Learning an optimal feedback operator semiglobally stabilizing semilinear parabolic equations

TL;DR

This paper develops and analyzes feedback operators for stabilizing semilinear parabolic equations, utilizing finite-dimensional projections and deep neural networks to achieve semiglobal stability and optimize control energy.

Contribution

It introduces a class of monotone feedback operators based on orthogonal projections and demonstrates their effectiveness, including an approach using deep neural networks for optimal control.

Findings

Feedback operators stabilize semilinear parabolic equations.

Deep neural networks can compute optimal feedback minimizing energy.

Numerical simulations confirm stabilizing performance.

Abstract

Stabilizing feedback operators are presented which depend only on the orthogonal projection of the state onto the finite-dimensional control space. A class of monotone feedback operators mapping the finite-dimensional control space into itself is considered. The special case of the scaled identity operator is included. Conditions are given on the set of actuators and on the magnitude of the monotonicity, which guarantee the semiglobal stabilizing property of the feedback for a class semilinear parabolic-like equations. Subsequently an optimal feedback control minimizing the quadratic energy cost is computed by a deep neural network, exploiting the fact that the feedback depends only on a finite dimensional component of the state. Numerical simulations demonstrate the stabilizing performance of explicitly scaled orthogonal projection feedbacks, and of deep neural network feedbacks.