Stabilization of nonautonomous parabolic equations by a single moving actuator

TL;DR

This paper demonstrates that a single moving actuator can stabilize nonautonomous parabolic equations in rectangular domains, using a control strategy that switches between static actuators and employs a receding horizon approach.

Contribution

It introduces a novel stabilization method using a moving actuator derived from static actuators and a switching control strategy for parabolic equations.

Findings

Moving control effectively stabilizes the system.

Numerical results confirm the stabilization performance.

Switching control based on static actuators is feasible.

Abstract

It is shown that an internal control based on a moving indicator function is able to stabilize the state of parabolic equations evolving in rectangular domains. For proving the stabilizability result, we start with a control obtained from an oblique projection feedback based on a finite number of static actuators, then we used the continuity of the state when the control varies in relaxation metric to construct a switching control where at each given instant of time only one of the static actuators is active, finally we construct the moving control by traveling between the static actuators. Numerical computations are performed by a concatenation procedure following a receding horizon control approach. They confirm the stabilizing performance of the moving control.