# Non-local control in the conduction coefficients: well posedness and convergence to the local limit

## TL;DR

This paper investigates non-local diffusion laws for optimal material distribution, showing convergence to local models and demonstrating that non-local problems often attain global minima where local models are ill-posed, supported by theoretical and numerical analysis.

## Contribution

It introduces a novel parametrization of non-local properties, analyzes the well-posedness and convergence to local limits, and compares the behaviors of local and non-local optimal control problems.

## Key findings

- Non-local models attain global minima in many cases.
- Non-local diffusion converges to the generalized Laplacian with SIMP.
- Numerical examples validate the theoretical insights.

## Abstract

We consider a problem of optimal distribution of conductivities in a system governed by a non-local diffusion law. The problem stems from applications in optimal design and more specifically topology optimization. We propose a novel parametrization of non-local material properties. With this parametrization the non-local diffusion law in the limit of vanishing non-local 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 non-local counterpart attains its global minima in many practical situations, as we demonstrate in this work. In spite of this qualitatively different behaviour, we are able to partially characterize the relationship between the non-local 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.