Analysis of optimal control problems of semilinear elliptic equations by BV-functions
aa r X i v : . [ m a t h . O C ] O c t ANALYSIS OF OPTIMAL CONTROL PROBLEMS OF SEMILINEARELLIPTIC EQUATIONS BY BV-FUNCTIONS ∗ EDUARDO CASAS † AND
KARL KUNISCH ‡ Abstract.
Optimal control problems for semilinear elliptic equations with control costs in thespace of bounded variations are analysed. BV-based optimal controls favor piecewise constant, andhence ’simple’ controls, with few jumps. Existence of optimal controls, necessary and sufficientoptimality conditions of first and second order are analysed. Special attention is paid on the effectof the choice of the vector norm in the definition of the BV-seminorm for the optimal primal andadjoined variables.
AMS subject classifications.
Key words. optimal control, bounded variation functions, sparsity, first and second orderoptimality conditions, semilinear elliptic equations
1. Introduction.
This paper is dedicated to the study of the optimal controlproblem(P) min u ∈ BV ( ω ) J ( u ) = 12 Z Ω | y − y d | dx + α Z ω |∇ u | + β (cid:16) Z ω u ( x ) dx (cid:17) + γ Z ω u ( x ) dx, where y is the unique solution to the Dirichlet problem (cid:26) − ∆ y + f ( x, y ) = uχ ω in Ω ,y = 0 on Γ . (1.1)The control domain ω is an open subset of Ω. We assume that α > β ≥ γ ≥ y d ∈ L (Ω), and Ω is a bounded domain in R n , n = 2 or 3, with Lipschitz boundaryΓ. Additionally we make the following hypothesis:if n = 3 , then γ > BV ( ω ) denotes the space of functions of bounded variation in ω and R ω |∇ u | stands for the total variation of u . The assumptions on the nonlinear term f ( x, y ) inthe state equation will be formulated later. By introducing the penalty term involvingthe mean of u when β > γ = 0, in dependence on the order of the nonlinearity f it can be necessary to choose β > y = y ( u )which is as close to y d as possible. Comparing with the common formulation of using L ( ω ) or L p ( ω ) control-cost functionals, with p > f , theselater functionals will produce smooth optimal controls which may be more intricate ∗ The first author was supported by Spanish Ministerio de Econom´ıa y Competitividad underproject MTM2014-57531-P. The second was supported by the ERC advanced grant 668998 (OCLOC)under the EUs H2020 research program. † Departamento de Matem´atica Aplicada y Ciencias de la Computaci´on, E.T.S.I. Industriales y deTelecomunicaci´on, Universidad de Cantabria, 39005 Santander, Spain ([email protected]). ‡ Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria ([email protected]). 1
E6. CASAS AND K. KUNISCH to realize in practice than controls which result from the BV − formulation. Piecewiseconstant behavior of the optimal controls can also be obtained by introducing bilat-eral bounds a ≤ u ( x ) ≤ ¯ b together with only the tracking term in (P). In this case wecan expect optimal controls which exhibit bang-bang structure. If an L ( ω ) controlcost term is added then the optimal control will be of the form bang-zero-bang. Butthis type of behavior is distinctly different from that which is allowed in (P), since thevalue of the piecewise constants plateaus is not prescribed. This is distinctly differentfrom the bilaterally constraint case where the optimal control typically assumes oneof the extreme values a or ¯ b . This in turn can lead to unnecessarily high control costs.Possibly one of the first papers where this was pointed out, but not systematicallyinvestigated is [15]. In [9] semilinear parabolic equations with temporally dependentBV-functions as controls were investigated. Thus we were focusing on controls whichare optimally switching in time. The analysis for this case is simpler and exploitsspecific properties of BV-functions in dimension one. Numerically the simple structureof the controls which is obtained for BV-constrained control problems was alreadydemonstrated in [5, 9] and a recent master thesis [19]. BV-seminorm control costsare also employed in [8], where the control appears as coefficient in the p -Laplaceequation. Beyond these papers the choice of the control costs related to BV-norms orBV-seminorms has not received much attention in the optimal control literature yet.In mathematical image analysis, to the contrary, the BV-seminorm has receiveda tremendous amount of attention. The beginning of this activity is frequently datedto [22]. Let us also mention the recent paper [2] which gives interesting insight intothe structure of the subdifferential of the BV-seminorm. Fine properties of BV-functions, in the context of image reconstruction problems, in particular the staircasing effect were, analyzed for the one-dimensional case in [21], and in higher dimen-sions in [20, 14], for example. In [13] the authors provided a convergence analysisfor BV-regularized mathematical imaging problems by finite elements, paying specialattention to the choice of the vector norm in the definition of the BV-seminorm.Let us also compare the use of the BV-term in (P) with the efforts that havebeen made for studying optimal control problems with sparsity constraints. Theseformulations involve either measure-valued norms of the control or L -functionalscombined with pointwise constraints on the control. We cite [5, 7] from among themany results which are now already available. The BV-seminorm therefore can alsobe understood as a sparsity constraint for the first derivative.Let us briefly describe the structure of the paper. Section 2 contains an analysisof the state equation and the smooth part of the cost-functional. The non-smoothpart of the cost-functional is investigated in Section 3. Special attention is given tothe consequences which arise from the specific choice which is made for the vectornorm in the variational definition of the BV-seminorm. In particular, we considerthe Euclidean and the infinity norms. Existence of optimal solutions and first orderoptimality conditions are obtained in Section 4. Second order sufficient optimalityconditions are provided in Section 5. Finally in Section 6 we consider (P) with anadditional H ( ω ) regularisation term and investigate the asymptotic behavior as theweight of the H ( ω ) regularisation tends to 0.
2. Analysis of the state equation and the cost functional.
We recall thata function u ∈ L ( ω ) is a function of bounded variation if its distributional derivatives ∂ x i u , 1 ≤ i ≤ n , belong to the Banach space of real and regular Borel measures M ( ω ). ptimal Control by BV-Functions µ ∈ M ( ω ), its norm is given by k µ k M ( ω ) = sup { Z ω z dµ : z ∈ C ( ω ) and k z k C ( ω ) ≤ } = | µ | ( ω ) , where C ( ω ) denotes the Banach space of continuous functions z : ¯ ω −→ R such that z = 0 on ∂ω , and | µ | is the total variation measure associated with µ . On the productspace M ( ω ) n we define the norm k µ k M ( ω ) n = sup { Z ω z dµ : z ∈ C ( ω ) n and | z ( x ) | ≤ ∀ x ∈ ω } , where | · | is a norm in R n .On BV ( ω ) we consider the usual norm k u k BV ( ω ) = k u k L ( ω ) + k∇ u k M ( ω ) n , that makes BV ( ω ) a Banach space; see [1, Chapter 3] or [18, Chapter 1] for details.We recall that the total variation of u is given by k∇ u k M ( ω ) n = sup { Z ω div z u dx : z ∈ C ∞ ( ω ) n and | z ( x ) | ≤ ∀ x ∈ ω } . We also use the notation Z ω |∇ u | = k∇ u k M ( ω ) n , as already employed in (P). For these topologies ∇ : BV ( ω ) −→ M ( ω ) n is a linearcontinuous mapping.In the sequel we will denote a u = 1 | ω | Z ω u ( x ) dx and ˆ u = u − a u for every u ∈ BV ( ω ) . By using [1, Theorem 3.44] it is easy to deduce that there exists a constant C ω suchthat k u k := | a u | + k∇ u k M ( ω ) n ≤ max (cid:0) , | ω | (cid:1) k u k BV ( ω ) ≤ C ω k u k . (2.1)In addition, we mention that BV ( ω ) is the dual space of a separable Banach space.Therefore every bounded sequence { u k } ∞ k =1 in BV ( ω ) has a subsequence convergingweakly ∗ to some u ∈ BV ( ω ). The weak ∗ convergence u k ∗ ⇀ u implies that u k → u strongly in L ( ω ) and ∇ u k ∗ ⇀ ∇ u in M ( ω ) n ; see [1, pages 124-125]. We will also usethat BV ( ω ) is continuously embedded in L p ( ω ) with 1 ≤ p ≤ nn − , and compactlyembedded in L p (0 , T ) for every p < nn − ; see [1, Corollary 3.49]. From this propertywe deduce that the convergence u k ∗ ⇀ u in BV ( ω ) implies that u k → u strongly inevery L p (0 , T ) for all p < nn − .We make the following assumption on the nonlinear term of the state equation.We assume that f : Ω × R −→ R is a Borel function, of class C with respect to the E6. CASAS AND K. KUNISCH last variable, and satisfies for almost all x ∈ Ω f ( · , ∈ L ˆ p (Ω) with ˆ p > n , (2.2) ∂f∂y ( x, y ) ≥ ∀ y ∈ R , (2.3) ∀ M > ∃ C M : (cid:12)(cid:12)(cid:12)(cid:12) ∂f∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C M ∀| y | ≤ M, (2.4) ∀ M > ∀ ρ > ∃ ε > (cid:12)(cid:12)(cid:12)(cid:12) ∂ f∂y ( x, y ) − ∂ f∂y ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ if | y − y | < ε and | y | , | y | ≤ M. (2.5)Let us observe that if f is an affine function, f ( x, y ) = c ( x ) y + d ( x ), then(2.2)-(2.5) hold if c ≥ c ∈ L ∞ (Ω), and d ∈ L ˆ p (Ω).By using these assumptions, the following theorem can be proved in a standardway; see, for instance, [26, § Proposition 2.1.
