Homogenization of an optimal control problem for nonlocal semilinear elasticity with soft inclusions

This paper investigates the asymptotic analysis of an optimal control problem (OCP) posed on a high-contrast elastic medium with soft periodic inclusions, governed by a semilinear elasticity system with a nonlocal term. The domain consists of a connected matrix phase and a soft inclusion phase. The model depends on two independent small parameters: the periodicity $\epsilon >0$ and the contrast $\delta >0$, and the distributed control acts only in the inclusion region. We consider an $L^2$-tracking cost on the displacement and analyze the limit as $(\epsilon ,\delta )\to (0,0)$ in the regime $$\underset{(\epsilon ,\delta )\to (0,0)}{\lim }\frac{\delta }{\epsilon }=\kappa \in (0,+\mathrm{\infty }]\mathrm{.}$$ First, we derive the homogenized (limit) state system associated with this scaling. We then formulate the limit OCP and prove that the limit of the microscopic optimal controls is an optimal control for the limit problem, using a $\mathrm{\Gamma }$-convergence approach.