Stabilizability of Parabolic Equations by Switching Controls Based on Point Actuators

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 is 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 stabilizing properties are analyzed. Numerical simulations validate the approach, illustrating its stabilizing and switching properties.