For every u ∈ L ˆ p ( ω ) the state equation (1.1) has a uniquesolution y u ∈ C σ ( ¯Ω) ∩ H (Ω) for some σ ∈ (0 , . In addition, for every M > thereexists a constant K M such that k y u k C σ (¯Ω) + k y u k H (Ω) ≤ K M ∀ u ∈ L ˆ p ( ω ) : k u k L ˆ p ( ω ) ≤ M. (2.6)In the sequel we will denote Y = C ( ¯Ω) ∩ H (Ω) and S : L ˆ p ( ω ) −→ Y the mappingassociating to each control u the corresponding state S ( u ) = y u . We have the followingdifferentiability property of S . Proposition 2.2.
The mapping S : L ˆ p ( ω ) −→ Y is of class C . For all elements u, v and w of L ˆ p ( ω ) , the functions z v = S ′ ( u ) v and z vw = S ′′ ( u )( v, w ) are the solutionsof the problems − ∆ z + ∂f∂y ( x, y u ) z = vχ ω in Ω ,z = 0 on Γ , (2.7) and − ∆ z + ∂f∂y ( x, y u ) z + ∂ f∂y ( x, y u ) z v z w = 0 in Ω ,z = 0 on Γ , (2.8) respectively. The proof is a consequence of the implicit function theorem. Let us give a sketch.We define the space V = { y ∈ Y : ∆ y ∈ L ˆ p (Ω) } endowed with the norm k y k V = k y k C (¯Ω)) + k y k H (Ω) + k ∆ y k L ˆ p (Ω) . ptimal Control by BV-Functions V is a Banach space. Now we introduce the mapping F : V × L ˆ p (Ω) −→ L ˆ p (Ω)by F ( y, u ) = − ∆ y + f ( x, y ) − u. From (2.4) we deduce that F is of class C and ∂ F ∂y ( y, u ) z = − ∆ z + ∂f∂y ( x, y ) z. From the monotonicity condition (2.3), we obtain that ∂ F ∂y ( y, u ) : V −→ L ˆ p (Ω) is anisomorphism. Hence, the implicit function theorem and Proposition 2.1 with ω = Ωimply the existence of a C mapping ˆ S : L ˆ p (Ω) −→ Y associating to every element u its corresponding state ˆ S ( u ) = y u . When ω Ω, we use that S = ˆ S ◦ S ω , where S ω : L ˆ p ( ω ) −→ L ˆ p (Ω) is defined by S ω u = uχ ω . Hence the chain rule leads to theresult.Next, we separate the smooth and the non smooth parts in J : J ( u ) = F ( u ) + αG ( u ) with F ( u ) = 12 Z Ω | y u − y d | dx + β (cid:0) Z ω u ( x ) dx (cid:1) + γ Z ω u ( x ) dx and G ( u ) = g ( ∇ u ) , where g : M ( ω ) n −→ R is given by g ( µ ) = k µ k M ( ω ) n . In the rest of this section westudy the differentiability of F . From Proposition 2.2 and the chain rule the followingproposition can be obtained. Proposition 2.3.
The functional F : L ( ω ) −→ R is of class C . The deriva-tives of F are given by F ′ ( u ) v = Z ω h ϕ u ( x ) + γu ( x ) + β (cid:0) Z ω u ( s ) ds (cid:1)i v ( x ) dx, (2.9) and F ′′ ( u )( v, w ) = Z Ω (cid:0) − ϕ u ∂ f∂y ( x, y u ) (cid:1) z v z w dx + γ Z ω vw dx + β (cid:16) Z ω v dx (cid:17)(cid:16) Z ω w dx (cid:17) (2.10) with z v = S ′ ( u ) v , z w = S ′ ( u ) w , and ϕ u ∈ Y the adjoint state which satisfies − ∆ ϕ u + ∂f∂y ( x, y u ) ϕ u = y u − y d in Ω ,ϕ u = 0 on Γ . (2.11)The C ( ¯Ω) regularity of ϕ u follows from the assumptions on y d ∈ L (Ω) and thefact that y u ∈ L ∞ (Ω). Remark If n = 2 , since BV ( ω ) is embedded in L ( ω ) , then the functional F : BV ( ω ) −→ R is well defined and it is of class C with derivatives given by (2.9) and (2.10) . However, if n = 3 , then BV ( ω ) is only embedded in L / ( ω ) . Hence, forelements u ∈ BV ( ω ) Proposition 2.1 is not applicable and, therefore, the functional F is not defined in BV ( ω ) . To deal with the case n = 3 we introduced the assumption (1.2) , i.e. γ > . Hence, the functional F : BV ( ω ) ∩ L ( ω ) −→ R is well defined andof class C . E6. CASAS AND K. KUNISCH
The assumption (1.2) can be avoided if we suppose that the nonlinearity f ( x, y ) has only polynomial growth of arbitrary order in y . In this case, Propositions 2.1 and2.2 hold if we change Y to Y q = L q (Ω) ∩ H (Ω) with q < ∞ arbitrarily big. We recallthat for a right hand side of the state equation belonging to L / (Ω) the solution ofthe state equation does not belong to L ∞ (Ω) , in general, even for linear equations.However, since L / (Ω) ⊂ W − , (Ω) , we can use [25, Theorem 4.2] to deduce that y u ∈ L q (Ω) ∀ q < ∞ . To analyze the semilinear case one can follow the classicalapproach of truncation of the nonlinear term, Schauder’s fix point theorem, and L q -estimates from the linear case combined with the monotonicity of the nonlinear term.Finally, since γ = 0 , we have that the functional F : BV ( ω ) −→ R is of class C .Remark In the state equation, the Laplace operator − ∆ can be replaced by amore general linear elliptic operator with bounded coefficients. All the results provedin this paper hold for these general operators.
