Stabilizability of parabolic equations by switching controls based on point actuators

TL;DR

This paper demonstrates that switching controls with finite Dirac delta actuators can stabilize nonautonomous parabolic equations, using a receding horizon approach and numerical simulations to validate effectiveness.

Contribution

It introduces a novel switching control strategy with finite point actuators for stabilizing parabolic equations, including practical implementation via receding horizon methods.

Findings

Finite Dirac delta actuators can stabilize parabolic equations.

A receding horizon framework effectively implements switching controls.

Numerical simulations confirm stabilization and switching capabilities.

Abstract

It is shown that a switching control involving a finite number of Dirac delta actuators is able to steer the state of a general class of nonautonomous parabolic equations to zero as time increases to infinity. The strategy is based on a recent feedback stabilizability result, which utilizes control forces given by linear combinations of appropriately located Dirac delta distribution actuators. Then, the existence of a stabilizing switching control with no more than one actuator active at each time instant is established. For the implementation in practice, the stabilization problem is formulated as an infinite horizon optimal control problem, with cardinality-type control constraints enforcing the switching property. Subsequently, this problem is tackled using a receding horizon framework. Its suboptimality and stabilizabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties.