Homogenization of an optimal control problem for nonlocal semilinear elasticity with soft inclusions
Amartya Chakrabortty, Abu Sufian
TL;DR
The paper addresses a two-parameter homogenization problem for a nonlocal semilinear elasticity OCP in a high-contrast medium with soft periodic inclusions. It combines periodic unfolding and Γ-convergence to derive a κ-dependent homogenized state system and a corresponding limit OCP, proving convergence of microscopic optimal controls to a solution of the limit problem. Key contributions include the construction of the two-scale limit energy ${\bf T}_\kappa$, the cell problems yielding the homogenized tensor $A^{\text{hom}}$, and the derivation of a two-scale limit adjoint framework that characterizes the limit control via $\widehat{\Theta}=-(1/\gamma)( {\bf v}_0+\frac{1}{\kappa}\widehat{Z})$. The findings provide a rigorous reduced-order model for design optimization of metamaterials with soft inclusions, capturing global amplitude effects through the nonlocal term and scale-separated or fully coupled limits depending on $\kappa$.
Abstract
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 $\varepsilon>0$ and the contrast $δ>0$, 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 $(\varepsilon,δ)\to(0,0)$ in the regime \[ \lim_{(\varepsilon,δ)\to(0,0)}\fracδ{\varepsilon}=κ\in(0,+\infty]. \] 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 $Γ$-convergence approach.