3. Analysis of the functional G . Now, we analyze the functional G . Wealready expressed G as the composition G = g ◦ ∇ . Concerning the functional g ,we note that it is Lipschitz continuous and convex. Hence, it has a subdifferentialand a directional derivative, which are denoted by ∂g ( µ ) and g ′ ( µ ; ν ), respectively.Before giving an expression for g ′ ( µ ; ν ), we have to specify the norm that we use in R n . Indeed, in the definition of the norm k µ k M ( ω ) n we have considered a genericnorm | · | in R n . The choice of the specific norm strongly influences the structure ofthe optimal controls. In this paper, we focus on the Euclidean and the | · | ∞ norms,which lead to different properties for g , that we consider separately in the followingtwo subsections. To illustrate one aspect, let us observe that the use of the | · | ∞ normon R n in the definition of k · k M ( ω ) n implies that k µ k M ( ω ) n = n X j =1 k µ j k M ( ω ) ∀ µ ∈ M ( ω ) n . (3.1)In particular, it holds that Z ω |∇ u | = n X j =1 k ∂ x j u k M ( ω ) ∀ u ∈ BV ( ω ) . However, for the Euclidean norm we have, in general, that k µ k M ( ω ) n = (cid:16) n X j =1 k µ j k M ( ω ) (cid:17) / . (3.2)Indeed, the identity (3.1) is an immediate consequence of the definitions of the norms k·k M ( ω ) and k·k M ( ω ) n . To verify (3.2) we give an example. Let us fix n different points { ξ i } ni =1 in ω and take ε > B ε ( ξ i ) are disjoint. Now,applying Uryshon’s lemma, cf. [23, Lemma 2.12], we get functions z i ∈ C ( ω ) suchthat 0 ≤ z i ( x ) ≤ ∀ x ∈ ω , z i ( ξ i ) = 1 and supp( z i ) ⊂ B ε ( ξ i ). We set z = ( z , . . . , z n )and µ = ( δ ξ , . . . , δ ξ n ). Then, since | z ( x ) | ≤ ∀ x ∈ ω , we have k µ k M ( ω ) n ≥ n X i =1 Z ω z i ( x ) dµ i ( x ) = n X i =1 z i ( ξ i ) = n. On the other hand, we get (cid:16) n X j =1 k µ j k M ( ω ) (cid:17) / = √ n. ptimal Control by BV-Functions | · | . In order to give an expression for g ′ ( µ ; ν ), let us introduce some notation. We recall that if µ ∈ M ( ω ) n , its associatedtotal variation measure is defined as a positive scalar measure as follows | µ | ( A ) = sup n ∞ X k =1 | µ ( E k ) | : { E k } k ⊂ B are pairwise disjoint and A = ∞ [ k =1 E k o , where B is the σ -algebra of Borel sets in ω , and | µ ( E k ) | denotes the Euclidean normin R n of the vector µ ( E k ). Let us denote by h µ the Radon-Nikodym derivative of µ with respect to | µ | . Thus we have h µ ∈ L ( ω, | µ | ) , | h µ ( x ) | = 1 for | µ |− a.e. x ∈ ω and µ ( A ) = Z A h µ ( x ) d | µ | ( x ) ∀ A ∈ B . Given a second vector measure ν ∈ M ( ω ) n , the following Lebesgue decompositionholds: ν = ν a + ν s , dν a = h ν d | µ | , where ν a and ν s are the absolutely continuous andsingular parts of ν with respect to | µ | , and h ν is the Radon-Nikodym derivative of ν with respect to | µ | . Then, the following identity is fulfilled k ν k M ( ω ) n = k ν a k M ( ω ) n + k ν s k M ( ω ) n = Z ω | h ν ( x ) | d | µ | ( x ) + k ν s k M ( ω ) n . The reader is referred to [1, Chapter 1].Now, we analyze the subdifferential ∂g ( µ ). It is well known that an element λ ∈ ∂g ( µ ) if h λ, ν − µ i + k µ k M ( ω ) n ≤ k ν k M ( ω ) n ∀ ν ∈ M ( ω ) n . (3.3)This is equivalent to the next two relations h λ, µ i = k µ k M ( ω ) n , (3.4) h λ, ν i ≤ k ν k M ( ω ) n ∀ ν ∈ M ( ω ) n . (3.5)Observe that λ belongs to the dual of M ( ω ) n , which is not a distributional space. Inthe special case where λ ∈ C ( ω ) n , we can establish some precise relations between λ and µ . Before proving these relations, let us mention that here we have k z k C ( ω ) n = sup {| z ( x ) | : x ∈ ω } ∀ z ∈ C ( ω ) n . Proposition 3.1. If λ ∈ C ( ω ) n ∩ ∂g ( µ ) , then k λ k C ( ω ) n ≤ . Moreover, if µ = 0 , then the following properties hold1. k λ k C ( ω ) n = 1 , and2. supp( µ ) ⊂ { x ∈ ω : | λ ( x ) | = 1 } . Proof . The inequality k λ k C ( ω ) n ≤ µ = 0,then (3.4) implies 1. To prove 2. we use (3.4) as follows Z ω d | µ | ( x ) = k µ k M ( ω ) n = h λ, µ i = Z ω λ ( x ) dµ ( x ) = Z ω λ ( x ) · h µ ( x ) d | µ | ( x ) . Then, using that | λ ( x ) | ≤ ∀ x ∈ ω and | h µ ( x ) | = 1 | µ | -a.e. in ω we deduce fromthe identity Z ω d | µ | ( x ) = Z ω λ ( x ) · h µ ( x ) d | µ | ( x ) E6. CASAS AND K. KUNISCH that λ ( x ) · h µ ( x ) = 1 | µ | -a.e. in ω . Using again that | h µ ( x ) | = 1, | µ | -a.e., we concludethat λ ( x ) = h µ ( x ), | µ | -a.e. Therefore, we have that | µ | (cid:0) { x ∈ ω : | λ ( x ) | < } (cid:1) = 0 , which implies 2.Next we study the directional derivatives of g . Proposition 3.2.
Let µ, ν ∈ M ( ω ) n , then g ′ ( µ ; ν ) = Z ω h ν dµ + k ν s k M ( ω ) n , (3.6) where ν = ν a + ν s = h ν d | µ | + ν s is the Lebesgue decomposition of ν respect to | µ | .Proof . As above, let us write dµ = h µ d | µ | . Then we have g ′ ( µ ; ν ) = lim ρ ց k µ + ρν k M ( ω ) n − k µ k M ( ω ) n ρ = lim ρ ց k µ + ρν a k M ( ω ) n + k ρν s k M ( ω ) n − k µ k M ( ω ) n ρ = lim ρ ց ρ (cid:18)Z ω | h µ ( x ) + ρh ν ( x ) | d | µ | ( x ) − Z ω | h µ ( x ) | d | µ | ( x ) (cid:19) + k ν s k M ( ω ) n = Z ω lim ρ ց | h µ ( x ) + ρh ν ( x ) | − | h µ ( x ) | ρ d | µ | ( x ) + k ν s k M ( ω ) n = Z ω h µ ( x ) · h ν ( x ) | h µ ( x ) | d | µ | ( x ) + k ν s k M ( ω ) n = Z ω h ν dµ + k ν s k M ( ω ) n . Since the quotients are dominated by | h ν | , we applied Lebesgue’s dominated con-vergence theorem above. Moreover, we use that | h µ ( x ) | = 1 | µ | -a.e. in ω in the lastequality and also to justify the differentiability of the norm | · | at every h µ ( x ) with x in the support of | µ | .Now, we come back to the mapping G . To this end, let us recall that the adjointoperator ∇ ∗ is defined by ∇ ∗ : [ M ( ω ) n ] ∗ −→ BV ( ω ) ∗ , h∇ ∗ λ, u i BV ( ω ) ∗ ,BV ( ω ) = h λ, ∇ u i [ M ( ω ) n ] ∗ , M ( ω ) n . Proposition 3.3.
The following identities hold for all u ∈ BV ( ω ) : ∂G ( u ) = ∂ ( g ◦ ∇ )( u ) = ∇ ∗ ∂g ( ∇ u ) , (3.7) G ′ ( u ; v ) = ( g ◦ ∇ ) ′ ( u ; v ) = Z ω h v d ( ∇ u ) + k ( ∇ v ) s k M ( ω ) n , (3.8) where ∇ v = h v d |∇ u | + ( ∇ v ) s is the Lebesgue decomposition of ∇ v with respect to |∇ u | .Proof . Since ∇ : BV ( ω ) −→ M ( ω ) n is a linear and continuous mapping and g : M ( ω ) n −→ R is convex and continuous, we can apply the chain rule [16, Chapter I,Proposition 5.7] to deduce that ∂ ( g ◦ ∇ )( u ) = ∇ ∗ ∂g ( ∇ u ), which immediately leads to(3.7).To verify (3.8) it is enough to observe that( g ◦ ∇ ) ′ ( u ; v ) = g ′ ( ∇ u ; ∇ v )and to apply (3.6). ptimal Control by BV-Functions | · | ∞ norm. The use of | · | ∞ norm implies that k z k C ( ω ) n = sup {| z ( x ) | ∞ : x ∈ ω } ∀ z ∈ C ( ω ) n . We recall that every scalar real measure µ ∈ M ( ω ) admits a Jordan decomposition µ = µ + − µ − , where µ + and µ − are positive measures with disjoint supports. Further,if h µ is the Radon-Nikodym derivative of µ with respect to | µ | , then µ + = h + d | µ | and µ − = h − d | µ | , where h = h + − h − is the decomposition of h in positive and negativeparts. Proposition 3.4. If λ ∈ C ( ω ) n ∩ ∂g ( µ ) , then k λ j k C ( ω ) ≤ for all j = 1 , . . . , n .Moreover, if µ j = 0 , then the following properties hold1. k λ j k C ( ω ) = 1 , and2. supp( µ + j ) ⊂ { x ∈ ω : λ j ( x ) = +1 } and supp( µ − j ) ⊂ { x ∈ ω : λ j ( x ) = − } .Proof . Inserting (3.1) in (3.4) and (3.5) we get n X i =1 h µ i , λ i i = n X i =1 k µ i k M ( ω ) , (3.9) n X i =1 h ν i , λ i i ≤ n X i =1 k ν i k M ( ω ) ∀ ν ∈ M ( ω ) n . (3.10)Let us fix 1 ≤ j ≤ n and take in (3.10) ν i = 0 for every i = j and ν j = ± δ x with x ∈ ω arbitrary. Then, we obtain ± λ j ( x ) = h ν j , λ j i ≤ k ν j k M ( ω ) = 1 . This proves that | λ j ( x ) | ≤ ∀ x ∈ ω for every j . Now, we assume that µ j = 0. From(3.9) we infer n X i =1 k µ i k M ( ω ) = n X i =1 h µ i , λ i i ≤ n X i =1 k µ i k M ( ω ) k λ i k C ( ω ) ≤ n X i =1 k µ i k M ( ω ) . This implies that k λ i k C ( ω ) = 1 for every i such that µ i = 0. Hence, 1. holds. Thesecond part was proved in [6, Lemma 3.4].Now, we compute the directional derivatives of g ′ ( µ ; ν ). Then, we have the fol-lowing expression which is similar but different from the one obtained in Proposition3.2. Proposition 3.5.
