Input-to-State Stabilization of 1-D Parabolic PDEs under Output Feedback Control

TL;DR

This paper develops an output feedback control method for 1-D parabolic PDEs with boundary disturbances, ensuring input-to-state stability using kernel-based backstepping and Lyapunov techniques.

Contribution

It introduces a control design using time-invariant kernels and Lyapunov methods to achieve input-to-state stabilization for parabolic PDEs with boundary disturbances.

Findings

Control scheme stabilizes PDEs with boundary disturbances

Lyapunov method confirms stability in $L^inity$-norm

Numerical simulations validate theoretical results

Abstract

This paper addresses the problem of input-to-state stabilization for a class of parabolic equations with time-varying coefficients, as well as Dirichlet and Robin boundary disturbances. By using time-invariant kernel functions, which can reduce the complexity in control design and implementation, an observer-based output feedback controller is designed via backstepping. By using the generalized Lyapunov method, which can be used to handle Dirichlet boundary terms, the input-to-state stability of the closed-loop system under output feedback control, as well as the state estimation error system, is established in the spatial $L^\infty$-norm. Numerical simulations are conducted to confirm the theoretical results and to illustrate the effectiveness of the proposed control scheme.