Optimal control of a quasilinear parabolic equation and its time discretization
aa r X i v : . [ m a t h . O C ] F e b Optimal control of a quasilinear parabolic equationand its time discretization
Luise Blank , Johannes Meisinger In this paper we discuss the optimal control of a quasilinear parabolic stateequation. Its form is leaned on the kind of problems arising for examplewhen controlling the anisotropic Allen-Cahn equation as a model for crystalgrowth. Motivated by this application we consider the state equation asa result of a gradient flow of an energy functional. The quasilinear termis strongly monotone and obeys a certain growth condition and the lowerorder term is non-monotone. The state equation is discretized implicitly intime with piecewise constant functions. The existence of the control-to-stateoperator and its Lipschitz-continuity is shown for the time discretized aswell as for the time continuous problem. Latter is based on the convergenceproof of the discretized solutions. Finally we present for both the existenceof global minimizers. Also convergence of a subsequence of time discreteoptimal controls to a global minimizer of the time continuous problem canbe shown. Our results hold in arbitrary space dimensions.
Key words. quasilinear parabolic equation, Allen-Cahn equation, anisotropy, optimalcontrol, implicit discretization, convergence analysis
AMS subject classification.
In many areas the optimal control of an interface evolution towards an anisotropic shapeis desired. For example in chemistry or materials science one wishes to steer the solidifica-tion process of crystals [8, 18, 21, 32]. For the time evolution of shapes phase-field modelshave shown great promise in many application areas and anisotropies can be incorpo-rated (see e.g. [16] and references therein). In this ansatz the interface is modeled witha diffuse interface layer, and an order parameter y —the so called phase-field—reflectsthe pure phases with the values ±
1, e.g. the liquid phase for y ≈ y ≈ −
1, and the diffuse interface with values between − Department of Mathematics, University of Regensburg, D-93040 Regensburg, Germany([email protected], [email protected]) ( y ) := Z Ω A ( ∇ y ) + ψ ( y ) d x (1)then determines the time evolution of the shape, and with it the state equation for thecontrol problem. Here the first term represents the surface energy where A : R d → R isan (an-)isotropy function, and the potential ψ can be thought of being symmetric andto have its global minima at ≈ ±
1. Let us mention that typically the energy involvesa variable ε > L -gradient flow with a smoothpotential we obtain the Allen-Cahn equation. For further introduction to phase fieldmodels we refer to [17] and references therein. The following analysis and numericalansatz will not only be valid for the Allen-Cahn equations but can be applied in generalto differential equations arising from a gradient flow of energies of the form eq. (1).The goal is now to determine the distributed control u driving the solution y of thegradient flow equation ∂ t y − ∇ · A ′ ( ∇ y ) + ψ ′ ( y ) = u (2)from an initial configuration y at time t —say t = 0—to a given target function y Ω ata given final time T (or a target function y Q over the whole time horizon as consideredin eq. (46)). Hence the optimal control problem is described by the following setting:Let Ω ⊂ R d be a bounded Lipschitz domain and y Ω ∈ L (Ω) be a given target function.Let a final time 0 < T < ∞ be given and denote the space-time cylinder by Q := [0 , T ] × Ωand its boundary by Σ := [0 , T ] × ∂ Ω. Our objective is to find for a given initial state y ∈ H (Ω) a solution to the optimal control problemmin J ( u, y ) := 12 k y ( T ) − y Ω k L (Ω) + λ k u k L ( Q ) (3)subject to the quasilinear, possibly nonsmooth parabolic state equation Z Q ∂ t yη + A ′ ( ∇ y ) T ∇ η + ψ ′ ( y ) η = Z Q uη ∀ η ∈ L (0 , T ; H (Ω)) y (0) = y in Ω (4)where u ∈ L ( Q ) ∼ = L (0 , T ; L (Ω)) and y ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; H (Ω) ′ ). Notethat J is well defined due to the embedding L (0 , T ; H (Ω)) ∩ H (0 , T ; H (Ω) ′ ) ֒ → C ([0 , T ]; L (Ω)).In contrast to the well-studied isotropic Allen-Cahn equation where A ′ = Id , in theanisotropic case A : R d → R is an absolutely 2-homogeneous function. As a consequence A ′ is not differentiable at 0 in general. Let us mention that roughly speaking y isconstant in the pure phases, and hence ∇ y ≈ A ′ . To allow theuse of possibly regularized anisotropies A we relax the requirement of 2-homogeneity.Moreover we allow for various potentials ψ .2 ssumptions 1.1. Assume A ∈ C ( R ) with A ′ is strongly monotone and fulfills thegrowth condition | A ′ ( p ) | ≤ C | p | .Furthermore let ψ ∈ C ( R ) be bounded from below and can be approximated by f n satis-fying f n ∈ C ( R ) , f n → ψ in C loc , − c ≤ f n ≤ c ( ψ + 1) , f ′′ n ≥ − C ψ , | f ′′ n | ≤ C n , (5) with c, C ψ , C n ≥ and ψ ( y ) ∈ L (Ω) for the given initial data y ∈ H (Ω) . The assumptions on ψ in particular imply that it holds (cid:0) ψ ′ ( y ) − ψ ′ ( y ) , y − y (cid:1) ≥ − C ψ | y − y | ∀ y , y ∈ R . (6)Some examples of A and ψ with respect to Allen-Cahn equations are mentioned inRemark 2.8.In this paper we study the existence of an optimal control to eq. (3)–eq. (4) in arbitraryspace dimension and of the corresponding in time discretized control problem. Herean implicit time discretization using piecewise constant functions is employed. Thistogether with regularizing A provides differentiability of the control-to-state operator(see the follow-up paper [4]) which is helpful for numerical approaches solving the controlproblem. It is not clear whether semi-implicit discretization provide differentiability.To the best of our knowledge there does not exist any mathematical treatment on theoptimal control of anisotropic phase-field models so far. Optimal control of isotropicAllen-Cahn variational equations are studied e.g. in [3, 5, 14, 20, 34] and of Cahn-Hilliard variational equalities in [24, 15, 25] and references therein. Let us mentionresults given in the context of