Optimal Control Problems for Nonlinear Variational Evolution Inequalities

and Applied Analysis 3 and satisfying ‖x‖ L 2 (0,T;V)∩W 1,2 (0,T;V ∗ ) ≤ C 1 ( 󵄨󵄨󵄨󵄨x0 󵄨󵄨󵄨󵄨 + ‖k‖L2(0,T;V∗)) , (17) where C 1 is a constant depending on T. Letf be a nonlinear single valuedmapping from [0,∞)× V intoH. (F) We assume that 󵄨󵄨󵄨󵄨f (t, x1) − f (t, x2) 󵄨󵄨󵄨󵄨 ≤ L 󵄩󵄩󵄩󵄩x1 − x2 󵄩󵄩󵄩󵄩 , (18) for every x 1 , x 2 ∈ V. Let Y be another Hilbert space of control variables and take U = L2(0, T; Y) as stated in the Introduction. Choose a bounded subset U of Y and call it a control set. Let us define an admissible controlUad as Uad = {u ∈ L 2 (0, T; Y) : u is strongly measurable function satisfying u (t) ∈ U for almost all t} . (19) Noting that the subdifferential operator ∂φ is defined by ∂φ (x) = {x ∗ ∈ V ∗ ; φ (x) ≤ φ (y) + (x ∗ , x − y) , y ∈ V} , (20) the problem (1) is represented by the following nonlinear functional differential problem onH: x 󸀠 (t) + Ax (t) + ∂φ (x (t)) ∋ f (t, x (t)) + Bu (t) ,

Recently, initial and boundary value problems for permanent magnet technologies have been introduced via variational inequalities in [1,2] and nonlinear variational inequalities of semilinear parabolic type in [3,4].The papers treating the variational inequalities with nonlinear perturbations are not many.First of all, we deal with the existence and a variation of constant formula for solutions of the nonlinear functional differential equation (1) governed by the variational inequality in Hilbert spaces in Section 2.
Based on the regularity results for solution of (1), we intend to establish the optimal control problem for the cost problems in Section 3.For the optimal control problem of systems governed by variational inequalities, see [1,5].We refer to [6,7] to see the applications of nonlinear variational inequalities.Necessary conditions for state constraint optimal control problems governed by semilinear elliptic problems have been obtained by Bonnans and Tiba [8] using methods of convex analysis (see also [9]).
Let   stand for solution of (1) associated with the control  ∈ U. When the nonlinear mapping  is Lipschitz continuous from R ×  into , we will obtain the regularity for solutions of (1) and the norm estimate of a solution of the above nonlinear equation on desired solution space.Consequently, in view of the monotonicity of , we show that the mapping   →   is continuous in order to establish the necessary conditions of optimality of optimal controls for various observation cases.
In Section 4, we will characterize the optimal controls by giving necessary conditions for optimality.For this, it is necessary to write down the necessary optimal condition due to the theory of Lions [9].The most important objective of such a treatment is to derive necessary optimality conditions 2 Abstract and Applied Analysis that are able to give complete information on the optimal control.
Since the optimal control problems governed by nonlinear equations are nonsmooth and nonconvex, the standard methods of deriving necessary conditions of optimality are inapplicable here.So we approximate the given problem by a family of smooth optimization problems and afterwards tend to consider the limit in the corresponding optimal control problems.An attractive feature of this approach is that it allows the treatment of optimal control problems governed by a large class of nonlinear systems with general cost criteria.

