Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions I

Abstract

In this article we study an optimal control problem for a nonlinear monotone Dirichlet problem where the controls are taken as matrix-valued coefficients in L ∞ (Ω; R N ×N). For the exemplary case of a tracking cost functional, we derive first order optimality conditions. This is the first part out of two articles. This first part is concerned with the general case of matrix-valued coefficients under some hypothesis, while the second part focuses on the special class of diagonal matrices.

Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

“…In this case (see [12] and [13] ) the first-order optimality conditions can be represented as follows…”

Optimal Control in Matrix-Valued Coefficients for Nonlinear Monotone Problems: Optimality Conditions II

“…In [2] we have derived first-order optimality conditions for optimal control problem (1)-( 4) and carried out their realization under some additional assumptions. We introduced the notion of a quasi-adjoint state ψ ε to an optimal solution y 0 ∈ W 1,p 0 (Ω) that was proposed for linear problems by Serovajskiy [5]) and showed that an optimality system can be recovered in an explicit form if the mapping U ad U → ψ ε (U) possesses the so-called H-property with respect to the pair of spaces L ∞ (Ω; R N ×N ), W 1,p 0 (Ω) .…”

“…As a result, we show that each of the variational problems for the corresponding quasi-adjoint states has a unique solution, and these solutions form a weakly convergent sequence ψ ε θ ,θ ∈ H p U 0 , y θ (Ω) θ→0 in the variable space. This property suffices in order to establish that the optimality system for the problem (1)-( 4), that was derived in [2], remains valid even if H-property does not hold for the quasi-adjoint states.…”

“…Optimality conditions. To derive the optimality conditions for given optimal control problem (10)-(12) in [2], we used the following concept of quasi-adjoint states, that was first introduced for linear problems by Serovajskiy [5].…”

“…As was shown in [2], in order to derive the optimality conditions to problem (10)-(12) in a correct way, the following property of quasi-adjoint states had to be fulfilled. Definition 2.4.…”