Nonlocal Control in the Conduction Coefficients: Well-Posedness and Convergence to the Local Limit

Abstract

We consider a problem of optimal distribution of conductivities in a system governed by a nonlocal diffusion law. The problem stems from applications in optimal design and more specifically topology optimization. We propose a novel parametrization of nonlocal material properties. With this parametrization the nonlocal diffusion law in the limit of vanishing nonlocal interaction horizons converges to the famous and ubiquitously used generalized Laplacian with SIMP (solid isotropic material with penalization) material model. The optimal control problem for the limiting local model is typically ill-posed and does not attain its infimum without additional regularization. Surprisingly, its nonlocal counterpart attains its global minima in many practical situations, as we demonstrate in this work. In spite of this qualitatively different behavior, we are able to partially characterize the relationship between the nonlocal and the local optimal control problems. We also complement our theoretical findings with numerical examples, which illustrate the viability of our approach to optimal design practitioners