Let µ, ν ∈ M ( ω ) n , then g ′ ( µ ; ν ) = Z ω h ν dµ + k ν s k M ( ω ) n = n X j =1 n Z ω h ν j dµ j + k ( ν j ) s k M ( ω ) o , (3.11) where ν j = ( ν j ) a + ( ν j ) s = h ν j d | µ j | + ( ν j ) s is the Lebesgue decomposition of ν j withrespect to | µ j | for ≤ j ≤ n .Proof . For the proof it is enough use (3.1) to obtain g ′ ( µ ; ν ) = lim ρ ց k µ + ρν k M ( ω ) n − k µ k M ( ω ) n ρ = n X i =1 lim ρ ց k µ i + ρν i k M ( ω ) n − k µ i k M ( ω ) n ρ . Then, we proceed as in the proof of [10, Proposition 3.3].0
E6. CASAS AND K. KUNISCH
With the same proof we infer that Proposition 3.3 is also true for the | · | ∞ normwith (3.8) being interpreted as follows G ′ ( u ; v ) = ( g ◦ ∇ ) ′ ( u ; v ) = Z ω h v d ( ∇ u ) + k ( ∇ v ) s k M ( ω ) n = n X j =1 n Z ω h v,j d ( ∂ x j u ) + k ( ∂ x j v ) s k M ( ω ) o , (3.12)where ∂ x j v = h v,j | ∂ x j u | + ( ∂ x j v ) s is the Lebesgue decomposition of ∂ x j v with respectto | ∂ x j u | .
4. Existence of an optimal control and first order optimality conditions.
The proof of the existence of an optimal control follows the lines of [9, Theorem 3.1]with the obvious modifications.
Theorem 4.1.
Let us assume that one of the following assumptions hold.1. β + γ > .2. There exist q ∈ [1 , and C > such that ∂f∂y ( x, y ) ≤ C (1 + | y | q ) for a.a. x ∈ Ω and ∀ y ∈ R . Then, problem (P) has at least one solution. Moreover, if f is affine with respect to y , the solution is unique. Now, we prove the first order optimality conditions satisfied by any local minimumof (P).
Theorem 4.2.
Let ¯ u be a local solution of (P) . Then, there exists ¯ λ ∈ ∂g ( ∇ ¯ u ) such that α h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n + Z ω (cid:16) ¯ ϕ + γ ¯ u + β Z ω ¯ u dz (cid:17) v dx = 0 ∀ v ∈ BV ( ω ) ∩ L ( ω ) , (4.1) where ¯ ϕ ∈ H (Ω) ∩ C ( ¯Ω) is the adjoint state corresponding to ¯ u .Proof . Let us denote by ¯ ϕ ∈ C ( ¯Ω) ∩ H (Ω) the adjoint state corresponding to thelocal solution ¯ u . Given v ∈ BV ( ω ) ∩ L ( ω ), from the local optimality of ¯ u and theconvexity of G we deduce for every 0 < ρ < ≤ J (¯ u + ρv ) − J (¯ u ) ρ = F (¯ u + ρv ) − F (¯ u ) ρ + α G (¯ u + ρv ) − G (¯ u ) ρ ≤ F (¯ u + ρv ) − F (¯ u ) ρ + α [ G (¯ u + v ) − G (¯ u )] . Passing to the limit as ρ → v ∈ BV ( ω )0 ≤ Z ω (cid:16) ¯ ϕ ( x ) + γ ¯ u ( x ) + β Z ω ¯ u ds (cid:17) v ( x ) dx + α [ G (¯ u + v ) − G (¯ u )] . Replacing v by u − ¯ u , this inequality can be written − α Z ω (cid:16) ¯ ϕ + γ ¯ u + β Z ω ¯ u ds (cid:17) ( u − ¯ u ) dx + G (¯ u ) ≤ G ( u ) ∀ u ∈ BV ( ω ) ∩ L ( ω ) . ptimal Control by BV-Functions − α (cid:16) ¯ ϕ + γ ¯ u + β Z ω ¯ u ds (cid:17) ∈ ∂G (¯ u ) = ∇ ∗ ∂g ( ∇ ¯ u ) . Hence, there exists ¯ λ ∈ ∂g ( ∇ ¯ u ) ⊂ [ M ( ω ) n ] ∗ such that h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n = − α Z ω h ¯ ϕ + β Z ω ¯ u ds i v dx ∀ v ∈ BV ( ω ) ∩ L ( ω ) , which implies (4.1).Since ¯ λ ∈ M ( ω ) n and M ( ω ) n is not a distribution space, sometimes it can be moreconvenient to handle a different optimality system involving distributional spaces,mainly if we think of the numerical analysis. To this end, we present the followingequivalent optimality conditions. Theorem 4.3.
Let us assume that n = 2 . Given ¯ u ∈ BV ( ω ) , let ¯ y and ¯ ϕ be theassociated state and adjoint state. Then, there exists ¯ λ ∈ ∂g ( ∇ ¯ u ) satisfying (4.1) ifand only if there exists ¯Φ ∈ C ( ω ) n such that α h∇ v, ¯Φ i M ( ω ) n ,C ( ω ) n + Z ω h ¯ ϕ + γ ¯ u + β Z ω ¯ u ds i v dx = 0 ∀ v ∈ BV ( ω ) , (4.2) h∇ v, ¯Φ i M ( ω ) n ,C ( ω ) n ≤ k∇ v k M ( ω ) n ∀ v ∈ M ( ω ) n , (4.3) h∇ ¯ u, ¯Φ i M ( ω ) n ,C ( ω ) n = k∇ ¯ u k M ( ω ) n . (4.4) Proof . Assume that ¯ λ ∈ ∂g ( ∇ ¯ u ) satisfies (4.1). We define a linear form T in M ( ω ) n as follows D ( T ) = {∇ v : v ∈ BV ( ω ) } and T ( µ ) = h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n if µ = ∇ v. From (3.4) and (3.5) we have T ( ∇ ¯ u ) = k∇ ¯ u k M ( ω ) n , (4.5) T ( µ ) ≤ k µ k M ( ω ) n ∀ µ ∈ D ( T ) . (4.6)We prove that T is weakly ∗ continuous on its domain. Let { µ k } k ⊂ D ( T ) and µ ∈ D ( T ) be such that µ k ∗ ⇀ µ in M ( ω ) n . By definition of D ( T ) there existselements { v k } k ⊂ BV ( ω ) and v ∈ BV ( ω ) such that µ k = ∇ v k and µ = ∇ v . Withoutloss of generality we assume that the integrals of each v k and v in ω are zero. Using(2.1), we know that { v k } k is bounded in BV ( ω ). From the continuity of the embedding BV ( ω ) ⊂ L ( ω ) due to n = 2 and the convergence ∇ v k ∗ ⇀ ∇ v in M ( ω ) n , we obtainthat v k ⇀ v in L ( ω ). Therefore, we get with (4.1)lim k →∞ T ( µ k ) = lim k →∞ h ¯ λ, ∇ v k i [ M ( ω ) n ] ∗ , M ( ω ) n = lim k →∞ − α Z ω h ¯ ϕ + γ ¯ u + β Z ω ¯ u ds i v k dx (4.7)= − α Z ω h ¯ ϕ + γ ¯ u + β Z ω ¯ u ds i v dx = h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n = T ( µ ) , which implies the weak ∗ continuity of T . Hence, there exists a weakly ∗ continuouslinear form T : M ( ω ) n −→ R extending T ; [24, Theorem 3.6]. In this case, we knowthat T can be identified with an element ¯Φ ∈ C ( ω ) n , i.e. T ( µ ) = h µ, ¯Φ i M ( ω ) n ,C ( ω ) n = Z ω ¯Φ dµ ∀ µ ∈ M ( ω ) n ;2 E6. CASAS AND K. KUNISCH see [3, Proposition 3.14]. The function ¯Φ fulfills (4.2)–(4.4). Indeed, (4.2) follows fromthe definition of T and (4.1), and (4.3)-(4.4) are the same as (4.5)-(4.6).Reciprocally, assume that ¯Φ ∈ C ( ω ) n satisfies (4.2)–(4.4). This time we definethe linear operator D ( T ) = {∇ v : v ∈ BV ( ω ) } and T ( µ ) = h∇ v, ¯Φ i M ( ω ) n ,C ( ω ) n if µ = ∇ v. From (4.3) we know that T is a continuous operator in D ( T ) for the strong topologyof M ( ω ) n , and k T k [ M ( ω ) n ] ∗ ≤
1. Hence, the Hahn-Banach theorem implies theexistence of an operator ¯ λ ∈ [ M ( ω ) n ] ∗ extending T and such that k ¯ λ k [ M ( ω ) n ] ∗ ≤ h ¯ λ, ∇ ¯ u i = k∇ ¯ u k M ( ω ) n , h ¯ λ, ν i ≤ k ν k M ( ω ) n ∀ ν ∈ M ( ω ) n . Hence, we have ¯ λ ∈ ∂g ( ∇ ¯ u ); see (3.3)–(3.5). Finally, (4.1) follows from (4.2) and thedefinition of T . This concludes the proof. Remark
Theorem 4.3 is still valid in dimension n = 3 if we take γ = 0 andwe assume that the nonlinearity of f ( x, y ) has a polynomial growth of arbitrary orderwith respect to the variable y ; see Remark 2.4. Indeed, let us observe that the limit ( ?? ) is still valid because v k ⇀ v in L / (Ω) and ¯ ϕ + β R ω ¯ u ds is a continuous functionin ¯Ω .Remark It would be interesting to prove the existence of a function ¯Φ ∈ C ( ω ) n ∩ ∂g ( ∇ ¯ u ) satisfying (4.3) – (4.5) . Indeed, Theorem 4.3 does not guarantee that k Φ k C ( ω ) n ≤ . In this hypothetic case, we could deduce from Propositions 3.1 and3.4 the following sparsity structure of ∇ ¯ u .1. For the | · | norm, if ∇ ¯ u = 0 we have k ¯Φ k C ( ω ) n = 1 and supp( ∇ ¯ u ) ⊂ { x ∈ ω : | ¯Φ( x ) | = 1 } .
