OPTIMAL CONTROL OF SOME QUASILINEAR MAXWELL EQUATIONS OF PARABOLIC TYPE

. An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak G(cid:226)teaux-di(cid:27)erent-iability of the control-to-state mapping are proved. Based on these results, (cid:28)rst-order necessary optimality conditions and an associated adjoint calculus are derived.

1. Introduction. We consider the optimal control of a system of quasilinear evolution Maxwell equations that models the behavior of magnetic elds in a vector potential formulation. The state equation is the non-degenerate parabolic equation (1.1) Here, Ω ⊂ R 3 is a bounded and simply connected domain with a connected Lipschitz boundary. We further assume that the electric conductivity σ : Ω → R + is a positive constant, (1.2) By n(x), we denote the outward normal direction in the point x ∈ ∂Ω. If the magnetic reluctivity ν : Ω × R 3 → R + is also a positive constant, then (1.1) is a standard linear evolution equation. However, as in Bachinger et al. [2], we allow ν to be a nonlinear function so that equation (1.1) becomes quasilinear. The mapping s → ν(x, s)s expresses the so-called |B|-|H| curve.
In the application to the magnetization processes we have in mind, the real quantity of interest is the magnetic induction B : Ω × (0, T ) → R 3 that will be represented here by a vector potential y : Ω × (0, T ) → R 3 obtained from equation (1.1). For the given right-hand side f : Ω × (0, T ) → R 3 we shall require some regularity properties. For instance, we assume div f = 0 in Ω. We will briey sketch some special types of associated (control) functions f at the end of this paper.
Our paper contributes to the fast developing theory of optimal control problems that include, as part of the control system, Maxwell's equations. For instance, Maxwell's equations appear in the control of processes of magneto-hydrodynamics (MHD). We mention examplarily [3], [6], [7], [8].
In these papers, the Maxwell equations are considered in a steady state or timeharmonic setting. The time harmonic approach is also used in [2], [5], [9], [10], [17], [18]. For the linear transient case, we mention the paper [4] on the optimal control of the full (linear) time dependent Maxwell system, where the control function is composed as a product of two functions depending on the time and on the space variable, respectively.
The main novelty of this paper is the consideration of quasilinear Maxwell equations of parabolic type in the context of optimal control. Quasilinear Maxwell equations have already been considered in [2]. The extension of optimality conditions to the quasilinear case causes specic diculties related to the existence and uniqueness of solutions to the state equation and the dierentiability of the controlto-state mapping. It turns out that we only have weak Gâteaux-dierentiability. However, since the objective functional is quadratic, this is sucient for deriving rst-order necessary optimality conditions.
Our approach extends results of the seminal paper [19] on the optimal control of certain quasilinear elliptic Maxwell equations. In this paper, main ideas were introduced that we were able to adopt in the context of non-stationary quasilinear systems. For proving existence and uniqueness of the solutions to our quasilinear parabolic Maxwell equations, we rely on results of [15]. In this context, we also mention the monography [14], where dierent important mathematical principles for nonlinear partial dierential equations are discussed.
For convenience, we also introduce a mapping N : Throughout this paper, vector valued functions will be written in boldface and we write L 2 (E) := L 2 (E) 3 for suitable measurable sets E. The associated inner product and norm will be denoted by (·, ·) E and · E , respectively; if E is equal to Ω, we will drop the index. In particular, we have (· , ·) := (· , ·) Ω and · := · Ω . Moreover, we will write a b, if a generic positive constant C exists such that a ≤ C b holds.
