Optimal control of the principal coefficient in a scalar wave equation

TL;DR

This paper develops an optimal control framework for scalar wave equations with coefficients as controls, incorporating total variation and multi-bang penalties, and proposes numerical methods for solution.

Contribution

It introduces a novel control approach with total variation and multi-bang penalties, providing existence, regularity, and numerical solution strategies.

Findings

Existence of optimal controls with discontinuous coefficients.

Enhanced regularity results for the state variable.

Effective numerical algorithms using finite elements and primal-dual splitting.

Abstract

We consider optimal control of the scalar wave equation where the control enters as a coefficient in the principal part. Adding a total variation penalty allows showing existence of optimal controls, which requires continuity results for the coefficient-to-solution mapping for discontinuous coefficients. We additionally consider a so-called "multi-bang" penalty that promotes controls taking on values pointwise almost everywhere from a specified discrete set. Under additional assumptions on the data, we derive an improved regularity result for the state, leading to optimality conditions that can be interpreted in an appropriate pointwise fashion. The numerical solution makes use of a stabilized finite element method and a nonlinear primal-dual proximal splitting algorithm.