Abstract
In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix-valued L∞ (Ω;RN×N)-controls in coecients and a nonlinear equation of Hammerstein type, where solution nonlinearly depends on L∞ -control. Since problems of this type have no solutions in general, we make a special assumption on the coecients of the state equations and introduce the class of so-called solenoidal admissible controls. Using the direct method in calculus of variations, we prove the existence of an optimal control. We also study the stability of the optimal control problem with respect to the domain perturbation. In particular, we derive the sucient conditions of the Mosco-stability for the given class of OCPs.
Highlights
The aim of this paper is to prove the existence result for an optimal control problem (OCP) governed by the system of a nonlinear monotone elliptic equation with homogeneous Dirichlet boundary conditions and a nonlinear equation of Hammerstein type, and to provide sensitivity analysis of the considered optimization problem with respect to the domain perturbations
Systems with distributed parameters and optimal control problems for systems described by PDE, nonlinear integral and ordinary dierential equations have been widely studied by many authors
The optimal control problem we study here is to minimize the discrepancy between a given distribution zd ∈ Lp(Ω) and a solution of Hammerstein equation z = z(U, v, y), choosing appropriate coecients (U, v) ∈ Uad × Vad, i.e. ˆ
Summary
The aim of this paper is to prove the existence result for an optimal control problem (OCP) governed by the system of a nonlinear monotone elliptic equation with homogeneous Dirichlet boundary conditions and a nonlinear equation of Hammerstein type, and to provide sensitivity analysis of the considered optimization problem with respect to the domain perturbations. It should be mentioned here that solution uniqueness is not typical for equations of Hammerstein type or optimization problems associated with such objects (see [1]). Such property would require rather strong assumptions on operators B and F , which is rather restrictive in view of numerous applications (see [25]). In view of the variational properties of Moscostable problems we can replace the rough domain Ω by a family of moreregular domains {Ωε}ε>0 ⊂ D forming some admissible perturbation and to approximate the original problem by the corresponding perturbed problems [12]
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