The divergence constraint is dened in distributional sense, i.e., for f ∈ L 2 (Ω), we say that For later use, we introduce the following spaces: The space H(div=0, Ω) is equipped with the norm of L 2 (Ω), while V and W(0, T ) are equipped with their known natural norms. Note that W(0, T ) is continuously embedded into C([0, T ]; H) (see for instance Theorem 3.10 from [16] In order to prove the existence and uniqueness of a strong solution to (1.1), we introduce the (nonlinear) operator A in H as follows: We recall [15, p. 158] that an operator A : H ⊃ D(A) → H in a Hilbert space H is said to be accretive (or monotone) , if and maximal accretive, if in addition Range (A + I) = H holds. Lemma 2.4. The operator A is maximal accretive in H. Proof. Let us start with the accretiveness. Indeed for y, z ∈ D(A), we have y − z is in H 0 (curl, Ω) and F(·, curl y) − F(·, curl z) belongs to H(curl, Ω), hence by Green's formula we can write (A(y) − A(z), y − z) = (curl(F(·, curl y) − F(·, curl z)), y − z) = (F(·, curl y) − F(·, curl z), curl(y − z)).
By the property (2.5), we deduce that and hence A is accretive.
Let us proceed with the maximality. Namely, we have to show that I + A is surjective. In other words, for all f ∈ H, the equation If such a solution exists, multiplying by v ∈ V and integrating by parts as before, we nd that The right-hand side of (2.8) denes an element of V while its left-hand side denes a monotone, hemicontinuous map T from V into V . Hence by [15, Corollary II.2.2], problem (2.8) has a unique solution u ∈ V, if T is coercive, that is Here and in all what follows, we denote by · , · the pairing between V and V . But this property directly follows from (2.5), since it yields Now, as u, f are divergence free, the identity (2.8) extends to any v ∈ H 0 (curl, Ω), i.e., As C ∞ 0 (Ω) 3 is included (and dense) in H 0 (curl, Ω), we conclude that u + curl(F(·, curl u)) = f in the distributional sense. This means that u belongs to D(A) and satises u + Proof. As the boundary of Ω is assumed to be connected, Corollary 3.19 in [1] guarantees that for all u ∈ H 0 (curl, Ω) such that div u ∈ L 2 (Ω). As a consequence, we deduce that (2.10) Therefore the map (2.11) As before, f being divergence free, u also satises and therefore curl(F(·, curl u)) = f holds in the distributional sense; hence u belongs to D(A) and satises Then by taking u ∈ D(A) such that A(u) = f , we deduce that From (2.7) (applied with z = 0 and the fact that A(0) = 0), we obtain that curl u = 0.
The regularity y ∈ L ∞ (0, T ; H) is stated in the proof of Theorem IV.4.1 of [15]. Furthermore by (2.12) (as A is single-valued), Ay = −y + ωy + f that also belongs to L ∞ (0, T ; H) since each term has this regularity. Weak solutions come from the following energy estimates. Lemma 2.8. Let y 0 , z 0 ∈ D(A) and f , g : [0, T ] −→ H be absolutely continuous. Let y (resp. z) be the strong solutions of (1.1) corresponding to an initial datum y 0 (resp. z 0 ) and right-hand sides f (resp. g). Then these solutions satisfy (2. 13) Proof. Multiplying the equation Integrating by parts in space, we get (2.14) Note that the same identity holds for z with g instead of f . Hence, making the dierence between (2.14) and the same identity with z instead of y, we nd that Taking v = y − z, we obtain in particular Integrating by parts in time, we nally obtain By (2.5), we deduce that Dropping the rst term of this left-hand side, we get where (2.10) was used in this last estimate. Applying Young's inequality, we nd (2.16) Again by (2.10), the estimate (2.16) yields (2.17) These two estimates prove (2.13). Corollary 2.9. Under the assumptions of Lemma 2.8, we have Proof. In (2.15), using (2.6) and Cauchy-Schwarz's inequality we nd that Hence by the estimate (2.13), we conclude that The conclusion directly follows.
At this stage, we are able to dene the notion of weak solutions.
