Stabilization of Quasilinear Parabolic Equations by Cubic Feedback at Boundary with Estimated Region of Attraction

TL;DR

This paper develops a boundary feedback control method for quasilinear parabolic PDEs that prevents finite-time blow-up, guarantees stability, and estimates the region of attraction, even with complex nonlinearities.

Contribution

It introduces a novel cubic feedback boundary control approach that handles nonlinearities and estimates the region of attraction for quasilinear parabolic PDEs.

Findings

Guarantees exponential stability of the origin.

Ensures convergence of solutions to zero in multiple norms.

Shows the region of attraction can grow unbounded with diffusion.

Abstract

For quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow up in open loop, i.e., under null boundary conditions, we provide an estimate of the region of attraction under cubic feedback laws applied at the boundary, using boundary measurements. We guarantee: 1-L 2 and H 1 exponential stability of the origin with an estimate of the region of attraction. 2-Convergence of the H 2 and the C 1 norms of the solutions to zero. 3-Existence and uniqueness of complete classical solutions. 4-Positivity of the solutions starting from positive initial conditions. Unlike existing approaches, our framework handles nonlinear state-dependent diffusion, convection, and (destabilizing) reaction. The cubic terms are used to enlarge our estimate of the region of attraction. The size of the region of attraction is shown, in many cases, to grow unboundedly as diffusion increases. Finally, our controllers can be implemented as Neumann, Dirichlet, or mixed-type boundary conditions.