the desired state y d and the desired control u d canbe chosen at will in Theorem 2.2 cannot be dropped. If, e.g., u d is fixed tobe zero, then it is perfectly possible that a problem of the type (P) is uniquelysolvable for all y d ∈ Y even if the control-to-state mapping S is non-affine.An example of such a configuration can be found in [9, Corollary 5.3] .(ii) The nonuniqueness of global minimizers in Theorem 2.2 implies that numeri-cal solution algorithms for problems of the type (P) may produce sequences ofiterates with several accumulation points and that termination criteria whichconsider the distance between successive iterates cannot be expected to reliablydetect stationarity. The instability of the solutions in Theorem 2.3 furthershows that numerical errors and small inaccuracies in the problem data mayprevent a proper identification of a global optimum.(iii) Theorem 2.3 shows that, in the situation of Assumption 1.1, every function F : Y × U → U with the property F ( y d , u d ) ∈ arg min u ∈ U k S ( u ) − y d k pY + k u − u d k pU ∀ ( y d , u d ) ∈ Y × U is discontinuous. There thus does not exist a continuous selection from the setof optimal controls of (P) (in the sense of set-valued analysis). Theorem 2.3further illustrates that, in the presence of nonlinearity, adding a Tikhonov-typeregularization term to an objective function may fail to properly regularize aninverse problem. We conclude this paper with some tangible examples of problems that are coveredby Theorems 2.2 and 2.3. (Note that the following list is far from exhaustive.)
C. CHRISTOF AND D. HAFEMEYER
Example 2.5 (Finite-Dimensional Tracking-Type Problems).
Consider afinite-dimensional optimization problem of the form min y ∈ R l , u ∈ R m
12 ( y − y d ) T A ( y − y d ) + ν u − u d ) T B ( u − u d ) s.t. y = S ( u ) (6) with some l, m ∈ N , an arbitrary but fixed Tikhonov parameter ν > , symmetricpositive definite matrices A ∈ R l × l and B ∈ R m × m , vectors y d ∈ R l and u d ∈ R m ,and a non-affine, continuous mapping S : R m → R l . Then, by defining Y := R l , k y k Y := (cid:18) y T Ay (cid:19) / , U := R m , k u k U := (cid:16) ν u T Bu (cid:17) / , p := 2 , we can recast (6) as a problem of the form (P) that satisfies all of the conditions inAssumption 1.1 (as one may easily check). Theorems 2.2 and 2.3 are thus applicableto (6) , and we may deduce that there exist choices of the tuple ( y d , u d ) for which (6) possesses more than one global solution and that the solution set of (6) does notadmit a continuous selection. Note that problems of the type (6) arise very frequentlyin optimal control when a continuous tracking-type problem is discretized, e.g., bymeans of the finite element method, cf. [8, Section 5.1] and [12, Sections 4.3, 5.3] . Example 2.6 (Optimal Control of a Nonsmooth Semilinear Elliptic PDE).
Consider an optimal control problem of the form min 12 k y − y d k L (Ω) + ν k u − u d k L (Ω) w . r . t . y ∈ H (Ω) , u ∈ L (Ω) , s . t . − ∆ y + max(0 , y ) = u in Ω , (7) where Ω ⊂ R m , m ∈ N , is a bounded domain, y d ∈ L (Ω) and u d ∈ L (Ω) are given, ν > is an arbitrary but fixed Tikhonov parameter, L (Ω) and H (Ω) are defined asin [2] , ∆ is the distributional Laplacian, and the function max(0 , · ) : R → R acts asa Nemytskii operator. Then, it follows from [8, Proposition 2.1, Corollary 3.8] that (7) possesses a well-defined and weak-to-weak continuous control-to-state mapping S : L (Ω) → L (Ω) , u y . Further, the map S is also non-affine. Indeed, if wechoose an arbitrary but fixed z ∈ H (Ω) ∩ H (Ω) that