Boundary Stabilization of Quasilinear Parabolic PDEs that Blow Up in Open Loop for Arbitrarily Small Initial Conditions

TL;DR

This paper introduces a new boundary control method for quasilinear parabolic PDEs that prevents finite-time blow-up, ensuring stability and positivity of solutions even with nonlinear terms, and demonstrates effectiveness through simulations.

Contribution

The paper presents a novel boundary stabilization framework for nonlinear PDEs with blow-up behavior, including explicit region of attraction estimates and adaptable boundary controllers.

Findings

Region of attraction expands with increased diffusion

Controllers prevent finite-time blow-up in simulations

Ensures positivity and exponential stability of solutions

Abstract

We propose a novel framework for stabilization, with an estimate of the region of attraction, of quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow-up phenomena when null boundary inputs are imposed. Using Neumann-type boundary controllers, which are cubic polynomials in boundary measurements, we ensure L2 exponential stability of the origin with an estimate of the region of attraction, boundedness and exponential decay towards zero of the state's max norm, well-posedness, as well as positivity of solutions starting from positive initial conditions. Unlike existing methods, our approach handles nonlinear state-dependent diffusion, convection, and reaction terms. In many cases, our estimate of the size of the region of attraction is shown to expand unboundedly as diffusion increases. Our controllers can be implemented as Neumann, Dirichlet, or mixed-type boundary conditions. Numerical simulations validate the effectiveness of our approach in preventing finite-time blow up.