anisotropic Allen-Cahn equations. One possible anisotropywas introduced in a pioneering paper by Kobayashi [27] and existence and uniquenessof a solution are studied in [9, 31, 37]. For quite general anisotropies the solution ofAllen-Cahn equations with obstacle potential is analyzed in [19]. Among others they use2-homogeneity of A , an approximation of the potential similar to eq. (5) and an implicittime discretization (without showing convergence of the discretization). Explicit andsemi-implicit approximations are discussed in the survey paper [16], where also manyreferences are given. For convex Kobayashi anisotropies several time discretizations areconsidered in [22]. In [2, 1] particular suggestions for the anisotropies are given and anefficient semi-implicit method using a particular linearization of A ′ and a convex/concavesplitting is presented and energy stability is shown. Also several numerical experimentsare shown comparing the anisotropies.Literature to optimal control of quasilinear parabolic equations of the form eq. (4) isstill in its infancy. Most literature known to us treat quasilinearities with coefficientsdepending on x, t and on the function y but not on its gradient [7, 26, 11, 29, 30]. Forquasilinearities involving spatial derivatives of y see for example [33, 10]. In particularlet us mention that the latter reference contains the most similar problem to ours, as theauthors require a rather general quasilinearity with some particular polynomial growthcondition. However they require the nonlinearity ψ ′ to be monotone. All the literature3isted here assumes the quasilinear term to be rather well behaved, in particular none ofits derivatives shall be singular at the origin. In the present context to our knowledgesuch difficulties have only been considered for elliptic equations [12].The outline of the paper is the following.As a first step we study the state equations. Therefore we introduce in section 2 the timediscretization. Then we discuss the existence and uniqueness of the solution of the dis-cretized state equation as well as the Lipschitz-continuity of the control-to-state operator.Furthermore, for a set of bounded controls we obtain bounds on the states indepedent ofthe discretization level. Using these results we consider the limit with respect to the timediscretization and obtain corresponding results for the in time continuous state equationeq. (4). Consequently we have also convergence of the discretization. In addition weshow energy stability of the discretization.Finally in section 3 the existence of the controls in the time continuous and time discretecase is shown. In addition the convergence of a subsequence of time discrete optimalcontrols to an optimal control of the original problem is obtained. These results holdnot only for aiming at an end time state but also for steering to a state over the wholetime horizon. First we introduce the time discretization. Then a certain boundedness property likein [19] is shown which is essential not only for the existence of the solution of the stateequation but also for proving the existence of an optimal control and the convergenceof the solution of the discretized problem to the time continuous solution. To obtainthis result, the potential ψ is approximated (as e.g. in [19] and [13]) with a sequenceof functions f n with bounded second derivatives, such that the dominated convergencetheorem can be used. Following the lines of [19] we have no restriction on the spacedimension d .The existence of the time continuous problem will then be shown by taking the limitwith respect of the time resolution which also shows convergence of the discretizationmethod. (In [19] first the limit in the time discretization and then in the approximationof ψ is taken).From now on if no subscripts are provided, with ( · , · ) and k · k we mean the L /l -scalarproduct and norm respectively. The space should be clear from the context. For aBanach space V we will denote its dual by V ′ and the duality product by h· , ·i .Next we introduce a time discretization and show the existence of a solution of thediscretized state equation. We divide the interval [0 , T ] into subintervals I j := ( t j − , t j ]for j = 1 , . . . , N with 0 = t < t < . . . < t N = T and define τ j := t j − t j − and4 := max j t j . The state equation we discretize in time with a discontinuous-Galerkinmethod (dG(0)). Therefore, let us define Y τ := { y τ : Q → R | y τ ( t, . ) ∈ H (Ω) , y τ ( ., x ) constant in I j for j = 1 , . . . , N } , (7) U τ := { u τ : Q → R | u τ ( t, . ) ∈ L (Ω) , u τ ( ., x ) constant in I j for j = 1 , . . . , N } , (8)and for each interval we label the constant by a subscript, e.g. y j := y τ | I j . The vectorcontaining these constants will be denoted in boldface, e.g. y := ( y j ) j =1 ,...,N ∈ H (Ω) N .The time-discretized variant of eq. (3)–eq. (4) is then given bymin Y τ × U τ J ( y τ , u τ ) = 12 k y N − y Ω k + λ N X j =1 τ j k u j k (9)subject to the time-discretized state equation( y j , ϕ ) + τ j ( A ′ ( ∇ y j ) , ∇ ϕ ) + τ j ( ψ ′ ( y j ) , ϕ ) = τ j ( u j , ϕ ) + ( y j − , ϕ ) ∀ ϕ ∈ H (Ω) (10)with j = 1 , . . . , N and y τ (0 , . ) := y ∈ H (Ω) is given.We note that the state equation could have arised equally well from an implicit Eulerdiscretization and we will use the notation ∂ − τt y with ∂ − τt y τ | I j := τ j ( y j − y j − ) . One may favour a splitting approach for ψ or an approximation of the quasilinear term A as in [1, 2] instead of the fully implicit method. However, to our knowledge thereexists no convergence proof for these discretizations of the state equation to the timecontinuous one in the limit τ →
0. Even more important is that for solving the controlproblem we aim at differentiability of the control to state operator, which we will show inthe forthcoming paper [4] for d ≤
3. For a semi-implicit discretization of the quasilinearterm it is unsure that this property holds. Moreover, the additional computational costis nearly negligible for solving the optimal control problem.The first step is given by the subsequent lemma.