Theorem 2.11. Let T > 0 be xed and assume that y 0 ∈ H and f ∈ L 2 (0, T ; H). Then problem (1.1) has a unique weak solution y that satises (2. 19) Proof. By Cororally 2.5, there exists a sequence {y 0,n } in D(A) such that Then by Theorem 2.7, for all n ∈ N, problem (1.1) with right-hand side f n and an initial datum y 0,n has a unique strong solution y n . Further, owing to Lemma 2.8 and its Corollary 2.9 applied to y n and y m , we have (2.20) Hence there exists y ∈ W(0, T ) such that y n → y in W(0, T ), as n → ∞.
Starting from (2.14) satised by y n and passing to the limit, we nd that y satises (2.19). Finally the estimate (2.19) follows from (2.18) with y = y n and 3. Optimal control. We will discuss the following optimal control problem that is dened upon the state equation (1.1). We consider the objective functional where λ T , λ Q , and λ f are nonnegative constants with λ T + λ Q > 0, while y Q ∈ L 2 (div=0, Q) and y T ∈ H(div=0, Ω) are given functions. Here, the control function f stands for a distributed current density f : The optimal control problem is where y f denotes the solution of the equation (1.1) associated with the control f , and the set of admissible controls F ad ⊂ L 2 (0, T ; H) is assumed to be non-empty, convex and closed.
Lemma 3.1. The control-to-state mapping G : f → y f for the equation (1.1) is weakly-strongly continuous from L 2 (div = 0, Q) to L 2 (0, T ; V). This means that Proof. We begin with the variational formulation for weak solutions of (1.1) and write for short y n := y fn , The sequence {f n } is bounded in L 2 (0, T ; H), hence, by Theorem 2.11, the sequence {y n } is bounded in W(0, T ) so that we can assume y n y in W(0, T ) with some y ∈ W(0, T ). The embedding W(0, T ) ⊂ L 2 (0, T ; H) is compact so that, after extracting a subsequence again, we can also assume the strong convergence y n → y in L 2 (0, T ; H). Inserting y n − y as test function in the weak formulation for y n , we get Adding suitable terms to both sides, we proceed by  The rst integral of the right-hand side of (3.3) tends to zero, since {y n } converges strongly to y in L 2 (0, T ; H) and {f n } is bounded. The second integral converges to zero, because {y n } converges weakly in W(0, T ), hence also weakly in L 2 (0, T ; V). Also the third integral tends to zero, since {curl y n } converges weakly to curl y in L 2 (Q). In view of all this, the right-hand side of (3.3) converges to zero, hence this holds also for the left-hand side.
We have already proved before that T 0 σ ∂(y n − y) ∂t , y n − y dt hence we deduce that {y n } converges to y in L 2 (0, T ; V).

Reminding (2.22) we can pass to the limit in (3.2) to nd
so that y is a weak solution associated with f . By uniqueness, we have y = y f , and thus y n = y fn → y f (strongly) in L 2 (0, T ; V) as n → ∞.
Let us next prove that our optimal control problem is well-posed, i.e. that there exists at least one optimal control. Thanks to Lemma 2.8, we know that the controlto-state mapping G : f → y f is continuous from L 2 (div=0, Q) ⊃ F ad to W(0, T ). Theorem 3.2 (Existence of an optimal control). If, in addition to the former assumptions, F ad is also bounded or if λ f is positive, then the optimal control problem (OCP) admits at least one optimal controlf ∈ F ad . Proof. Let {f n } ∞ n=1 ⊂ F ad be an inmal sequence, i.e. J(y n , f n ) = J(G(f n ), f n ) → j, as n → ∞, where If F ad is bounded, then its closedness and convexity imply that F ad is weakly sequentially compact. Therefore, we can assume w.l.o.g. that f n f in L 2 (0, T ; H). Thanks to Lemma 3.1, we have y n → yf in L 2 (0, T ; V). Moreover, an inspection of the proof of this Lemma shows that y n ȳ := yf in W(0, T ), hence y n (T ) ȳ(T ) in L 2 (Ω). The lower semicontinuity of the functional J nally yields that and hencef is an optimal control.