is positive almost everywherein Ω (such a z exists by [8, Lemma A.1] ) and if we define u := 2( − ∆ z + z ) ∈ L (Ω) and u := 2∆ z ∈ L (Ω) , then we clearly have S ( u ) = 2 z , S ( u ) = − z , and S ( u ) + 12 S ( u ) = 0 = S ( z ) = S (cid:18) u + 12 u (cid:19) , where the inequality = S ( z ) follows immediately from the PDE in (7) and ourassumption z > a.e. in Ω . Since (7) can be recast as a problem of the form (P) (with Y := L (Ω) , U := L (Ω) , p := 2 , and appropriately rescaled norms) andsince Hilbert spaces are trivially uniformly convex and uniformly smooth, we maynow conclude that the optimal control problem (7) satisfies all of the conditions inAssumption 1.1. Theorems 2.2 and 2.3 are thus applicable and it follows that (7) is not uniquely solvable for certain choices of the tuple ( y d , u d ) ∈ L (Ω) × L (Ω) and that the solution set of (7) does not admit a continuous selection. Note thatthe above setting is precisely the one considered in [8] . N THE NONUNIQUENESS AND INSTABILITY OF SOLUTIONS 7
Example 2.7 ( L p -Boundary Control for a Signorini-Type VI). Consider anoptimal control problem of the form min 1 p k y − y d k pL p (Ω) + νp k u − u d k pL p ( ∂ Ω) w . r . t . y ∈ H (Ω) , u ∈ L p ( ∂ Ω) , s . t . y ∈ K, Z Ω ∇ y · ∇ ( v − y ) + y ( v − y )d x ≥ Z ∂ Ω u ( v − y )d s ∀ v ∈ K, (8) where Ω ⊂ R m , m ∈ N , is a bounded Lipschitz domain with boundary ∂ Ω , y d ∈ L p (Ω) and u d ∈ L p ( ∂ Ω) are given, ν > is an arbitrary but fixed Tikhonov parameter, p isan exponent that satisfies p ∈ [2 , ∞ ) for m ≤ and p ∈ [2 , m/ ( m − for m ≥ , L p ( ∂ Ω) , L p (Ω) , and H (Ω) are defined as in [2] , ∇ is the weak gradient, and K is the set of all elements of H (Ω) whose trace is nonnegative a.e. on ∂ Ω . Then,using [19, Theorem II-2.1] , the Sobolev embeddings, see [22, Theorem 2-3.4] , and thecompactness of the trace operator, see [22, Theorem 2-6.2] , it is easy to check thatthe elliptic variational inequality in (8) possesses a well-defined and weak-to-weakcontinuous solution operator S : L p ( ∂ Ω) → H (Ω) ֒ → L p (Ω) , u y . To see thatthis S is non-affine, we note that, for every a.e.-positive control u ∈ L p ( ∂ Ω) , thetrace of S ( u ) has to be positive a.e. on a set of positive surface measure. Indeed,if the latter was not the case for an a.e.-positive control u , then the variational in-equality in (8) and the inclusion H (Ω) ⊂ K would imply that y = S ( u ) ∈ H (Ω) is also the solution of − ∆ y + y = 0 in Ω , y = 0 on ∂ Ω . This, however, would yield y = 0 and, as a consequence, ≥ Z ∂ Ω uv d s = Z ∂ Ω | uv | d s ∀ v ∈ K which is a contradiction. The trace of S ( u ) thus has to be positive on a non-negligiblesubset of ∂ Ω for all a.e.-positive u ∈ L p ( ∂ Ω) as claimed. Since we trivially have S (0) = 0 and since S ( u ) has to be an element of K for all u by the definition of S ,it now follows immediately that S ( u ) + S ( − u ) = S (0) holds for all u ∈ L p ( ∂ Ω) thatare positive a.e. on ∂ Ω . In combination with our previous observations on S andthe fact that L q -spaces are uniformly convex and uniformly smooth for < q < ∞ (see [20, Theorems 5.2.11, 5.5.12] ), this shows that (8) satisfies the conditions inAssumption 1.1 (with Y := L p (Ω) , U := L p ( ∂ Ω) , and again appropriately rescalednorms). We may thus again invoke Theorems 2.2 and 2.3 to deduce that (8) is notuniquely solvable for certain tuples ( y d , u d ) ∈ L p (Ω) × L p ( ∂ Ω) and that the solutionset of (8) does not admit a continuous selection. Example 2.8 (Distributed Control of the Parabolic Obstacle Problem).