2. For the | · | ∞ norm, for any ≤ j ≤ n such that if ∂ x j ¯ u = 0 we have k ¯Φ j k C ( ω ) = 1 , and supp([ ∂ x j u ] + ) ⊂ { x ∈ ω : ¯Φ j ( x ) = +1 } , supp([ ∂ x j ¯ u ] − ) ⊂ { x ∈ ω : ¯Φ j ( x ) = − } .
5. Second order optimality conditions.
The goal of this section is to provenecessary and sufficient second order optimality conditions for problem (P). In thewhole section, ¯ u will denote a fixed element of BV ( ω ) ∩ L ( ω ) satisfying the optimalityconditions given in Theorem 4.2. As in Section 3, we will distinguish the cases wherethe norms | · | and | · | ∞ in R n are used in the definition of the measure k∇ u k M ( ω ) n . | · | ∞ norm. As pointed out in (3.1), the use of the | · | ∞ norm in R n leads to the identity k∇ v k M ( ω ) n = n X j =1 k ∂ x j v k M ( ω ) = n X j =1 n Z ω | h v,j | d | ∂ x j ¯ u | + k ( ∂ x j v ) s k M ( ω ) o (5.1) ∀ v ∈ BV ( ω ), where ∂ x j v = h v,j | ∂ x j ¯ u | + ( ∂ x j v ) s is the Lebesgue decomposition of ∂ x j v with respect to the measure | ∂ x j ¯ u | , 1 ≤ j ≤ n . Moreover, for every 1 ≤ j ≤ n thereexists a Borel function ¯ h j such that | ¯ h j ( x ) | = 1 , | ∂ x j ¯ u |− a.e. , and ∂ x j ¯ u = ¯ h j | ∂ x j ¯ u | . (5.2) ptimal Control by BV-Functions h v = ( h v, , . . . , h v,n ) and ¯ h = (¯ h , . . . , ¯ h n ).First, we state the second order necessary optimality conditions. To this end wedefine the cone of critical directions C ¯ u as the closure in L ( ω ) of the cone E ¯ u = { v ∈ BV ( ω ) ∩ L ( ω ) : F ′ (¯ u ) v + αG ′ (¯ u ; v ) = 0and h v,j ∈ L ( | ∂ x j ¯ u | ) , ≤ j ≤ n } . (5.3)Then, we have the following result. Theorem 5.1. If ¯ u is a local minimum of (P) , then F ′′ (¯ u ) v ≥ ∀ v ∈ C ¯ u .Proof . We will prove the result for every v ∈ E ¯ u . Then, the theorem follows byusing the continuity of quadratic from v ∈ L ( ω ) → F ′′ (¯ u ) v ∈ R . Given v ∈ E ¯ u and ρ > ω ρ,j = { x ∈ ω : ρ | h v,j ( x ) | ≤ } ≤ j ≤ n. We have with Schwarz inequality | ∂ x j ¯ u | ( ω \ ω ρ,j ) ≤ ρ Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u |≤ ρ q | ∂ x j ¯ u | ( ω \ ω ρ,j ) (cid:16) Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u | (cid:17) / , which implies q | ∂ x j ¯ u | ( ω \ ω ρ,j ) ≤ ρ (cid:16) Z ω \ ω ρ,j | h v.j ( x ) | d | ∂ x j ¯ u | (cid:17) / ≤ j ≤ n. (5.4)Taking into account (5.2) we get for 1 ≤ j ≤ n | ¯ h j ( x ) + ρh v,j ( x ) | − | ¯ h j ( x ) | ρ = h v,j ( x )¯ h j ( x ) [ | ∂ x j ¯ u | ] − a.e. x ∈ ω ρ,j . Using this identity and (5.1) we get G (¯ u + ρv ) − G (¯ u ) ρ = n X j =1 n Z ω ρ,j | ¯ h j + ρh v,j | − | ¯ h j | ρ d | ∂ x j ¯ u | + k ( ∂ x j v ) s k M ( ω ) n o + n X j =1 Z ω \ ω ρ,j | ¯ h j + ρh v,j | − | ¯ h | ρ d | ∂ x j ¯ u | = n X j =1 n Z ω ρ,j ( h v,j ¯ h j ) d | ∂ x j ¯ u | + k ( ∂ x j v ) s k M ( ω ) n o + n X j =1 Z ω \ ω ρ,j | ¯ h j + ρh v,j | − | ¯ h j | ρ d | ∂ x j ¯ u | = n X j =1 n Z ω h v,j d∂ x j ¯ u + k ( ∂ x j v ) s k M ( ω ) n o + n X j =1 n Z ω \ ω ρ,j | ¯ h j + ρh v,j | − | ¯ h j | ρ d | ∂ x j ¯ u | − Z ω \ ω ρ,j ( h v,j ¯ h j ) d | ∂ x j ¯ u | o . E6. CASAS AND K. KUNISCH
Now, from (3.12), (5.2), Schwarz inequality, and (5.4) we infer G (¯ u + ρv ) − G (¯ u ) ρ ≤ G ′ (¯ u ; v ) + 2 n X j =1 Z ω \ ω ρ,j | h v,j | d | ∂ x j ¯ u |≤ G ′ (¯ u ; v ) + 2 n X j =1 q | ∂ x j ¯ u | ( ω \ ω ρ,j ) (cid:16) Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u | (cid:17) / ≤ G ′ (¯ u ; v ) + 4 ρ n X j =1 Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u | . Next we use the local optimality of ¯ u . By a Taylor expansion of F around ¯ u andusing that v ∈ E ¯ u , we get for ρ > ≤ J (¯ u + ρv ) − J (¯ u ) = ρ [ F ′ (¯ u ) v + αG ′ (¯ u ; v )]+ ρ (cid:16) F ′′ (¯ u + θ ρ v ) v + 8 α n X j =1 Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u | (cid:17) = ρ (cid:16) F ′′ (¯ u + θ ρ v ) v + 8 α n X j =1 Z ω \ ω ρ,j | h v,j ( x ) | d | ∂ x j ¯ u | (cid:17) with 0 ≤ θ ρ ≤
1. Dividing the above expression by ρ /
2, passing to the limit as ρ →
0, and taking into account that h v.j ∈ L ( | ∂ x j ¯ u | ) and | ∂ x j ¯ u | ( ω \ ω ρ,j ) →
0, weconclude that F ′′ (¯ u ) v ≥ C τ ¯ u = { v ∈ BV ( ω ) ∩ L ( ω ) : F ′ (¯ u ) v + αG ′ (¯ u ; v ) ≤ τ k z v k L (Ω) } , (5.5)where τ > z v = S ′ (¯ u ) v . The reader is referred to [4] for some second orderconditions based on these cones; see also [11] and [12]. Let us observe that (4.1) andthe inequality G ′ (¯ u ; v ) ≥ h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n imply that ∀ v ∈ BV ( ω ) ∩ L ( ω ) F ′ (¯ u ) v + αG ′ (¯ u ; v ) ≥ F ′ (¯ u ) v + α h ¯ λ, ∇ v i [ M ( ω ) n ] ∗ , M ( ω ) n = 0 . (5.6) Theorem 5.2.
