Optimal Control Problem for the Weak Nonlinear Equation of Thin Plate With Control at the Coefficient of Lowest Term

The paper deals with an inverse problem of determining the right-hand side of the linear equation of oscillations of thin plates. The problem is reduced to the optimal control problem. Differentiability of the functional is studied. Necessary condition of optimality is derived.


Introduction
It is known that the oscillations of thin plates are described by the differential equations with partial derivatives of the fourth order. For more detail, the reader is referred to (Komkov, 1975) and (Arman, 1977). Therefore, the study of optimal control problems for the equation of the thin plate is of great theoretical and practical importance, see (Kabanikhin, 2009). One of the approaches for solving the inverse problems is the optimization method. The essence of this method lies in the fact that the inverse problem is reduced to the optimal control problem and this new problem is investigated by the methods of optimal control theory.

Formulation of the Problem
Our needs to find the pair of functions , T are given positive numbers.
As a generalized solution of the problem (1) This problem we reduce to the following optimal control problem: to find the minimum of the functional we denote the generalized solution of the problem (1)-(3) corresponding to the control ) , ( y x υ . We regularize the problem (1)-(3), (6) by the following way: instead of the functional (6) consider the next one where 0  α is a positive number.
In the considered problem all condition for existence of optimal control are fulfilled (Lions, 1972). Therefore the new optimal control problem (1)-(3), (7) also has unique solution.

Existence of the Optimal Control
Theorem 1. Under the imposed conditions on the problem data, there exists an optimal control in problem (1)-(3), (7).
Proof. Let's   ad n U υ  be a minimizing sequence, i.e.
Therefore by (8), for solutions of problem (1)-(3) corresponding to n  , we obtain the estimation By (8) and (9), property of weak compactness in the Hilbert spaces and imbedding theorem, it is possible to consider, that as Take into account last relations in the definition of the generalized solution for the problem (1) provides the minimum to functional (7), i.e. is an optimal control.
The completes the proof.

Differentiability of the Functional (7) and Necessary and Sufficient Optimality Conditions
Let us introduce the adjoint problem to (1)-(3), (7) problem for the given control .
To derive the necessary conditions for optimality in the considered problem we take two arbitrary admissible controls .
For this purpose, we use Faedo-Galerkin's method. Take the basis  and the approximate solution for the problem (14)-(16) search in the form Multiplying both sides of (17) If to integrate this equality over t by the imposed conditions, we get where C the constant independent on the estimating quantities and admissible controls.
Due to equivalency of the norms in By virtue of well-known inequality (Ladijenskaya, 1973) Application of the Gronwall's lemma leads to As follows from this inequality from the sequence   Theorem 2. Let's all conditions of the Theorem 1 be satisfied. Then functional (7) is continuously Frechet differentiable on ad U and its differential in the point is the remainder term.
Since u  is a generalized solution of the problem (13)-(16), for arbitrary function is a solution of the problem (10)-(12), for any function  (22) and (23), respectively. After subtracting (22) from (23), we obtain Then from (21) and (24) Then from formula for increment of the functional (25) follows that differential of functional (7) Then as follows from (27) Since the functional (7) is strongly convex in ad U and the problem (1)-(3) is linear, condition (28) is also sufficient for the optimality.
Thus the following theorem is proved.