Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain

TL;DR

This paper investigates how the shape of a domain can be used to control solutions of heat and wave equations, demonstrating controllability results for both continuous and discretized models.

Contribution

It establishes new controllability results for shape-based control of parabolic and hyperbolic PDEs, including discretized approximations.

Findings

Approximate controllability for linearized parabolic problem

Exact local controllability for hyperbolic equations

Controllability results extend to finite difference discretizations

Abstract

In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution into an open subdomain and at a given time, when the right hand side source term is known. The mapping that associates this trace to the shape of the domain is nonlinear. We show (i) an approximate controllability property for the linearized parabolic problem and (ii) an exact local controllability property for the linearized and the nonlinear equations in the hyperbolic case. We then address the same questions in the context of a finite difference spatial semi-discretization in both the parabolic and hyperbolic problems. In this discretized case again we prove a local controllability result for the parabolic problem, and an exact controllability for the hyperbolic case, applying a local surjectivity theorem together with a unique continuation property of the underlying adjoint discrete system.