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Learning an optimal feedback operator semiglobally stabilizing semilinear parabolic equations

Karl Kunisch, Sérgio S. Rodrigues, Daniel Walter

TL;DR

The paper develops a semiglobal exponential stabilization framework for semilinear parabolic equations using finite-dimensional actuators and projection-based feedback $\mathcal{K}_M(P_{\mathcal U_M}y)$. It proves a main stabilization theorem under general operator and nonlinearity assumptions, and provides explicit linear feedbacks (scaled identity and oblique projections) that achieve stabilization for large enough $M$ and monotone strength $\overline\lambda$. To address practical design, it introduces a learning-based approach that parametrizes $\mathcal{K}_M$ with neural networks to minimize a quadratic energy cost while enforcing monotonicity via a penalty, enabling computation of near-optimal feedbacks in the reduced space $\mathcal U_M$. Numerical experiments on parabolic PDEs demonstrate that projection-based controllers stabilize for sufficiently large $M$ and $\lambda$, and that neural-network-based feedbacks can yield lower actuation energy and favorable transient behavior, highlighting a viable path to scalable, data-driven optimal control for nonlinear PDEs.

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.

Learning an optimal feedback operator semiglobally stabilizing semilinear parabolic equations

TL;DR

The paper develops a semiglobal exponential stabilization framework for semilinear parabolic equations using finite-dimensional actuators and projection-based feedback . It proves a main stabilization theorem under general operator and nonlinearity assumptions, and provides explicit linear feedbacks (scaled identity and oblique projections) that achieve stabilization for large enough and monotone strength . To address practical design, it introduces a learning-based approach that parametrizes with neural networks to minimize a quadratic energy cost while enforcing monotonicity via a penalty, enabling computation of near-optimal feedbacks in the reduced space . Numerical experiments on parabolic PDEs demonstrate that projection-based controllers stabilize for sufficiently large and , and that neural-network-based feedbacks can yield lower actuation energy and favorable transient behavior, highlighting a viable path to scalable, data-driven optimal control for nonlinear PDEs.

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.

Paper Structure

This paper contains 24 sections, 9 theorems, 141 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Towards taming the curse of dimensionality
  3. Illustrating example.
  4. On the literature
  5. Contents
  6. Notation
  7. Assumptions
  8. The main stabilizability result
  9. Auxiliary results
  10. Proof of Theorem \ref{['T:main']}
  11. On the existence and uniqueness of the solutions for the controlled system
  12. On the explicit linear feedback operators
  13. Learning an optimal feedback control
  14. Feedback control as learning problem
  15. Practical realization
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let Assumptions A:A0sp--A:monotone hold true, and let $R>0$ and $\mu>0$ be given. Then if $M\in{\mathbb N}_0$ and $\overline\lambda>0$ are large enough, the solution of system sys-y-intro-K satisfies In particular, $t\mapsto \left|y(t)\right|_{H}$ is strictly decreasing at time $t=s$, if $\left|y(s)\right|_{H}\ne0$. Furthermore, $M$ and $\overline\lambda$ can be chosen as $M=\overline C_{\left[R

Figures (8)

  • Figure 1: Supports of actuators in the rectangle $\Omega^\times\subset{\mathbb R}^2$.
  • Figure 2: Actuators locations and triangulations of $\Omega$
  • Figure 3: Free dynamics norm behavior.
  • Figure 4: The case of $9$ actuators.
  • Figure 5: The case of $4$ and $16$ actuators.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.7
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • ...and 18 more