Input-to-state stabilization of $1$-D parabolic equations with Dirichlet boundary disturbances under boundary fixed-time control

TL;DR

This paper develops a boundary feedback control method for 1-D parabolic equations that achieves input-to-state stability with boundary disturbances while maintaining fixed-time stability, using backstepping and Lyapunov techniques.

Contribution

It introduces a novel control scheme combining backstepping and splitting methods to ensure ISS and FTS for parabolic PDEs with boundary disturbances.

Findings

The proposed controller guarantees ISS under Dirichlet boundary disturbances.

Fixed-time stability is achieved with a prescribed fixed time.

Numerical simulations confirm the effectiveness of the control scheme.

Abstract

This paper addresses the problem of stabilization of $1$-D parabolic equations with destabilizing terms and Dirichlet boundary disturbances. By using the method of backstepping and the technique of splitting, a boundary feedback controller is designed to ensure the input-to-state stability (ISS) of the closed-loop system with Dirichlet boundary disturbances, while preserving fixed-time stability (FTS) of the corresponding disturbance-free system, for which the fixed time is either determined by the Riemann zeta function or freely prescribed. To overcome the difficulty brought by Dirichlet boundary disturbances, the ISS and FTS properties of the involved systems are assessed by applying the generalized Lyapunov method. Numerical simulations are conducted to illustrate the effectiveness of the proposed scheme of control design.