Lemma 2.1.
Let A fulfill the conditions in Assumptions 1.1. Furthermore let y ∈ H (Ω) and u τ ∈ U τ . Then for every f ∈ C ( R d ) with f bounded from below, | f ′′ | bounded and f ( y ) ∈ L (Ω) there exists for τ > small enough a y τ ∈ Y τ which is asolution of y τ (0) = y in Ω and for all j = 1 , . . . , N it holds ∀ η ∈ H (Ω) : Z Ω ∂ − τt y j ( η − y j ) + A ( ∇ η ) − A ( ∇ y j ) + f ′ ( y j )( η − y j ) − u j ( η − y j ) ≥ . (11) Moreover, given a Λ > then for all u τ , y , A, f fulfilling in addition k u τ k L (0 ,T ; L (Ω)) , k y k L (Ω) ≤ Λ , Λ − ( − | p | ) ≤ A ( p ) and A ′ ( p ) T p ≤ Λ | p | as well as Z Ω ( A ( ∇ y ) + f ( y )) ≤ Λ and f ≥ − Λ , f ′′ ≥ − Λ , (12) there exist for all τ small enough solutions y τ of eq. (11) satisfying k ∂ − τt y τ k L (0 ,T ; L (Ω)) + k y τ k L ∞ (0 ,T ; H (Ω)) + k f ′ ( y τ ) k L (0 ,T ; L (Ω)) ≤ C (Λ) . (13)5 roof. We consider directly the bounds defined by Λ since for fixed u τ , y , A and f thisconstant can be chosen appropriately. In particular for A the constant can be foundsince the growth condition induces A ′ (0) = 0 and then the strong monotonicity provides A ′ ( p ) T p ≤ c | p | and c ( − | p | ) ≤ A ( p ) for some constants c > f and f ′ induce Nemytskii operators f : L (Ω) → L (Ω) and f ′ : L (Ω) → L (Ω) due to the bounds on f ′′ .Starting with y := y (0), define y j ∈ H (Ω) successively for j ≥ j,τ ( η ) := Z Ω (cid:16) τ j | η − y j − | + A ( ∇ η ) + f ( η ) − u j η (cid:17) (14)where the integrands are strongly convex for τ + f ′′ ( s ) ≥ τ − C >
0. Since | f ′ ( x ) | ≤ c (1 + | x | ) one can apply the dominated convergence theorem and obtains the first ordercondition. Hence y j fulfills for all η ∈ H (Ω) Z Ω ( A ( ∇ η ) − A ( ∇ y j )) ≥ Z Ω − y j − y j − τ j ! ( η − y j ) − f ′ ( y j )( η − y j ) + u j ( η − y j ) . (15)and therefore eq. (11). The summation of Φ l,τ ( y l ) ≤ Φ l,τ ( y l − ) yields Z t j Z Ω 12 | ∂ − τt y τ | + Z Ω ( A ( ∇ y j ) + f ( y j )) ≤ Z Ω ( A ( ∇ y ) + f ( y )) + Z t j Z Ω u τ ∂ − τt y τ ≤ C (Λ) + Z t j Z Ω | ∂ − τt y τ | . (16)Using the assumptions Λ − ( − | p | ) ≤ A ( p ), − f ≤ Λ, k y k L (Ω) ≤ Λ as well as y j = y + R t j ∂ − τt y τ , we obtain k ∂ − τt y τ k L (0 ,T ; L (Ω)) + k∇ y τ k L ∞ (0 ,T ; L (Ω)) + k y τ k L ∞ (0 ,T ; L (Ω)) ≤ C (Λ) . (17)Then, choosing η := y j − δf ′ ( y j ) , δ > Z Ω f ′ ( y j ) ≤ Z Ω − y j − y j − τ j f ′ ( y j ) + u j f ′ ( y j ) − δ ( A ( ∇ y j ) − A ( ∇ ( y j − δf ′ ( y j )))) . (18)To the third integral we can apply the mean value theorem pointwisely almost every-where, with the intermediate point of 1 and 1 − δf ′′ ( y ) denoted by ξ δ ( y ). Note that dueto the boundedness of f ′′ also ξ δ ( · ) is bounded and ξ δ → δ → ≤ A ′ ( p ) T p ≤ Λ | p | , − f ≤ Λ and − f ′′ ≤ Λ as well as dominated convergenceto obtain Z Ω f ′ ( y j ) ≤ Z Ω − y j − y j − τ j f ′ ( y j ) + u j f ′ ( y j ) + C (Λ) |∇ y j | ≤ Z Ω (cid:16) y j − y j − τ j (cid:17) + f ′ ( y j ) + u j + f ′ ( y j ) + C (Λ) |∇ y j | . and hence with eq. (17) N X j =1 τ j k f ′ ( y j ) k / ≤ C (Λ) . (19)6 emark 2.2. As mentioned the strong monotonicity of A ′ and the growth condition | A ′ ( p ) | ≤ C | p | induces A ′ (0) = 0 , A ′ ( p ) T p ≤ c | p | and c ( − | p | ) ≤ A ( p ) . Furthermorealso A ( p ) ≤ c (1 + | p | ) holds. Hence A ( ∇ η ) ∈ L ( Q ) and (using Young’s inequality) A ′ ( ∇ η ) T ∇ ξ ∈ L ( Q ) for all η, ξ ∈ L (0 , T ; H (Ω)) . It also induces the pointwise esti-mate | A ′ ( ∇ y + sδ ∇ ξ ) T ∇ ξ | ≤ C ′ ( |∇ y | + |∇ ξ | ) , for ≤ sδ ≤ providing an integrablemajorant, which allows to take the limit δ ց for the integral below. Hence we obtain lim δ ց δ Z Q ( A ( ∇ y + δ ∇ ξ ) − A ( ∇ y )) = lim δ ց Z Q Z A ′ ( ∇ y + sδ ∇ ξ ) T ∇ ξds = Z Q A ′ ( ∇ y ) T ∇ ξ. (20) The same holds respectively for integration over Ω . Together with the monotonicity of A ′ this enables the usual steps of the proof that solving the variational inequality eq. (11) isequivalent to solving the variational equality eq. (10) with f instead of ψ . With lemma 2.1 at hand we can show the existence of a unique weak solution to timediscretized state equation eq. (10).
