# Semiglobal exponential stabilization of nonautonomous semilinear parabolic-like systems

## TL;DR

This paper demonstrates that an explicit nonlinear feedback controller using oblique projection can semiglobally stabilize nonautonomous semilinear parabolic systems with polynomial nonlinearities, even with limited actuators and large initial conditions.

## Contribution

It introduces a novel explicit oblique projection nonlinear feedback method for stabilizing complex parabolic systems with time-dependent dynamics.

## Key findings

- Stability achieved with finite actuators covering small subdomains.
- Applicable to systems with arbitrarily large initial conditions.
- Explicit actuator placement guarantees stability in rectangular domains.

## Abstract

It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depend on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.