Constrained evolution for a quasilinear parabolic equation

TL;DR

This paper develops a feedback control law for a quasilinear parabolic PDE, proving well-posedness and demonstrating finite-time convergence to convex sets under certain conditions.

Contribution

It introduces a novel control approach for quasilinear parabolic equations with obstacle constraints, establishing well-posedness and finite-time set attainment.

Findings

Proved well-posedness and regularity of the modified PDE

Established finite-time convergence to convex sets with large control factors

Demonstrated the control law's effectiveness for obstacle problems

Abstract

In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the Cauchy-Neumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set of the space of square-integrable functions. Then, we consider convex sets of obstacle or double-obstacle type and prove rigorously the following property: if the factor in front of the feedback control is sufficiently large, then the solution reaches the convex set within a finite time and then moves inside it.