Theorem 2.3.
Let Assumptions 1.1 be fulfilled. If τ = max j τ j < /C ψ then for every u τ ∈ U τ the time discretized state equation eq. (10) has a unique solution y τ ∈ Y τ .The solution operator is denoted by S τ : U τ → Y τ .Furthermore, given a ¯ c > then for all k u τ k L (0 ,T ; L (Ω)) ≤ ¯ c it holds independent of τ k ∂ − τt y τ k L (0 ,T ; L (Ω)) + k y τ k L ∞ (0 ,T ; H (Ω)) + k ψ ′ ( y τ ) k L (0 ,T ; L (Ω)) ≤ C A,ψ,y (¯ c ) . (21) Proof.
We consider the approximation of ψ by f n according to Assumption 1.1. Then,due to ψ ( y ) ∈ L (Ω) for given y ∈ H (Ω) and − c ≤ f n ≤ ( c + 1) ψ , − C ψ ≤ f ′′ n one canfind Λ depending only on A, ψ, y large enough such that k y k L (Ω) , Z Ω ( A ( ∇ y ) + f n ( y )) , − inf t ∈ R f n ( t ) , − inf t ∈ R f ′′ n ( t ) ≤ Λ , Λ − | p | ≤ A ( p ) and A ′ ( p ) T p ≤ Λ | p | . (22)We denote by y j,n the solutions of eq. (11) with f = f n which exist according to lemma 2.1and remark that they exist for τ < C ψ where the integrands of eq. (14) are stronglyconvex due to τ + f ′′ n ( s ) ≥ τ − C ψ > . Also lemma 2.1 provides the estimates eq. (13),i.e. for all τ and n it holds k ∂ − τt y τ,n k L (0 ,T ; L (Ω)) + k y τ,n k L ∞ (0 ,T ; H (Ω)) + k f ′ n ( y τ,n ) k L (0 ,T ; L (Ω)) ≤ C (Λ , ¯ c ) . (23)Then f n → ψ in C for n → ∞ and the weak-lower semicontinuity of A allows to takefor a subsequence of y j,n the limit n → ∞ for all terms in eq. (23) and in eq. (11) toobtain eq. (21) and that for all η ∈ H (Ω) it holds Z Ω ( A ( ∇ η ) − A ( ∇ y j )) ≥ Z Ω − y j − y j − τ j ! ( η − y j ) − ψ ′ ( y j )( η − y j ) + u j ( η − y j ) . (24)7inally we can go over to the equality eq. (10) by the reasoning from Remark 2.2.The uniqueness of the solution of eq. (10) can be shown for each time step separatelyone after another. For this purpose assume the existence of two solutions. Subtractingtheir defining equations, testing with their difference and using the strong monotonicityof A ′ and of s + τ n ψ ′ ( s ) due to τ < /C ψ shows that the H -norm of their differencevanishes.With a further (minor) restriction on the maximal time step τ we obtain Lipschitz-continuity. Theorem 2.4.
Let Assumptions 1.1 and τ ≤ C ψ hold. Then the mapping ˜ S τ :( y , u τ ) y τ where y τ is the solution of equation eq. (10) , is Lipschitz-continuous, i.e.more precisely it holds k y (1) τ − y (2) τ k L ∞ (0 ,T ; L (Ω)) + k∇ y (1) τ − ∇ y (2) τ k L (0 ,T ; L (Ω)) ≤≤ C A,ψ,T (cid:16) k y (1)0 − y (2)0 k L (Ω) + k u (1) τ − u (2) τ k L (0 ,T ; H (Ω) ′ ) (cid:17) (25) where y ( i ) τ = ˜ S τ ( y ( i )0 , u ( i ) τ ) for i = 1 , .Proof. We note down the differences by a prescript δ , e.g. δy τ := y (1) τ − y (2) τ . With ( a − b ) ≤ ( a − b ) a , testing the defining equalities eq. (10) with δy j and using that A ′ is strongly monotone as well as eq. (6), we obtain (cid:16) k δy j k − k δy j − k (cid:17) + τ j C A k∇ δy j k ≤ ( δy j − δy j − , δy j ) + τ j (cid:16) A ′ ( ∇ y (1) j ) − A ′ ( ∇ y (2) j ) , ∇ δy j (cid:17) = τ j ( δu j , δy j ) − τ j (cid:16) ψ ′ ( y (1) j ) − ψ ′ ( y (1) j ) , δy j (cid:17) ≤ τ j ǫ k δu j k H ′ + τ j ǫ k δy j k H + τ j C ψ k δy j k . In the last step we used scaled Young’s inequality with < ǫ < min(1 , C A ) . We nowsum over j = 1 , . . . , J and get k δy J k + ˜ C A J X j =1 τ j k∇ δy j k ≤ k δy k + J X j =1 τ j ǫ k δu j k H ′ + ˜ C ψ J X j =1 τ j k δy j k (26)for all ≤ J ≤ N τ . Here we defined ˜ C A = C A − ǫ and ˜ C ψ := ǫ + 2 C ψ Omitting thegradient term on the left and absorbing the J -th term from the right, we obtain k δy J k ≤ − ˜ C ψ τ J ) k δy k + J X j =1 τ j ǫ k δu j k H ′ + ˜ C ψ − ˜ C ψ τ J J − X j =1 τ j k δy j k ≤ C ψ,τ k δy k + N τ X j =1 τ j ǫ k δu j k H ′ + C ψ,τ ˜ C ψ J − X j =1 τ j k δy j k , C ψ,τ := − ˜ C ψ τ > . To this we apply thediscrete Gronwall Lemma which yields k δy J k ≤ k δy k + N τ X j =1 τ j ǫ k δu j k H ′ C ψ,τ exp C ψ,τ ˜ C ψ J − X j =1 τ j ≤ k δy k + N τ X j =1 τ j ǫ k δu j k H ′ C ψ,τ exp (cid:16) C ψ,τ ˜ C ψ T (cid:17) . (27)Inserting this into eq. (26) we finally get for all J = 1 , . . . , N τ ˜ C A J X j =1 τ j k∇ δy j k ≤ k δy k + N τ X j =1 τ j ǫ k δu j k H ′ (cid:16) C ψ,τ ˜ C ψ T exp (cid:16) C ψ,τ ˜ C ψ T (cid:17)(cid:17) , (28)which together with eq. (27) and the boundedness of C ψ,τ independently of τ yields theinequality eq. (25).A similar result as that of theorem 2.3 could also be obtained by using results on mono-tone operators, see e.g. [28]. Together with an argument formerly found by Stampacchiaone would obtain the regularity y j ∈ L ∞ (Ω) ∩ H (Ω) at each step of our time regular-ization [36, 13]. These results are applicable if τ is sufficiently small such that the term y j + τ j ψ ′ ( y j ) becomes monontonic. However this regularity comes with restriction onthe space dimension d .Our approach also allows taking the limit τ → providing the existence of a solution inthe time-continuous case, as is demonstrated in the following theorem. Theorem 2.5.
