Perturbations of time optimal control problems for a class of abstract parabolic systems

TL;DR

This paper investigates how solutions to abstract parabolic time optimal control problems behave as the system generators converge, with applications to PDEs with oscillating coefficients and homogenization, showing convergence of optimal controls.

Contribution

It establishes the convergence of solutions and optimal controls for parabolic systems with oscillating coefficients to those of the homogenized system, including new results on time and norm optimal control problems.

Findings

Solutions converge in usual norms as generators approach the limit

Optimal controls for oscillating systems tend to those of the homogenized system

New theoretical results on time and norm optimal control problems

Abstract

In this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge, in an appropriate sense, to a given strictly negative operator. Our main application to PDEs systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients, as we encounter in homogenization theory. Our main results assert that, provided that the target is a closed ball centered at the origin and of positive radius, the solutions of the time optimal control problems for the systems with oscillating coefficients converge, in the usual norms, to the solution of the corresponding problem for the homogenized system. In order to prove our main theorem, we provide several new results, which could be of a broader interest, on time and norm optimal control problems.