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

M C Belhadjoudja, M Maghenem, E Witrant, M Krstic

TL;DR

The paper addresses stabilization of quasilinear parabolic PDEs that can blow up in open loop by developing cubic Neumann-type boundary feedback laws. It proves $L^2$ exponential stability of the origin with a computable region-of-attraction, along with boundedness and exponential decay of the $L^$ norm, well-posedness, and positivity for nonnegative initial data. The feedback laws can be implemented as Neumann, Dirichlet, or mixed boundary controls, and the region of attraction may grow unboundedly with diffusion; the linear case emerges when nonlinear terms vanish. Numerical simulations validate blow-up prevention across cases with nonlinear diffusion, convection, and reaction, and illustrate the impact of boundary-gain choices on convergence speed.

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.

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

TL;DR

The paper addresses stabilization of quasilinear parabolic PDEs that can blow up in open loop by developing cubic Neumann-type boundary feedback laws. It proves exponential stability of the origin with a computable region-of-attraction, along with boundedness and exponential decay of the norm, well-posedness, and positivity for nonnegative initial data. The feedback laws can be implemented as Neumann, Dirichlet, or mixed boundary controls, and the region of attraction may grow unboundedly with diffusion; the linear case emerges when nonlinear terms vanish. Numerical simulations validate blow-up prevention across cases with nonlinear diffusion, convection, and reaction, and illustrate the impact of boundary-gain choices on convergence speed.

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.
Paper Structure (10 sections, 2 theorems, 61 equations, 3 figures)

This paper contains 10 sections, 2 theorems, 61 equations, 3 figures.

Table of Contents

  1. introduction
  2. Problem Formulation
  3. Class of systems
  4. Control-design problem
  5. Main results
  6. Statement of the main results
  7. Sketch of proof of Theorem \ref{['thm1']}
  8. Discussion
  9. Simulation Results
  10. Conclusion and Research Perspectives

Key Result

Theorem 1

Let Assumption ass1 be verified, and let $v_1$ and $v_o$ be given by v1 and v2, respectively. Then, Property prop1 holds with $\lambda_1,\lambda_o,\mu$ given by const1, const2, and const3 respectively, $\omega>0$ being any constant verifying omega_1-omega3, and where $\kappa_o$ is defined in init_func. Additionally, if Assumption ass2 holds, then Property prop2 (well-posedness) is also verified.

Figures (3)

  • Figure 1: Left: response of $\Sigma$ to $u_x(1)=u_x(0):=0$, with $\gamma_1:=0$. Right: response of $\Sigma$ to $u_x(1):=-\lambda_1 u(1)$ and $u_x(0):=\lambda_o u(0)$, with $\gamma_1:=0$.
  • Figure 2: Left: theoretical bound on $t\mapsto \int_{0}^{1}u(x)^2dx$, i.e. the right-hand side of \ref{['decay_L2']} (black); the map $t\mapsto \int_{0}^{1}u(x,t)^2dx$ for $k_1=k_o:=0$ (blue), $k_1=k_o:=1$ (red), and $k_1=k_o:=10$ (green). Right: theoretical bound on $t\mapsto |u(\cdot,t)|_{\infty}^2$, i.e. the right-hand side of \ref{['decay_max']} (black); the map $t\mapsto |u(\cdot,t)|_{\infty}^2$ for $k_1=k_o:=0$ (blue), $k_1=k_o:=1$ (red), and $k_1=k_o:=10$ (green).
  • Figure 3: Left: response of $\Sigma$ to $u_x(1)=u_x(0):=0$, with $\gamma_1:=1$. Right: response of $\Sigma$ to $u_x(1):=-\lambda_1 u(1)-\mu u(1)^3$ and $u_x(0):=\lambda_o u(0)+\mu u(0)^3$, with $\gamma_1:=1$.

Theorems & Definitions (8)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • Proposition 1
  • proof