Let Assumptions 1.1 hold. Then for every u ∈ L (0 , T ; L (Ω)) there exists a unique weaksolution y ∈ L ∞ (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) to eq. (4) , i.e. to y (0) = y a.e. in Ω and Z Q ∂ t yη + A ′ ( ∇ y ) T ∇ η + ψ ′ ( y ) η = Z Q uη ∀ η ∈ L (0 , T ; H (Ω)) . Moreover the solution depends Lipschitz-continuously on ( y , u ) . More precisely it holds k y − y k C ([0 ,T ]; L (Ω)) ∩ L (0 ,T ; H (Ω)) ≤ C ψ,A,T (cid:16) k y , − y , k L (Ω) + k u − u k L (0 ,T ; H (Ω) ′ ) (cid:17) (29) where y , y are the solutions to the data ( y , , u ) and ( y , , u ) respectively.Proof. Given u ∈ L (0 , T ; L (Ω)) we choose a sequence of discretizations u τ ∈ U τ with u τ ⇀ u in L (0 , T ; L (Ω)) for τ → . This allows for the choice of a ¯ c > with k u τ k ≤ ¯ c .9et y τ be the solution of eq. (10) corresponding to u τ . Then the estimates eq. (21) hold.Hence, for τ → there exists a (sub-)sequence satisfying y τ ⇀ y in L (0 , T ; H (Ω)) ,y τ ∗ ⇀ y in L ∞ (0 , T ; H (Ω)) ,y τ → y in L (0 , T ; L (Ω)) ,∂ − τt y τ ⇀ ∂ t y in L (0 , T ; L (Ω)) ,ψ ′ ( y τ ) ⇀ ψ ′ ( y ) in L (0 , T ; L (Ω)) (30)where the weak limits are identified using pointwise almost-everywhere convergence of y τ and continuity of ψ ′ . With that we can take the limit in the variational inequalityeq. (24) to obtain that y satisfies Z Q ∂ t y ( η − y )+ A ( ∇ η ) − A ( ∇ y )+ ψ ′ ( y )( η − y ) − u ( η − y ) ≥ ∀ η ∈ L (0 , T ; H (Ω)) . (31)Remark 2.2 yields that y solves also the variational equality eq. (4). Furthermore, usingweak ( ∗ ) lower-semicontinuity the solution y satisfies k ∂ t y k L (0 ,T ; L (Ω)) + k y k L ∞ (0 ,T ; H (Ω)) + k ψ ′ ( y ) k L (0 ,T ; L (Ω)) ≤ C A,ψ,y (¯ c ) . (32)The uniqueness follows from the Lipschitz-continuity shown further below.Recall that the choice of discretization (given by the choice of the intervals) was arbi-trary. Furthermore on the way to obtain y from y j,n we had to take subsequences twice.Whatever choice made we would have got a y satisfying the same variational inequality.However, this variational inequality has a unique solution, so the whole sequence has toconverge. Summarized, for all discretization, we get a sequence y j,n that for n → ∞ andthen τ → ( j → ∞ ) results in the same limit y satisfiying the variational inequality.Finally, we will turn our attention to the Lipschitz-continuity. Let y , y ∈ H (0 , T ; L (Ω)) ∩ L (0 , T ; H (Ω)) be two solutions belonging to ( y , , u ) and ( y , , u ) and define δy := y − y , δu := u − u as well as δy := y , − y , . Subtracting the defining equationsand testing with δy , we obtain for almost all t ∈ [0 , T ] ddt k δy k + ( A ′ ( ∇ y ) − A ′ ( ∇ y ) , ∇ δy ) = ( δu, δy ) − (cid:0) ψ ′ ( y ) − ψ ′ ( y ) , δy (cid:1) . Using strong monotonicity of A ′ , ( ψ ′ ( y ) − ψ ′ ( y ) , y − y ) ≥ − C ψ k y − y k as well asCauchy-Schwarz and scaled Young’s inequality yields ddt k δy k ≤ ddt k δy k + ( C A − ǫ ) k∇ δy k ≤ ǫ k δu k H (Ω) ′ + ( C ψ + ǫ ) k δy k , (33)with < ǫ < C A . Integrating from to s ≤ T to apply Gronwall’s inequality gives(note that y , y ∈ C ([0 , T ]; L (Ω)) by imbedding) k δy ( s ) k ≤ exp ((2 C ψ + ǫ ) T ) (cid:16) k δy (0) k + ǫ k δu k L (0 ,T ; H (Ω) ′ ) (cid:17) . (34)10ence k δy k L ∞ (0 ,T ; L (Ω)) ≤ C ψ,A,T (cid:16) k δy (0) k + ǫ k δu k L (0 ,T ; H (Ω) ′ ) (cid:17) and then eq. (33)yields k∇ δy k L (0 ,T ; L (Ω)) ≤ C ψ,A,T (cid:16) k δy (0) k + ǫ k δu k L (0 ,T ; H (Ω) ′ ) (cid:17) (35)with a generic constant C ψ,A,T depending only on ψ, A, T . Together the estimates implyeq. (29).Note that eq. (29) is the time-continuous equivalent to eq. (25) and the counterparts toeq. (34) and eq. (35) are eq. (27) and eq. (28), respectively. Especially note that theconstants equal if one sets τ = 0 there. However, since the convergence of y τ is notshown in L ∞ (0 , T ; L (Ω)) we cannot built the proof of eq. (29) on eq. (25).Including results from the preceding proof also the following statement regarding theconvergence of the time discretization holds. Theorem 2.6.
