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Optimal Control of 1D Semilinear Heat Equations with Moment-SOS Relaxations

Charlie Lebarbé, Emilien Flayac, Michel Fournié, Didier Henrion, Milan Korda

TL;DR

This work addresses boundary optimal control of a 1D semilinear heat equation with a polynomial nonlinearity by extending the moment-SOS framework to include a quadratic control cost and to extract a nonlinear boundary feedback from occupation-measure moments. The authors formulate the control problem as an infinite-dimensional LP in the space of occupation measures and implement finite-dimensional LMI relaxations that yield lower bounds and pseudo-moments, from which a control law is reconstructed as an integral of a polynomial in time, space, and state. They demonstrate that a linear moment-based controller can nearly match the optimal LQR performance in the linear case, and that a nonlinear moment-based controller can stabilize the nonlinear system when the LQR fails, highlighting robustness to nonlinearity and potential for scalable PDE control. The results suggest a viable, discretization-free alternative for nonlinear PDE control with boundary actuation and point toward extensions to higher dimensions and more complex fluid-structure problems.

Abstract

We use moment-SOS (Sum Of Squares) relaxations to address the optimal control problem of the 1D heat equation perturbed with a nonlinear term. We extend the current framework of moment-based optimal control of PDEs to consider a quadratic cost on the control. We develop a new method to extract a nonlinear controller from approximate moments of the solution. The control law acts on the boundary of the domain and depends on the solution over the whole domain. Our method is validated numerically and compared to a linear-quadratic controller.

Optimal Control of 1D Semilinear Heat Equations with Moment-SOS Relaxations

TL;DR

This work addresses boundary optimal control of a 1D semilinear heat equation with a polynomial nonlinearity by extending the moment-SOS framework to include a quadratic control cost and to extract a nonlinear boundary feedback from occupation-measure moments. The authors formulate the control problem as an infinite-dimensional LP in the space of occupation measures and implement finite-dimensional LMI relaxations that yield lower bounds and pseudo-moments, from which a control law is reconstructed as an integral of a polynomial in time, space, and state. They demonstrate that a linear moment-based controller can nearly match the optimal LQR performance in the linear case, and that a nonlinear moment-based controller can stabilize the nonlinear system when the LQR fails, highlighting robustness to nonlinearity and potential for scalable PDE control. The results suggest a viable, discretization-free alternative for nonlinear PDE control with boundary actuation and point toward extensions to higher dimensions and more complex fluid-structure problems.

Abstract

We use moment-SOS (Sum Of Squares) relaxations to address the optimal control problem of the 1D heat equation perturbed with a nonlinear term. We extend the current framework of moment-based optimal control of PDEs to consider a quadratic cost on the control. We develop a new method to extract a nonlinear controller from approximate moments of the solution. The control law acts on the boundary of the domain and depends on the solution over the whole domain. Our method is validated numerically and compared to a linear-quadratic controller.

Paper Structure

This paper contains 11 sections, 2 theorems, 20 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. Relaxation of PDE control problems into the space of occupation measures
  3. Problem statement
  4. Occupation measures
  5. LMI relaxations
  6. Extraction of a control law from the pseudo-moments
  7. Numerical simulations
  8. The linear-quadratic regulator
  9. The linear case
  10. The nonlinear case
  11. CONCLUSIONS

Key Result

Proposition II.1

For all $u \in L^\infty(0,1)$, there exists a time horizon $T_u > 0$ such that system eq:heat1D admits a unique weak solution $y \in V_u = L^\infty((0,T_u)\times(0,1))\cap C^0([0,T_u];H^{-1}(0,1))$, where $H^{-1}(0,1)$ is the dual of $H_0^1(0,1)$.

Figures (5)

  • Figure 1: Moment-based approach for the control of PDEs.
  • Figure 2: Definition of the domain and its boundaries. The red line indicates where the control is applied.
  • Figure 3: Numerical simulation of \ref{['eq:heat1D']} with $\eta = 0$ and the LQR controller.
  • Figure 4: Numerical simulation of \ref{['eq:heat1D']} with $\eta = 0$ and a linear control extracted from the moments.
  • Figure 5: Numerical simulation of \ref{['eq:heat1D']} with $\eta \neq 0$. Comparison of a nonlinear control law computed from the moments (top row) with the LQR controller (bottom row).

Theorems & Definitions (3)

  • Proposition II.1
  • Definition II.1
  • Proposition II.2