Consider an optimal control problem of the form min 12 k y ( T ) − y d k L (Ω) + ν k u − u d k L (0 ,T ; L ( D )) w . r . t . u ∈ L (0 , T ; L ( D )) , y ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) , s . t . y ( t ) ≥ ψ a.e. in Ω for a.a. t ∈ (0 , T ) , y (0) = 0 a.e. in Ω , Z T h ∂ t y − ∆ y − Bu, v − y i d t ≥ ∀ v ∈ L (0 , T ; H (Ω)) , v ( t ) ≥ ψ a.e. in Ω for a.a. t ∈ (0 , T ) , (9) C. CHRISTOF AND D. HAFEMEYER where Ω ⊂ R m , m ∈ N , is a bounded domain, D is an open, non-empty subset of Ω , T > is a given final time, ν > is an arbitrary but fixed Tikhonov parameter, theappearing Lebesgue-, Sobolev-, and Bochner spaces are defined as in [2] and [13] , y d ∈ L (Ω) and u d ∈ L (0 , T ; L ( D )) are given, ψ ∈ L (Ω) is a given function thatsatisfies ψ ≤ a.e. in Ω , ∂ t is the time derivative in the Sobolev-Bochner sense, ∆ isthe distributional Laplacian, B denotes the canonical embedding of L (0 , T ; L ( D )) into L (0 , T ; L (Ω)) , and h· , ·i denotes the dual pairing in H (Ω) . Then, using [3, Theorem 1.13, Equation (1.70)] , [6, Theorem 2.3] , and the lemma of Aubin-Lions, see [25, Theorem 10.12] , it is easy to check that the evolution variationalinequality in (9) possesses a well-defined weak-to-weak continuous solution map G : L (0 , T ; L ( D )) → H (0 , T ; L (Ω)) , u y . (Note that, in order to apply [3,Theorem 1.13] , one has to define the function Φ appearing in this theorem as in [3, Equation (4.9)] .) As H (0 , T ; L (Ω)) embeds continuously into C ([0 , T ]; L (Ω)) by [25, Theorem 10.9] , the above implies in particular that (9) possesses a well-defined weak-to-weak continuous control-to-state (or, in this context, more preciselycontrol-to-observation) operator S : L (0 , T ; L ( D )) → L (Ω) , u G ( u )( T ) , where G ( u )( T ) denotes the value of the C ([0 , T ]; L (Ω)) -representative of G ( u ) at the finaltime T . To see that the map S is non-affine, we proceed similarly to Examples 2.6and 2.7. Suppose that E is an open, non-empty set whose closure is contained in D ,and that ε ∈ (0 , T ) is fixed. Then, it follows from [8, Lemma A.1] that there existsa function z ∈ C ∞ c ((0 , T ] × Ω) that is positive in ( ε, T ] × E and zero everywherein (0 , T ] × Ω \ ( ε, T ] × E . If, for such a z , we define ˜ u := ( ∂ t z − ∆ z ) | (0 ,T ) × D ,where the vertical bar denotes a restriction, then it clearly holds S (˜ u ) = z ( T ) > a.e. in E . From the C ([0 , T ]; L (Ω)) -regularity and the properties of the solu-tions of the evolution variational inequality in (9) and the closedness of the set { v ∈ L (Ω) | v ≥ ψ a.e. in Ω } in L (Ω) , we further obtain that S ( α ˜ u ) ≥ ψ has tohold a.e. in Ω for all α ∈ R . In combination with the trivial identity S (0) = 0 and z ( T ) > a.e. in E , it now follows immediately that ψ ≤ S ( α ˜ u ) = S ( α ˜ u ) − S (0) = α ( S (˜ u ) − S (0)) = αS (˜ u ) = αz ( T ) cannot be true a.e. in Ω for all α ∈ R . This shows that the map S is indeednon-affine in the situation of (9) . In summary, we may now again conclude that (9) satisfies all of the conditions in Assumption 1.1 (with p := 2 , Y := L (Ω) , U := L (0 , T ; L ( D )) , and appropriately rescaled norms). Theorems 2.2 and 2.3thus apply to (9) , and we obtain that this optimal control problem is not uniquelysolvable for certain choices of the tuple ( y d , u d ) ∈ L (Ω) × L (0 , T ; L ( D )) and thatthe solution set of (9) does not admit a continuous selection. Acknowledgments.
We would like to thank Gerd Wachsmuth for making us awareof the concept of Chebyshev set.
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