# Local stabilization of an unstable parabolic equation via saturated controls

## TL;DR

This paper presents a novel saturated feedback control method that stabilizes unstable linear reaction-diffusion equations without requiring the uncontrolled system to be stable, using Lyapunov techniques and matrix inequalities.

## Contribution

It introduces a new control approach for unstable systems, extending stabilization techniques to more general cases with various control types and saturations.

## Key findings

- Effective stabilization of unstable systems demonstrated
- Region of attraction estimated via matrix inequalities
- Numerical simulations confirm control performance

## Abstract

We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of the uncontrolled system and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop system, given in terms of linear and bilinear matrix inequalities. We show that our results can be used with distributed as well as scalar boundary control, and with different types of saturations. The efficiency of the proposed method is demonstrated by means of numerical simulations.