Let now λ f be positive. The inmum j satises j ≤ J(y 0 , 0), hence we can restrict the search for an optimal solution to the set of all controls f with J(y f , f ) ≤ J(y 0 , 0). By an optimal solution must belong to the set F ad ∩ {f ∈ L 2 (0, T ; H) : f 2 ≤ 2(λ f ) −1 J(y 0 , 0)}. Again, this is a weakly sequentially compact set so that the proof can be nished in the same way as above. Note that by (2.6), the Jacobian matrix ∂F(x, s)/∂s satises Proof. We rst mention that the weak solution z∈ W(0, T ) of (3.5) is unique. This is a consequence of the inequality s ∂F ∂s (x, curl y f (x, t)) s ≥ ν |s| 2 ∀s ∈ R 3 , for a.a. (x, t) ∈ Q (3.6) that follows from [19], Proposition 3.7. Now we select a sequence {τ n } n∈N of nonzero real numbers tending to zero and consider the solutions y := y f and y τn := y f +τnh associated with the controls f and f + τ n h, respectively. Let us write for convenience y n := y τn . Subtracting the state equations for y n and y (written in strong form), we nd σ ∂(y n − y) ∂t + curl F(·, curl y n ) − F(·, curl y) = τ n h.

(3.7)
Testing the variational formulation with y n − y yields  We already know that v Q curl v Q , hence y n − y L 2 (0,T ;V) τ n h Q (3.9) and therefore y n − y Q + curl (y n − y) Q → 0, n → ∞.
Re-arranging the variational formulation of (3.11), we nd We will conrm below that r n curl v → 0 in L 2 (Q), n → ∞, up to a subsequence. Since τ −1 n (y n − y) z in W(0, T ), we can pass to the limit and obtain A simple inspection of the proof reveals that we have obtained a little bit more: Any subsequence of {τ −1 n (y n − y)} contains a weakly convergent subsequence, and all these subsequences converge weakly to the same limit z. Notice that z is a solution of the linearized equation and hence is unique. Therefore, the whole sequence {τ −1 n (y n − y)} converges weakly and we have proven the desired result of weak Gâteaux-dierentiability.
Moreover, since the embedding of W(0, T ) in L 2 (0, T ; H) is compact, we even know that all subsequences of {τ −1 n (y n − y)} contain a subsequence that converges strongly in L 2 (0, T ; H), again with the same limit. This yields the result on strong Gâteaux dierentiability in L 2 (0, T ; H).
It remains to conrm the strong convergence of a subsequence of r n curl v to 0 in L 2 (Q). Thanks to the estimate (3.4), all entries of r n are functions of L ∞ (Q). Moreover, we know that curl y n → curl y in L 2 (Q). Therefore, a subsequence {curl y n k } k tends to curl y almost everywhere in Q. In view of this, and since F is continuously dierentiable w.r. to s, the entries of r n k converge to zero a.e. in Q. We have a.e. in Q with some C > 0. The right-hand side is integrable on Q. Now, the pointwise convergence of r n k along with Lebesgue's dominated convergence theorem ensure that r n k curl v L 2 (Q) → 0, k → ∞.  It is well known that an optimal controlf minimizingĴ in F ad has to obey the variational inequalityĴ

Adjoint equation and necessary optimality conditions. Let us rst introduce the reduced objective functionalĴ
provided thatĴ is (strongly) Gâteaux-dierentiable atf. In our case, the mapping f → y f (T ) is only weakly dierentiable. Therefore, let us prove thatĴ is Gâteauxdierentiable although the control-to-state mapping is only weakly Gâteaux-dierentiable.
