Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws

TL;DR

This paper studies optimal control problems constrained by nonlocal conservation laws, proves the existence of minimizers, analyzes their convergence to local limits, and provides numerical illustrations.

Contribution

It establishes the existence of minimizers for nonlocal control problems and their convergence to local problems as the kernel approaches a delta function.

Findings

Existence of minimizers for a broad class of nonlocal control problems.

Convergence of nonlocal minimizers to local minimizers as the kernel becomes singular.

Numerical results illustrating the theoretical findings.

Abstract

This contribution considers optimal control problems subject to nonlocal conservation laws -- those in which the velocity depends nonlocally (i.e., via a convolution) on the solution -- and the so-called singular limit. First, the existence of minimizers is demonstrated for a broad class of optimal control problems, involving optimization over the initial datum, velocity, and nonlocal kernel for classical tracking-type $L^2$ cost functionals. Then, it is proven that the obtained minimizers converge to minimizers of the corresponding local optimal control problem when the kernel function of the convolution is of exponential type and approaches a Dirac distribution. Finally, some numerical results are presented.