Stabilization of an unstable wave equation using an infinite dimensional dynamic controller

TL;DR

This paper proposes an infinite dimensional boundary control law to stabilize an unstable wave equation, extending the domain to counteract anti-damping, achieving exponential stability with robustness to uncertainties.

Contribution

It introduces a novel infinite dimensional control method that stabilizes an anti-stable wave system by domain extension, outperforming backstepping in robustness and decay-rate control.

Findings

Achieves exponential stabilization of an anti-stable wave equation.

Robust to uncertainties in wave speed and damping coefficients.

Compared favorably to backstepping control approach.

Abstract

This paper deals with the stabilization of an anti-stable string equation with Dirichlet actuation where the instability appears because of the uncontrolled boundary condition. Then, infinitely many unstable poles are generated and an infinite dimensional control law is therefore proposed to exponentially stabilize the system. The idea behind the choice of the controller is to extend the domain of the PDE so that the anti-damping term is compensated by a damping at the other boundary condition. Additionally, notice that the system can then be exponentially stabilized with a chosen decay-rate and is robust to uncertainties on the wave speed and the anti-damped coefficient of the wave equation, with the only use of a point-wise boundary measurement. The efficiency of this new control strategy is then compared to the backstepping approach.