Let Assumptions 1.1 and u ∈ L (0 , T ; L (Ω)) hold. Then for everysequence of discretizations u τ ∈ U τ with u τ ⇀ u in L (0 , T ; L (Ω)) , the correspond-ing solutions y τ ∈ Y τ of the time discretized equation eq. (10) converge to the solution y ∈ H (0 , T ; L (Ω)) ∩ L ∞ (0 , T ; H (Ω)) of the continuous problem eq. (4) in the spacesspecified in eq. (30) . Furthermore we have as τ → t ∈ [0 ,T ] k y τ ( t ) − y ( t ) k L (Ω) → . (36) Proof.
It only remains to show the last convergence stated. Using the definitions fromeq. (7), for given y τ we define its linear interpolant z τ , i.e. z τ ( t ) | I j = y j − + ( t − t j − ) ∂ − τt y τ ( t j ) . Note that from eq. (21) we have k z τ k H (0 ,T ; L (Ω)) ∩ L ∞ (0 ,T ; H (Ω)) ≤ C independent from τ . By the compact imbedding L ∞ (0 , T ; H (Ω)) ∩ H (0; T ; L (Ω)) ֒ → C ([0 , T ]; L (Ω)) (see Aubin–Lions–Simon compactness theorem, e.g. in [35]) we deducethe existence of a z such that (possibly for a subsequence) z τ → z in C ([0 , T ]; L (Ω)) .In addition for t = βt j + (1 − β ) t j − with β ∈ (0 , we find k y τ ( t ) − z τ ( t ) k L (Ω) = (1 − β )( t j − t j − ) k ∂ − τt y τ ( t j ) k L (Ω) ≤ (1 − β ) τ k ∂ − τt y τ k L (0 ,T ; L (Ω)) ≤ C A,ψ,y τ (37)independent of t . Consequently it holds max t ∈ [0 ,T ] k y τ ( t ) − z ( t ) k L (Ω) → as τ → .Note that the limit z is uniquely given by y due to the pointwise a.e. convergence of y τ (cf. eq. (30)) and in addition the whole sequence converges.Since the state equation is a result of the gradient flow of the energy E given in eq. (1),this functional decreases in time when there is no input, i.e. u = 0 . The discretizationshall inherit this property. 11 heorem 2.7. Let Assumptions 1.1 and τ ≤ /C ψ hold. Then the scheme eq. (10) forthe state equation is energy stable, i.e. for u τ = 0 the energy functional E is decreasingin time.Proof. We test eq. (10) with the difference y j − y j − and obtain τ j k y j − y j − k + (cid:0) A ′ ( ∇ y j ) , ∇ y j − ∇ y j − (cid:1) + (cid:0) ψ ′ ( y j ) , y j − y j − (cid:1) = 0 (38)The convexity of A (recall A ′ is strongly monotone) yields ( A ′ ( ∇ y j ) , ∇ y j − ∇ y j − ) ≥ A ( ∇ y j ) − A ( ∇ y j − ) for the second term. The third term can be estimated by thefollowing relation ψ ′ ( y j )( y j − y j − ) ≥ ψ ( y j ) − ψ ( y j − ) − C ψ ( y j − y j − ) . (39)This follows from the fact that this holds for f n approximating ψ as in Assumptions 1.1.Collecting terms and using the definition of the Ginzburg-Landau energy eq. (1) one finds τ j − C ψ ! k y j − y j − k + [ E ( y j ) − E ( y j − )] ≤ (40)and thus E ( y j ) ≤ E ( y j − ) if τ ≤ /C ψ .The result from theorem 2.7 can be applied to the discretizations assumed in theorem 2.3and theorem 2.4 since it provides the less restricting assumption on the step length τ .Finally let us comment on possible choices the function A for the quasilinear term andthe function ψ in particular regarding to the application to optimal control of anisotropicAllen-Cahn equations. Remark 2.8.
