Unstable free boundary problems in optimal control theory: existence and regularity

TL;DR

This paper proves regularity and bang-bang properties of solutions in certain constrained optimal control problems related to mathematical physics and biology, using advanced free boundary analysis.

Contribution

It introduces a novel approach to analyze unstable free boundary problems in optimal control, establishing regularity and bang-bang solutions in complex settings.

Findings

Solutions are bang-bang, i.e., characteristic functions of sets.

Boundary of the optimal set is smooth up to a (d-2)-dimensional subset.

In 2D, the boundary is a finite union of smooth curves.

Abstract

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form ``maximise $\int \psi(\Theta_m)-c\int m$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=\chi_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem ``maximise $\int \psi(\Theta_m)$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=\chi_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau \& Weiss and Chanillo, Kenig \& To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.