Let ¯ u ∈ BV ( ω ) ∩ L ( ω ) satisfy the first order optimality conditionsstated in Theorem 4.2 and the second order condition ∃ δ > and ∃ τ > F ′′ (¯ u ) v ≥ δ k z v k L (Ω) ∀ v ∈ C τ ¯ u . (5.7) Then, there exist κ > and ε > such that J (¯ u ) + κ k y u − ¯ y k L ( ω ) ≤ J ( u ) ∀ u ∈ BV ( ω ) ∩ L ( ω ) : k u − ¯ u k L ( ω ) ≤ ε, (5.8) where y u = S ( u ) and ¯ y = S (¯ u ) .Proof . We follow the proof of [4, Theorem 3.6] with some changes. First, from [4,Lemma 2.7] we deduce the existence of ε > | [ F ′′ ( u ) − F ′′ (¯ u )] v | ≤ δ k z v k L (Ω) ∀ v ∈ L ( ω ) and all k u − ¯ u k L ( ω ) ≤ ε . (5.9)Moreover, from Proposition 2.2 we deduce the existence of a constant C > k z v k L (Ω) = k S ′ (¯ u ) v k L (Ω) ≤ C k v k L ( ω ) ∀ v ∈ L ( ω ) . (5.10) ptimal Control by BV-Functions K such that k y u k C (¯Ω) ≤ K if k u − ¯ u k L ( ω ) ≤ ε . From the adjoint state equation (2.11) and (2.3) we deduce that k ϕ u k C (¯Ω) ≤ K ′ for every k u − ¯ u k L ( ω ) ≤ ε and some constant K ′ . Finally, usingthese estimates, (2.4) and the expression (2.10) we infer the existence of a constant C > F ′′ ( u ) v ≥ γ k v k L ( ω ) − C k z v k L (Ω) for all k u − ¯ u k L ( ω ) ≤ ε and ∀ v ∈ L ( ω ) . (5.11)Now, we set ε = min n ε , τ ( δ + C ) C o with τ and δ given in (5.7). Let u ∈ BV ( ω ) ∩ L ( ω ) such that k u − ¯ u k L ( ω ) ≤ ε . Wedistinguish two cases. Case I: u − ¯ u ∈ C τ ¯ u . Making a Taylor expansion of F around ¯ u , using the convexityof G and (5.6), (5.7) and (5.9), we get for some 0 ≤ θ ≤ J ( u ) − J (¯ u ) ≥ [ F ′ (¯ u )( u − ¯ u ) + αG ′ (¯ u ; u − ¯ u )] + 12 F ′′ (¯ u + θ ( u − ¯ u ))( u − ¯ u ) ≥ F ′′ (¯ u )( u − ¯ u ) + 12 [ F ′′ (¯ u + θ ( u − ¯ u )) − F ′′ (¯ u )]( u − ¯ u ) ≥ δ k z u − ¯ u k L (Ω) − δ k z u − ¯ u k L (Ω) = δ k z u − ¯ u k L (Ω) . (5.12) Case II: u − ¯ u C τ ¯ u . This implies that F ′ (¯ u )( u − ¯ u ) + αG ′ (¯ u ; u − ¯ u ) > τ k z u − ¯ u k L (Ω) . (5.13)Moreover, from (5.10) and the definition of ε we infer k z u − ¯ u k L (Ω) ≤ C k u − ¯ u k L (Ω) ≤ τδ + C , and therefore δ + C τ k z u − ¯ u k L (Ω) ≤ . (5.14)Using again the convexity of G , (5.11), (5.13) and (5.14) we infer J ( u ) − J (¯ u ) ≥ [ F ′ (¯ u )( u − ¯ u ) + αG ′ (¯ u ; u − ¯ u )] + 12 F ′′ (¯ u + θ ( u − ¯ u ))( u − ¯ u ) ≥ τ k z u − ¯ u k L (Ω) − C k z u − ¯ u k L (Ω) ≥ δ + C k z u − ¯ u k L (Ω) − C k z u − ¯ u k L (Ω) = δ k z u − ¯ u k L (Ω) . (5.15)From (5.12) and (5.15) we deduce that [4, page 2364] J ( u ) − J (¯ u ) ≥ δ k z u − ¯ u k L (Ω) ∀ u ∈ BV ( ω ) ∩ L ( ω ) : k u − ¯ u k L ( ω ) ≤ ε. Finally, choosing ε still smaller, if necessary, we have that [4, page 2364]12 k y u − ¯ y k L (Ω) ≤ k z u − ¯ u k L (Ω) ∀ u ∈ BV ( ω ) ∩ L ( ω ) : k u − ¯ u k L ( ω ) ≤ ε. E6. CASAS AND K. KUNISCH
The last two inequalities imply (5.8) with κ = δ .We observe that (5.7) is a sufficient second order condition for strict local opti-mality of ¯ u in the L ( ω ) sense. Moreover, by using (5.8) we can prove stability ofthe optimal states with respect to perturbations in the data of the control problem.However, it does not provide information on the optimal controls. If γ > k y u − ¯ y k L (Ω) can be replaced by k u − ¯ u k L ( ω ) in (5.8).However, if γ = 0, then this is not possible; see [4]. Theorem 5.3.
Suppose that γ > and let ¯ u ∈ BV ( ω ) ∩ L ( ω ) satisfy the firstorder optimality conditions stated in Theorem 4.2 and the second order condition ∃ δ > and ∃ τ > F ′′ (¯ u ) v ≥ δ k v k L ( ω ) ∀ v ∈ C τ ¯ u . (5.16) Then, there exist κ > and ε > such that J (¯ u ) + κ k u − ¯ u k L ( ω ) ≤ J ( u ) ∀ u ∈ BV ( ω ) ∩ L ( ω ) : k u − ¯ u k L ( ω ) ≤ ε. (5.17) Proof . Using again [4, Lemma 2.7] along with (5.10) we infer the existence of ε > | [ F ′′ ( u ) − F ′′ (¯ u )] v | ≤ δ k v k L (Ω) ∀ v ∈ L ( ω ) and all k u − ¯ u k L ( ω ) ≤ ε. (5.18)Arguing similarly to (5.12), but using (5.16) and (5.18) we obtain for every u ∈ BV ( ω ) ∩ L ( ω ) such that k u − ¯ u k L ( ω ) ≤ ε and u − ¯ u ∈ C τ ¯ u J ( u ) − J (¯ u ) ≥ [ F ′ (¯ u )( u − ¯ u ) + αG ′ (¯ u ; u − ¯ u )] + 12 F ′′ (¯ u + θ ( u − ¯ u ))( u − ¯ u ) ≥ F ′′ (¯ u )( u − ¯ u ) + 12 [ F ′′ (¯ u + θ ( u − ¯ u )) − F ′′ (¯ u )]( u − ¯ u ) ≥ δ k u − ¯ u k L ( ω ) − δ k u − ¯ u k L ( ω ) = δ k u − ¯ u k L ( ω ) . (5.19)Thus, (5.17) holds with κ = δ assuming that u − ¯ u ∈ C τ ¯ u . Now, we argue by contra-diction, and we assume that there do not exist κ > ε > u ∈ BV ( ω ) ∩ L ( ω ) with k u − ¯ u k L ( ω ) ≤ ε . This implies that forevery integer k >