Proof. We havê The second and the third part ofĴ are obviously dierentiable. Notice that the mapping f → y, considered with range L 2 (0, T ; H), is (strongly) Gâteaux-dierentiable by Lemma 3.5. Therefore, it suces to prove the Lemma for the rst termĴ T (f ), The mapping G : f → y f is weakly Gâteaux-dierentiable with range in W(0, T ) by Lemma 3.5, and the mapping y → y(T ) is linear and continuous from W(0, T ) to L 2 (Ω).
Consider the sequence z n = (y n − y)/t n that converges to some z weakly in W(0, T ) as t n → 0, i.e.
To computeĴ T (f )h, we consider The embedding W(0, T ) ⊂ C([0, T ], H) is linear and continuous. Moreover, the mapping y → y(T ) is linear and continuous from W(0, T ) to L 2 (Ω). Let us denote the mapping y → y(T ) from W(0, T ) to L 2 (Ω) by E T . As a continuous linear mapping, E T is also weakly continuous. We proceed by By z n z in W(0, T ), the weak continuity of E T and the strong convergence of G(f + t n h)(T ) to G(f )(T ) in L 2 (Ω), cf. (3.8), we can pass to the limit and nd (3.14) In view of (3.14), the derivative ofĴ atf in the direction h ∈ L 2 (div=0, Q) is given bŷ (3.15) where z = G (f )h is the solution of the linearized equation (3.5).
By an adjoint equation, we are able to transform this expression to one, where the increment h appears explicitely. Denition 3.7 (Adjoint equation). Let f ∈ L 2 (div=0, Q) be given and let y f be the associated state. Then the equation is said to be the adjoint equation associated with y f . The unique solution of (3.16) is called adjoint state associated with f and denoted by ϕ f . Existence and uniqueness of ϕ f will be discussed below.
Let us briey conrm that the adjoint equation has a unique solution ϕ. This follows immediately by the estimate (3.6) that was used to prove the unique solvability of the linearized equation. Obviously, the same inequality is satised for the matrix ∂F ∂s (x, curl y f (x, t)) , hence the dierential operator in the adjoint equation (3.16) is coercive and the existence of a unique ϕ f of W(0, T ) is a fairly standard conclusion.
Theorem 3.8. Iff is optimal for the optimal control problem (3.1), then there exists a unique adjoint state ϕf ∈ W(0, T ) such that the variational inequality is found. Performing an integration by parts with respect to the time in (3.18) yields Inserting the terminal condition σ ϕf (T ) = λ T (yf (T ) − y T ), we arrive in view of the adjoint equation (3.16) at In view of the representation (3.15), this is equivalent tô The claim follows from the general variational inequality (3.13) and from our setting This variational inequality (3.17) is a quite general result that can be discussed in more detail for particular cases of F ad . We consider the following three particular cases: (i) At rst, the unconstrained case is of interest,  (ii) Let us dene for convenience the closed ball of L 2 (Ω) with radius r > 0, We may also consider, for given r > 0, the admissible set (3.20) (iii) Another interesting set is dened as follows: Fix k ∈ N, functions e i ∈ H with e i = 0, real numbers α i < β i , i = 1, . . . , k, and dene  Case (i). If F ad is given by (3.19), then obviously the variational inequality is equivalent tof for a.a. (x, t) ∈ Ω × (0, T ), (3.22) provided that λ f > 0. The positivity of λ f is needed anyway to guarantee the solvability of the optimal control problem. For λ f = 0, the problem is solvable, if the desired state is attainable.
If λ f > 0, then we nd in a standard way for a.a. t ∈ (0, T ) that In other words, we have almost everywherē where P Br : H → H denotes the L 2 (Ω)-projection operator onto B r (0).
Case (iii). Here, we expand the terms in the variational inequality as follows:  and for all real numbers v i with α i ≤ v i ≤ β i , i = 1, . . . , k. Obviously, this splits in k independent variational inequalities  since the functions e i were assumed to be mutually orthogonal. This last variational inequality is equivalent to relation (3.25). We refer to [16,Thm. 2.28] for an analogous situation.