# Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type

## TL;DR

This paper investigates the existence and stability of optimal controls in coupled nonlinear systems with non-smooth coefficients, introducing solenoidal admissible controls and establishing conditions for Mosco-stability.

## Contribution

It introduces the class of solenoidal admissible controls and proves existence and Mosco-stability of optimal controls for coupled Hammerstein-type systems.

## Key findings

- Existence of optimal control under special coefficient assumptions
- Introduction of solenoidal admissible controls
- Derivation of sufficient conditions for Mosco-stability

## Abstract

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with matrix- valued non-smooth controls in coefficients and a nonlinear equation of Ham- merstein type. Since problems of this type have no solutions in general, we make a special assumption on the coefficients of the state equation and in- troduce 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 sufficient conditions of the Mosco-stability for the given class of OCPs.