Abstract
In this paper we study an optimal control problem for a nonlinear monotone Dirichlet problem where the control is taken as -coefficient of -Laplacian. Given a cost function, the objective is to derive first-order optimality conditions and provide their substantiation. We propose some ideas and new results concerning the differentiability properties of the Lagrange functional associated with the considered control problem. The obtained adjoint boundary value problem is not coercive and, hence, it may admit infinitely many solutions. That is why we concentrate not only on deriving the adjoint system, but also, following the well-known Hardy-Poincare Inequality, on a formulation of sufficient conditions which would guarantee the uniqueness of the adjoint state to the optimal pair. MSC:35J70, 49J20, 49J45, 93C73.
Highlights
The aim of this paper is to derive a first-order optimality system for a nonlinear Dirichlet optimal control problem where the control is taken as an L∞-coefficient in a nonlinear state equation
With N ≥, yd ∈ W,p( ) is a given distribution, and y is the solution of a nonlinear Dirichlet problem by choosing an appropriate coefficient u ∈ L∞( ) of p-Laplacian
This has been pursued by Allaire [ ] and many other authors in recent years. Another restriction can be realized via regularity of the coefficients and hard constraints. This procedure has been followed first by Casas [ ] for a scalar problem, as one of the first papers in that direction, and later by Haslinger et al [ ] in the context of what has come to be known as Free Material Optimization (FMO)
Summary
The aim of this paper is to derive a first-order optimality system for a nonlinear Dirichlet optimal control problem where the control is taken as an L∞-coefficient in a nonlinear state equation. In the case when the control is considered in the coefficients of the main part of the state equation, the classical adjoint system often cannot be directly constructed due to the lack of differential properties of the solution to the boundary value problem with respect to control variables It was the main reason why Serovajskiy has proposed the concept of the so-called quasi-adjoint system [ ] and showed that optimality conditions for the linear elliptic control problem in coefficients can be derived, provided the mapping u → ψε(u) possesses the weakened continuity property. = p div |∇yθ – ∇yd|p– (∇yθ – ∇yd) in , ψθ ∈ W ,p( ), where the degeneration occurs in a natural way with respect to the states This concept was proposed for linear problems by Serovajskiy [ ], where it was shown that an optimality system for the optimal control problems in coefficients can be recovered in an explicit form if the mapping Aad u → ψε(u) possesses the so-called weakened continuity property. This property suffices in order to establish that the optimality system for problem ( . )-( . ) remains valid even if the matrix |∇y|p– A(u, y) has a degenerate spectrum
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