Global stabilizability to trajectories for the Schl\"ogl equation in a Sobolev norm

TL;DR

This paper studies how to stabilize the Schl"ogl equation to desired trajectories within a specific Sobolev norm, proposing a control method that is robust to different trajectories and includes numerical validation.

Contribution

It introduces a new stabilizing control approach for the Schl"ogl equation that is independent of the target trajectory and provides optimality conditions for constrained controls.

Findings

The proposed control stabilizes the Schl"ogl equation in the Sobolev norm.

Numerical simulations demonstrate the effectiveness of receding horizon controls.

The control method is robust to different trajectories and control bounds.

Abstract

The stabilizability to trajectories of the Schl\"ogl model is investigated in the norm of the natural state space for strong solutions, which is strictly contained in the standard pivot space of square integrable functions. As actuators a finite number of indicator functions are used and the control input is subject to a bound constraint. A stabilizing saturated explicit feedback control is proposed, where the set of actuators and the input bound are independent of the targeted trajectory. Further, the existence of open-loop optimal stabilizing constrained controls and related first-order optimality conditions are investigated. These conditions are then used to compute stabilizing receding horizon based controls. Results of numerical simulations are presented comparing their stabilizing performance with that of saturated explicit feedback controls.