The Assumptions 1.1 on A are fulfilled when1. A ∈ C ( R ) is convex, -homogeneous and satisfies A ( p ) > for p = 0 as in [19].In this case the analysis can actually be done with only A ∈ C ( R ) as describedin the reference. If A ( p ) = γ ( p ) then sufficient conditions on γ can be founde.g. in [22]. In the forthcoming paper [4] we will discuss optimal control includinganisotropies arising for γ ( p ) = P Ll =1 ( p T G l p ) / with symmetric positive definite G l ∈ R d × d and show the relevant properties of γ . For the Allen-Cahn equationsuch anisotropies are studied in [2, 1].2. A ( p ) = (cid:16)P Ll =1 ( p T G l p + δ ) / (cid:17) with symmetric positive definite matrices G l ∈ R d × d and δ > , which is a regularization of A given above. A is not 2-homogeneousbut in C ( R d ) and hence suitable for numerical optimal control approaches. Detailswill be found in the forthcoming article [4]. he Assumptions 1.1 on ψ are fulfilled in the following cases:1. if ψ ∈ C ( R ) is bounded from below, ψ ′′ ≥ − C ψ for some C ψ ≥ and lim t →±∞ ψ ′′ ( t ) =+ ∞ .Because in this case one can choose e.g. x − > large enough such that ψ ′′ ≥ , ψ ′ ≥ on [ x − , ∞ ) and define with x n := argmin x ∈ [ x n − +1 , ∞ ) ψ ′′ ( x ) the approx-imation on [0 , x n ] by f n := ψ , for x > x n as f n ( x ) := ψ ( x n ) + ψ ′ ( x n )( x − x n ) + ψ ′′ ( x n )( x − x n ) . Then use this construction respectively on ( −∞ , .2. for the double well potential ψ ( y ) = ( y − , since then the conditions in 1.hold.In addition to the regularity y ∈ L ∞ (0 , T ; H (Ω)) for the solution of eq. (4) shownin theorem 2.5 this potential yields together with the estimate eq. (32) also theregularity y ∈ L (0 , T ; L (Ω)) for all space dimensions.3. for regularizations of the obstacle potential ψ obst which is (1 − x ) on [ − , and ∞ elsewhere, as for example:• the regularization considered in [6] for analyzing the solution of the isotropicAllen-Cahn or Cahn-Hilliard variational inequalities. There ψ obst is regular-ized to ψ ∈ C by a smooth continuation with a cubic polynomial in a neigh-borhood ± (1 , δ ) and then by a quadratic polynomial (cf. formula (2.9)there).• the Moreau-Yosida regularization of ψ obst , i.e. ψ ∈ C with ψ ( x ) = (1 − x ) + s (min { x + 1 , } ) + s (max { x − , } ) . It is e.g. used in [24] to study theoptimal control of isotropic Allen-Cahn inequalities and to obtain a numericalapproach. Having shown the existence of solutions to the discretized eq. (10) and time-continuousstate equation eq. (4), we will further develop our results to show existence of solutionsto the pertinent control problems eq. (9) and eq. (3).
Theorem 3.1.
Let Assumptions 1.1 be fulfilled and max j τ j := τ < C ψ ) hold. Thenfor every y Ω ∈ L (Ω) the control problem eq. (9) –eq. (10) has at least one solution in U τ × Y τ .Proof. The requirements assure that theorem 2.3 is applicable and for every u τ ∈ U τ wefind a unique solution S τ ( u τ ) = y τ ∈ Y τ of eq. (10). Since the feasible set { ( u τ , y τ ) | y τ = S τ ( u τ ) for u τ ∈ U τ } is nonempty and the cost functional in eq. (9) is bounded from below13e can deduce the existence of an infimum ι and of a minimizing sequence (( u ( m ) τ , y ( m ) τ )) m with ι := lim m →∞ J ( u ( m ) τ , y ( m ) τ )) . If u ( m ) τ ∈ L (0 , T ; L (Ω)) was unbounded so would be J which would contradict its convergence to an infimum. Hence there exists a constant ¯ c τ > possibly depending on τ with k u ( m ) τ k L (0 ,T ; L (Ω)) ≤ ¯ c τ for all m and we can extracta weakly convergent subsequence denoted the way u ( m ) τ ⇀ u ∗ τ in L (0 , T ; L (Ω)) . Fromtheorem 2.3 we obtain independent from m k y ( m ) τ k L ∞ (0 ,T ; H (Ω)) + k ψ ′ ( y ( m ) τ ) k L (0 ,T ; L (Ω)) ≤ C A,ψ,y (¯ c τ ) . (41)This yields y ( m ) τ → y τ ∗ in L (0 , T ; L (Ω)) and almost everywhere, as well as ψ ′ ( y ( m ) τ ) ⇀ ψ ′ ( y ∗ τ ) in L (0 , T ; L (Ω)) possibly for a subsequence. Since U τ is finitedimensional in time and due to the compact imbedding L (Ω) ֒ → H (Ω) ′ we obtain u ( m ) τ → u ∗ τ in L (0 , T ; H (Ω) ′ ) . So the Lipschitz-continuity stated in theorem 2.4 inaddition yields y ( m ) τ → y τ ∗ in L (0 , T ; H (Ω)) . Now we can take the limit in the stateequation and obtain ( y ∗ j − y ∗ j − , ϕ ) + τ j ( A ′ ( ∇ y ∗ j ) , ∇ ϕ ) + τ j ( ψ ′ ( y ∗ j ) , ϕ ) = τ j ( u ∗ j , ϕ ) j = 1 , . . . , N. (42)The convergence of the second term arises from the fact that A ′ : L (Ω) → L (Ω) is aNemytskii operator. From eq. (42) we conclude that y ∗ τ = S τ ( u ∗ τ ) and hence ( u ∗ τ , y ∗ τ ) isfeasible and its optimality follows by using the weak lower-semicontinuity of J .Similarly we can show the existence of the optimal control in the time continuous settinggiven the control-to-state operator S : u → y and the estimates eq. (32) for y providedin the proof of theorem 2.5. Theorem 3.2.