0, there exists an element u k ∈ BV ( ω ) ∩ L ( ω ) with k u k − ¯ u k L ( ω ) ≤ k and J (¯ u ) + 12 k k u k − ¯ u k L ( ω ) > J ( u k ) . (5.20)From (5.19) we know that u k − ¯ u C τ ¯ u , hence with (5.14) F ′ (¯ u )( u k − ¯ u ) + αG ′ (¯ u ; u k − ¯ u ) > τ k z u k − ¯ u k L (Ω) ≥ δ + C k z u k − ¯ u k L (Ω) (5.21)for every k large enough. Using (5.11), (5.20) and (5.21) we obtain12 k k u k − ¯ u k L ( ω ) > J ( u k ) − J (¯ u ) ≥ [ F ′ (¯ u )( u k − ¯ u ) + αG ′ (¯ u ; u k − ¯ u )] + 12 F ′′ (¯ u + θ k ( u k − ¯ u ))( u k − ¯ u ) ≥ δ + C k z u − ¯ u k L (Ω) + γ k u k − ¯ u k L ( ω ) − C k z u k − ¯ u k L (Ω) ≥ γ k u k − ¯ u k L ( ω ) , with is a contradiction because γ > ptimal Control by BV-Functions | · | norm. Given an element v ∈ BV ( ω ), we considerthe Lebesgue decomposition of ∇ v with respect to the positive measure |∇ ¯ u | : ∇ v = h v d |∇ ¯ u | + ( ∇ v ) s . Hence, we have k∇ v k M ( ω ) n = Z ω | h v ( x ) | d |∇ ¯ u | + k∇ v ) s k M ( ω ) . (5.22)We also set ∇ ¯ u = ¯ h |∇ ¯ u | , where | ¯ h ( x ) | = 1 |∇ ¯ u | -a.e. in ω . Then, we have with (3.8) G ′ (¯ u ; v ) = Z ω (¯ h · h v ) d |∇ ¯ u | + k ( ∇ v ) s k M ( ω ) n . (5.23)Now, we define the cone of critical directions C ¯ u = { v ∈ BV ( ω ) ∩ L ( ω ) : F ′ (¯ u ) v + αG ′ (¯ u ; v ) = 0 and | h v | ∈ L ( |∇ ¯ u | ) } . (5.24)Then, we have the following second order necessary optimality conditions. Theorem 5.4. If ¯ u is a local minimum of (P) , then F ′′ (¯ u ) v + α Z ω (cid:16) | h v ( x ) | − (¯ h ( x ) · h v ( x )) (cid:17) d |∇ ¯ u | ≥ ∀ v ∈ C ¯ u . (5.25) Proof . For fixed v ∈ C ¯ u and given ρ >
0, we define ω ρ = { x ∈ ω : ρ | h v ( x ) | ≤ } . Arguing as in the proof of Theorem 5.1 we get the following inequality analogous to(5.4) q |∇ ¯ u | ( ω \ ω ρ ) ≤ ρ (cid:16) Z ω \ ω ρ | h v ( x ) | d |∇ ¯ u | (cid:17) / . (5.26)Using the differentiability of the | · | -norm x ∈ R n → | x | for every x = 0, the factthat | ¯ h ( x ) | = 1 |∇ ¯ u | -a.e., the Schwarz inequality, and (5.26) we get for 0 ≤ θ ρ ( x ) ≤ G (¯ u + ρv ) − G (¯ u ) ρ = Z ω ρ | ¯ h + ρh v | − | ¯ h | ρ d |∇ ¯ u | + k ( ∇ v ) s k M ( ω ) n + Z ω \ ω ρ | ¯ h + ρh v | − | ¯ h | ρ d |∇ ¯ u | = Z ω ρ h ¯ h · h v + ρ (cid:16) | h v | | ¯ h + θ ρ ρh v | − (¯ h + θ ρ ρh v ) · h v | ¯ h + θ ρ ρh v | (cid:17)i d |∇ ¯ u | + k ( ∇ v ) s k M ( ω ) n + Z ω \ ω ρ | ¯ h + ρh v | − | ¯ h | ρ d |∇ ¯ u |≤ Z ω (¯ h · h v ) d |∇ ¯ u | + ρ Z ω ρ h | h v | | ¯ h + θ ρ ρh v | − (¯ h + θ ρ ρh v ) · h v | ¯ h + θ ρ ρh v | i d |∇ ¯ u | + k ( ∇ v ) s k M ( ω ) n + 2 Z ω \ ω ρ | h v | d |∇ ¯ u |≤ Z ω (¯ h · h v ) d |∇ ¯ u | + k ( ∇ v ) s k M ( ω ) n + ρ n Z ω ρ h | h v | | ¯ h + θ ρ ρh v | − (¯ h + θ ρ ρh v ) · h v | ¯ h + θ ρ ρh v | i d |∇ ¯ u | + 8 Z ω \ ω ρ | h v | d |∇ ¯ u | o . E6. CASAS AND K. KUNISCH
Using this inequality and the local optimality of ¯ u , we infer with u ρ = ¯ u + θ ρ ρv ,0 ≤ θ ρ ≤ ≤ J (¯ u + ρv ) − J (¯ u ) = ρ h F ′ (¯ u ) v + α G (¯ u + ρv ) − G (¯ u ) ρ i + ρ F ′′ ( u ρ ) v ≤ ρ [ F ′ (¯ u ) v + αG ′ (¯ u ; v )] + ρ n F ′′ ( u ρ ) v + α Z ω ρ h | h v | | ¯ h + θ ρ ρh v | − (¯ h + θ ρ ρh v ) · h v | ¯ h + θ ρ ρh v | i d |∇ ¯ u | + 8 α Z ω \ ω ρ | h v | d |∇ ¯ u | o . Now, taking into account that v ∈ C ¯ u and dividing the above inequality by ρ / ≤ F ′′ ( u ρ ) v + α Z ω ρ h | h v | | ¯ h + θ ρ ρh v | − (¯ h + θ ρ ρh v ) · h v | ¯ h + θ ρ ρh v | i d |∇ ¯ u | + 8 α Z ω \ ω ρ | h v | d |∇ ¯ u | . Finally, using that |∇ ¯ u | ( ω \ ω ρ ) → ρ → | ¯ h ( x ) | = 1, and12 ≤ − ρ | h v ( x ) | ≤ | ¯ h + θ ρ ρh v | ≤ ρ | h v ( x ) | ≤ |∇ ¯ u | -a.e. in ω ρ , we pass to the limit as ρ → Remark
The reader can easily check that Theorems 5.2 and 5.3 also holdwhen the | · | norm is used. However, to reduce the gap between the necessary andsufficient conditions for optimality, we should prove that the conditions F ′′ (¯ u ) v + α Z ω (cid:16) | h v ( x ) | − (¯ h ( x ) · h v ( x )) (cid:17) d |∇ ¯ u | ≥ δ k z v k L (Ω) ∀ v ∈ C τ ¯ u and F ′′ (¯ u ) v + α Z ω (cid:16) | h v ( x ) | − (¯ h ( x ) · h v ( x )) (cid:17) d |∇ ¯ u | ≥ δ k v k L ( ω ) ∀ v ∈ C τ ¯ u imply (5.8) and (5.17) , respectively. This, however, remains as a challenge.