Regularity for Solutions
If  is identified with its dual space we may write  ⊂  ⊂  * densely and the corresponding injections are continuous.The norm on , , and  * will be denoted by || ⋅ ||, | ⋅ |, and || ⋅ || * , respectively.The duality pairing between the element  1 of  * and the element  2 of  is denoted by ( 1 ,  2 ), which is the ordinary inner product in  if  1 ,  2 ∈ .
For  ∈  * we denote (, ) by the value () of  at  ∈ .The norm of  as element of  * is given by Therefore, we assume that  has a stronger topology than  and, for brevity, we may regard that Let (⋅, ⋅) be a bounded sesquilinear form defined in × and satisfying Gårding's inequality where  1 > 0 and  2 is a real number.Let  be the operator associated with this sesquilinear form: (, ) =  (, ) , , ∈ .
Then − is a bounded linear operator from  to  * by the Lax-Milgram Theorem.The realization of  in  which is the restriction of  to is also denoted by .From the following inequalities where is the graph norm of (), it follows that there exists a constant  0 > 0 such that Thus we have the following sequence where each space is dense in the next one with continuous injection.
It is also well known that  generates an analytic semigroup () in both  and  * .For the sake of simplicity we assume that  2 = 0 and hence the closed half plane { : Re  ≥ 0} is contained in the resolvent set of .
If  is a Banach space,  2 (0, ; ) is the collection of all strongly measurable square integrable functions from (0, ) into  and  1,2 (0, ; ) is the set of all absolutely continuous functions on [0, ] such that their derivative belongs to  2 (0, ; ).([0, ]; ) will denote the set of all continuously functions from [0, ] into  with the supremum norm.If  and  are two Banach spaces, L(, ) is the collection of all bounded linear operators from  into , and L(, ) is simply written as L().Here, we note that by using interpolation theory we have  2 (0, ; ) ∩  1,2 (0, ;  * ) ⊂  ([0, ] ; ) . ( First of all, consider the following linear system: By virtue of Theorem 3.3 of [11] (or Theorem 3.1 of [12,13]), we have the following result on the corresponding linear equation of (13).
Lemma 2. Suppose that the assumptions for the principal operator  stated above are satisfied.Then the following properties hold.
Let  be another Hilbert space of control variables and take U =  2 (0, ; ) as stated in the Introduction.Choose a bounded subset  of  and call it a control set.Let us define an admissible control U ad as U ad = { ∈  2 (0, ; ) :  is strongly measurable function satisfying  (t) ∈  for almost all } . ( Noting that the subdifferential operator  is defined by Referring to Theorem 3.1 of [3], we establish the following results on the solvability of (1).
(A)  is symmetric and there exists ℎ ∈  such that for every  > 0 and any  ∈ () where Then for  ∈  2 (0, ; ),  ∈ L(, ), and  0 ∈ () ∩  (1) has a unique solution Remark 4. In terms of Lemma 1, the following inclusion is well known as seen in ( 9) and is an easy consequence of the definition of real interpolation spaces by the trace method (see [4,13]).

Necessary Conditions for Optimality
In this section we will characterize the optimal controls by giving necessary conditions for optimality.For this it is necessary to write down the necessary optimal condition  () ( − ) ≥ 0,  ∈ U ad (84) and to analyze (84) in view of the proper adjoint state system, where () denote the Gâteaux derivative of () at  = .Therefore, we have to prove that the solution mapping   → () is Gâteaux differentiable at  = .Here we note that from Theorem 6 it follows immediately that lim The solution map   → () of  2 (0, ; ) into  2 (0, ; ) ∩ ([0, ]; ) is said to be Gâteaux differentiable at  =  if for any  ∈  2 (0, ; ) there exists a () ∈ L( 2 (0, ; ),  2 (0, ; ) ∩ ([0, ]; ) such that The operator () denotes the Gâteaux derivative of () at  =  and the function () ∈  2 (0, ; )∩([0, ]; )) is called the Gâteaux derivative in the direction  ∈  2 (0, ; ), which plays an important part in the nonlinear optimal control problems.First, as is seen in Corollary 2.2 of Chapter II of [18], let us introduce the regularization of  as follows.
Lemma 10.For every  > 0, define where   = ( + ) −1 .Then the function   is Fréchet differentiable on  and its Frećhet differential   is Lipschitz continuous on  with Lipschitz constant  −1 .In addition, where () 0 () is the element of minimum norm in the set ().
Now, we introduce the smoothing system corresponding to (1) as follows.
Proof.We set  =  − .From Theorem 6, it follows immediately that Let the solution space W 1 of (1) of strong solutions is defined by Theorem 12. Let the assumptions (A), (F1), and (F2) be satisfied.Let  ∈ U ad be an optimal control for the cost function  in (61).Then the following inequality holds: With every control  ∈  2 (0, ; ), we consider the following distributional cost function expressed by where the operator  is bounded from  to another Hilbert space  and   ∈  2 (0, ; ).Finally we are given that  is a self adjoint and positive definite: Let   () stand for solution of (1) associated with the control  ∈  2 (0, ; ).Let U ad be a closed convex subset of  2 (0, ; ).
Theorem 13.Let the assumptions in Theorem 12 be satisfied and let the operators  and  satisfy the conditions mentioned above.Then there exists an element  ∈ U ad such that Furthermore, the following inequality holds: holds, where Λ  is the canonical isomorphism  onto  * and   satisfies the following equation: which is rewritten by (106).Note that  * ∈ ( * , ) and for  and  in  we have ( * Λ  , ) = ⟨, ⟩  , where duality pairing is also denoted by (⋅, ⋅).
Remark 14. Identifying the antidual  with  we need not use the canonical isomorphism Λ  .However, in case where  ⊂  * this leads to difficulties since  has already been identified with its dual.