If Assumptions 1.1 and y Ω ∈ L (Ω) hold, then there exists a solution tothe optimization problem eq. (3) –eq. (4) .Proof. As in the proof of theorem 3.1 we obtain a minimizing sequence ( u m , y m ) with y m = S ( u m ) where u m is bounded and consequently providing a constant ¯ c such thateq. (32) holds independently of m , i.e. k ∂ t y m k L (0 ,T ; L (Ω)) + k y m k L (0 ,T ; H (Ω)) + k ψ ′ ( y m ) k L (0 ,T ; L (Ω)) ≤ ¯ c. (43)From this we get a subsequence ( u m , y m ) with u m converging weakly to a ¯ u in L (0 , T ; L (Ω)) , and y m converging to a ¯ y weakly in L (0 , T ; H (Ω)) ∩ H (0 , T ; H (Ω) ′ ) ֒ → C ([0 , T ]; L (Ω)) ,strongly in L ( Q ) with ¯ y (0) = y and pointwise almost everywhere in Q . Moreover, ∂ t y m and ψ ′ ( y m ) converge weakly to ∂ t ¯ y , respectively to ψ ′ (¯ y ) in L (0 , T ; L (Ω)) . Inorder to obtain ¯ y = S (¯ u ) we need to be able to pass to the limit also in the A ′ -term ofeq. (4). Given the fact that A ′ : L ( Q ) → L ( Q ) is a Nemytskii operator it is sufficientto show the strong convergence ∇ y m → ∇ ¯ y in L (0 , T ; L (Ω)) . Then finally, the weaklower-semicontinuity of J provides (¯ y, ¯ u ) being a minimizer of J .14he time derivative is monotone if y m (0) − ¯ y (0) = 0 . Hence we have h ∂ t ¯ y, y m − ¯ y i ≤ h ∂ t y m , y m − ¯ y i , and y m = S ( u m ) yields ( A ′ ( ∇ y m ) , ∇ y m − ∇ ¯ y ) ≤ ( u m , y m − ¯ y ) − ( ψ ′ ( y m ) , y m − ¯ y ) − h ∂ t ¯ y, y m − ¯ y i . Recalling the convergence properties of y m together with k u m k + k ψ ′ ( y m ) k ≤ C , theright hand side vanishes in the limit m → ∞ . From strong monotonicity we obtain C k∇ y m − ∇ ¯ y k ≤ ( A ′ ( ∇ y m ) , ∇ y m − ∇ ¯ y ) − ( A ′ ( ∇ ¯ y ) , ∇ y m − ∇ ¯ y ) , where the second term on the right hand side vanishes in the limit by weak convergenceand we have just shown that the limit of the first one can be bounded by from above.This finally yields the desired strong convergence of ∇ y m in L ( Q ) .Note that for the convergence ∇ y m → ∇ y in L (0 , T ; L (Ω)) we could not use eq. (29)like we could use eq. (25) for theorem 3.1. The reason is that in the time continu-ous case we do not have the analogon to the compact imbedding L (Ω) N ֒ → H (Ω) ′ N ( L (0 , T ; L (Ω)) ֒ → L (0 , T ; H (Ω) ′ ) is not compact). Therefore we had to show theconvergence more directly.Finally we obtain a convergence result for the discrete optimal controls like e.g. in [38]or [23]. Theorem 3.3.
Let the previous assumptions on A and ψ hold. Consider a sequence ofglobal optimal controls ( u τ , y τ ) τ of eqs. (9) to (10) belonging to a sequence of discretiza-tions with τ → . Then there exists a subsequence with u τ → u in L (0 , T ; L (Ω)) where ( u, y ( u )) solves eqs. (3) to (4) .Proof. First we choose an arbitrary u ∗ ∈ L (0 , T ; L (Ω)) and a sequence u ∗ τ ∈ U τ with u ∗ τ → u ∗ in L (0 , T ; L (Ω)) . Hence y ∗ τ = S τ ( u ∗ τ ) is bounded in L ∞ (0 , T ; H (Ω)) dueto eq. (21). Now let ( u τ ) τ be the sequence of global minimizers to eq. (9) subject toeq. (10). Then J ( y τ , u τ ) ≤ J ( y ∗ τ , u ∗ τ ) ≤ c implies that ( u τ ) τ is bounded in L (0 , T ; L (Ω)) and we deduce a subsequence with u τ ⇀ u in L (0 , T ; L (Ω)) . Then theorem 2.6 yieldsthat for y τ = S τ ( u τ ) and y = S ( u ) we have the strong convergence y τ ( T, · ) → y ( T, · ) in L (Ω) . Respectively, given some arbitrary sequence ˜ u τ with ˜ u τ → ˜ u in L (0 , T ; L (Ω)) we obtain the latter also for ˜ y τ and ˜ y . This yields J ( y, u ) ≤ lim inf τ → J ( y τ , u τ ) ≤ lim inf τ → J (˜ y τ , ˜ u τ ) = J (˜ y, ˜ u ) . (44)Since ˜ u was arbitrary this yields the global optimality of u . Plugging in ˜ u = u yieldsthe convergence k u τ k → k u k and therefore with the weak convergence also the strongconvergence u τ → u in L (0 , T ; L (Ω)) . 15 emark 3.4. If instead of the cost functionals eq. (3) and eq. (9) one considers thecost functionals with a target function y Q in L (0 , T ; L (Ω)) given over the whole timehorizon min J ( u, y ) := 12 k y − y Q k L ( Q ) + λ k u k L ( Q ) (45) and its discrete counterpart min Y τ × U τ J τ ( y τ , u τ ) = 12 N X j =1 τ j k y j − y Q,j k + λ N X j =1 τ j k u j k , (46) with y Q,τ ∈ Y τ and y Q,τ → y Q in L (0 , T ; L (Ω)) , the theorems of this section still holdtrue with proofs following the same lines. Acknowledgements
The authors gratefully acknowledge the support by the RTG 2339 “Interfaces, ComplexStructures, and Singular Limits” of the German Science Foundation (DFG).
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