6. A regularization of problem (P) . Here we briefly discuss the effect of an H ( ω )-regularization term on the first order optimality conditions. For ǫ > ǫ ) min u ∈ H ( ω ) J ǫ ( u ) = J ( u ) + ǫ Z ω |∇ u ( x ) | dx, subject to (1.1), and denote a solution by u ǫ . Let us set J ǫ ( u ) = F ǫ ( u ) + G ( u ) , where F ǫ ( u ) = F ( u ) + ǫ R ω |∇ u | dx for u ∈ H ( ω ). We have F ′ ǫ ( u ) v = F ′ ( u ) v + ǫ Z ω ∇ u · ∇ v dx, and ∂G ( u ) = ∇ ∗ ∂g ( ∇ u ) for u ∈ H ( ω ) , ptimal Control by BV-Functions ∇ : H ( ω ) → L ( ω ) n , and g : L ( ω ) n → R is given by g ( v ) = k v k L ( ω ) n .We have the analog of Theorem 4.2, i.e. for every local solution u ǫ of (P ǫ ) there exists λ ǫ ∈ ∂G ( ∇ u ǫ ) such that α ( λ ǫ , ∇ v ) L ( ω ) n + F ′ ǫ ( u ǫ ) v = 0 , for all v ∈ H ( ω ) . (6.1)Let us focus on λ ǫ ∈ ∂g ( ∇ u ǫ ) next. It is equivalent to( λ ǫ , ∇ u ǫ ) = k∇ u ǫ k L ( ω ) n , and ( λ ǫ , v ) ≤ k v k L ( ω ) n for all v ∈ L ( ω ) n . (6.2) The use of the Euclidean norm | · | : Here (6.2) results in n X i =1 ( λ ǫ,i , ∂ x i u ǫ ) = Z ω ( n X i =1 | ∂ x i u ǫ | ) dx, and n X i =1 ( λ ǫ,i , v i ) ≤ Z ω ( n X i =1 | v i | ) dx, (6.3)for all v ∈ L ( ω ) n . The second expression in (6.3) implies that k λ ǫ k L ∞ ( ω, R n ) ≤ ∇ u ǫ = 0, k λ ǫ k L ∞ ( ω, R n ) = 1 and supp ∇ u ǫ ⊂ { x ∈ ω : | λ ǫ ( x ) | = 1 } . (6.4)The first claim follows from the equality in (6.3). This equality can also be expressedas R ω |∇ u ǫ | dx = R ω ( ∇ u ǫ · λ ǫ ) dx , which, together with | λ ( x ) | ≤ The use of the | · | ∞ -norm : In this case (6.2) results in n X i =1 ( λ ǫ,i , ∂ x i u ǫ ) = n X i =1 k ∂ x i u ǫ k L ( ω ) and n X i =1 ( λ ǫ,i , v i ) ≤ n X i =1 k v i k L ( ω ) , (6.5)for all v ∈ L ( ω ) n . This implies that k λ ǫ,j k L ∞ ( ω ) ≤ j = 1 , . . . , n and if ∂ x j u ǫ = 0 k λ ǫ,j k L ∞ ( ω ) = 1 , and supp ( ∂ x j u ǫ ) ± ⊂ { x ∈ ω : λ ǫ,j = ± } . (6.6)In fact, for any 1 ≤ j ≤ n , let ν i = 0 for all i = j and ν j = λ ǫ,j on S + j = { x : λ ǫ,j > } ,and equal to 0 otherwise. Then R S + j ( λ ǫ,j − λ ǫ,j )( x ) dx ≤
0, while the integrand isstrictly positive a.e. Hence meas ( S + j ) = 0. In an analogous form we exclude the case λ ǫ,j < −
1, and hence k λ ǫ,j k L ∞ ( ω ) ≤
1, for all j . Using the first expression in (6.5) wehave n X i =1 k ∂ x i u ǫ k L ( ω ) = n X i =1 ( λ ǫ,i , ∂ x i u ǫ ) ≤ n X i =1 k ∂ x i u ǫ k L ( ω ) , which implies (6.6). Asymptotic behavior:
Finally we consider the asymptotic behavior of (6.1), (6.2)as ǫ → + . From the inequality J ε ( u ε ) ≤ J (0) for all ε >
0, we deduce with (1.2)the boundedness of { u ε } ε in BV ( ω ) ∩ L ( ω ). Moreover, (6.4) and (6.6) imply theboundedness of { λ ε } ε in L ∞ ( ω ) n . Hence there exists (¯ u, ¯ λ ) ∈ ( BV ( ω ) ∩ L ( ω )) × L ∞ ( ω ) n such that on a subsequence ( u ǫ , λ ǫ ) ∗ ⇀ (¯ u, ¯ λ ) weakly ∗ in ( BV ( ω ) ∩ L ( ω )) × L ∞ ( ω ). Moreover y u ǫ → y ¯ u in L (Ω).Now, given an arbitrary element u ∈ H ( ω ), the optimality of u ε and the structureof J implies J (¯ u ) ≤ lim inf ε → J ( u ε ) ≤ lim sup ε → J ( u ε ) ≤ lim sup ε → J ε ( u ε ) ≤ lim sup ε → J ε ( u ) = J ( u ) . E6. CASAS AND K. KUNISCH
Since H (Ω) is dense in BV ( ω ) ∩ L ( ω ), the above inequality implies that ¯ u is asolution of (P) and J (¯ u ) = lim ε → J ( u ε ) = lim sup ε → J ε ( u ε ) = inf (P) = J (¯ u ) . (6.7)This implies that J ( u ε ) → J (¯ u ) and ε R ω |∇ u ε | dx →
0. Moreover, from the conver-gence properties of { u ε } ε and { y ε } ε we deduce thatlim ε → " k y u ε − y d k L (Ω) + β (cid:18)Z ω u ε dx (cid:19) = 12 k y ¯ u − y d k L (Ω) + β (cid:18)Z ω ¯ u dx (cid:19) , (6.8) Z ω |∇ ¯ u | ≤ lim inf ε → Z ω |∇ ¯ u ε | . (6.9)Combining (6.8) with the convergence J ( u ε ) → J (¯ u ) we inferlim ε → (cid:18) γ k u ε k L ( ω ) + α Z ω |∇ u ε | (cid:19) = γ k ¯ u k L ( ω ) + α Z ω |∇ ¯ u | . (6.10)If γ = 0 then this identity is reduced to R ω |∇ u ε | → R ω |∇ ¯ u | . Let us prove thatthis convergence property also holds for γ >
0. Using (6.10), the convergence u ε ⇀ ¯ u in L ( ω ), and (6.9) we obtain γ k ¯ u k L ( ω ) ≤ lim inf ε → k u ε k L ( ω ) ≤ lim sup ε → k u ε k L ( ω ) ≤ lim sup ε → (cid:18) γ k u ε k L ( ω ) + α Z ω |∇ u ε | (cid:19) − α lim inf ε → Z ω |∇ ¯ u ε |≤ (cid:18) γ k ¯ u k L ( ω ) + α Z ω |∇ ¯ u | (cid:19) − α Z ω |∇ ¯ u | = γ k ¯ u k L ( ω ) . Therefore, k u ε k L ( ω ) → k ¯ u k L ( ω ) holds. Combining this fact with the weak conver-gence we conclude that u ε → ¯ u strongly in L ( ω ). Inserting this in (6.10) it followsthat R ω |∇ u ε | → R ω |∇ ¯ u | .From (6.1) we have that α ( λ ǫ , ∇ v ) + Z ω (cid:16) ϕ ( u ǫ ) + γu ǫ + β Z ω u ǫ dz (cid:17) v dx − ǫ Z ω u ǫ ∆ v dx = 0 , ∀ v ∈ C ∞ ( ω ) . Taking the limit ǫ → α (¯ λ, ∇ v ) + Z ω (cid:16) ϕ (¯ u ) + γ ¯ u + β Z ω ¯ u dz (cid:17) v dx = 0 , ∀ v ∈ C ∞ ( ω ) , which corresponds to (4.1). Moreover, the above relation implies that ¯ λ ∈ L ( ω ),and − α div¯ λ + ϕ (¯ u ) + γ ¯ u + β Z ω ¯ u dz = 0 in L ( ω ) . (6.11)This relation can also be deduced from (4.1). Thus div ¯ λ from Section 4 coincideswith div ¯ λ obtained by regularisation and it is uniquely defined by (6.11). ptimal Control by BV-Functions ε R ω |∇ u ε | dx → ε → ( λ ǫ , ∇ u ǫ ) = − lim ε → α F ′ ǫ ( u ǫ ) u ǫ = − lim ε → α (cid:16) F ′ ( u ǫ ) u ǫ − ǫ Z ω |∇ u ǫ | dx (cid:17) = − α F ′ (¯ u )¯ u = − (div ¯ λ, ¯ u ) . Now, from (6.2) and the convergence R ω |∇ u ε | → R ω |∇ ¯ u | we inferlim ε → ( λ ǫ , ∇ u ǫ ) = k∇ ¯ u k M ( ω ) n . From the last two identities, and using again (6.2) along with the convergence λ ε ∗ ⇀ ¯ λ in L ∞ ( ω ) we obtain( − div ¯ λ, ¯ u ) = k∇ ¯ u k M ( ω ) n , and (¯ λ, v ) ≤ | v | L ( ω ) n for all v ∈ L ( ω ) n . This corresponds to h ¯ λ, ∇ ¯ u i [ M ( ω ) n ] ∗ , M ( ω ) n = k∇ ¯ u k M ( ω ) n , and h ¯ λ, ν i ≤ k ν k M ( ω ) n forall ν ∈ M ( ω ) n , which was obtained in Theorem 4.2 with ¯ λ ∈ ∂g ( ∇ ¯ u ) ⊂ [ M ( ω ) n ] ∗ .
7. Conclusions.
An analysis for BV-regularised optimal control problems asso-ciated to semilinear elliptic equations was provided. Existence, first order necessaryand second order sufficient optimality conditions were investigated. Special attentionwas given to the different cases which arise due to the choice of a particular vectornorm in the definition of the BV-seminorm. If (P) is additionally regularised by an H ( ω )-seminorm, then the set where the gradient of the optimal solution vanishes,can be characterised conveniently by an adjoint variable, see (6.4) and (6.6). Forthe original problem (P) without H ( ω )-seminorm regularisation, such a transparentdescription of the set where the measure |∇ ¯ u | vanishes is not available, rather it wasreplaced by the properties specified in Theorem 4.3. REFERENCES[1]
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