A Koopman Operator Approach to Data-Driven Control of Semilinear Parabolic Systems

TL;DR

The paper develops a data-driven framework to stabilize unknown semilinear parabolic PDEs with boundary input by lifting the dynamics through a finite set of Koopman eigenfunctionals, yielding a Koopman modal representation with diagonal eigendynamics and a state dependent input. It then applies a bilinearization and feedback linearization to place closed-loop eigenvalues, with stability guarantees that tolerate errors from data-driven eigenfunctionals estimation. The eigenfunctionals and eigenvalues are learned from state data using an extended dynamic mode decomposition, enabling a fully data-driven controller that stabilizes an unstable reaction-diffusion PDE with finite-time blow-up. Overall, the work extends Koopman-based spectral analysis to nonlinear PDE control, offering a scalable, data-driven approach with practical validation on a challenging PDE example.

Abstract

This paper is concerned with the data-driven stabilization of unknown boundary controlled semilinear parabolic systems. The nonlinear dynamics of the system are lifted using a finite number of eigenfunctionals of the Koopman operator related to the autonomous semilinear PDE. This results in a novel data-driven finite-dimensional model of the lifted dynamics, which is amenable to apply design procedures for finite-dimensional systems to stabilize the semilinear parabolic system. In order to facilitate this, a bilinearization of the lifted dynamics is considered and feedback linearization is applied for the data-driven stabilization of the semilinear parabolic PDE. This reveals a novel connection between the assignment of eigenfunctionals to the closed-loop Koopman operator and feedback linearization. By making use of a modal representation, exponential stability of the closed-loop system in the presence of errors resulting from the data-driven computation of eigenfunctionals and the bilinearization is verified. The data-driven controller directly follows from applying generalized eDMD to state data available for the semilinear parabolic PDE. An example of an unstable semilinear reaction-diffusion system with finite-time blow up demonstrates the novel data-driven stabilization approach.

Figures (3)

• Figure 1: The top plot shows the eigenvalues $\hat{\lambda}_i$, $i = 1,\ldots,5$ ( $\mathbin{}$ ) resulting from the eDMD as well as the exact principal eigenvalues $\lambda_i$, $i = 4,5$ ( ) and non-principal eigenvalues $\lambda_i = \mu_i\lambda_4$, $\mu_i \in \mathbb{N}$, $i = 1,2,3$ ( ) . The bottom plot displays the $L_2$-norms of the prediction error of the eigenpairs $(\hat{\lambda}_i, \hat{\varphi}_i[x]), i = 1, \dots, 5$. For the non-principal eigenvalues also $||\operatorname{e}^{\hat{\lambda}_i t}\hat{\varphi}^{\mu_i}_4[x(0)] - \hat{\varphi}^{\mu_i}_4[x(t)]||_2, i = 1, 2, 3$, is shown in black, since $\varphi_i[x] = \varphi^{\mu_i}_4[x]$.
• Figure 2: The left column shows the comparison of the transformed state trajectory $\Phi(w)$(\ref{['r']}) for the closed-loop bilinear system \ref{['blinappr']}, \ref{['sf']} and the IC $w(0) = \boldsymbol{\hat{\varphi}_2}[2.4g]$ with the corresponding solution $\tilde{w}^{\text{lin}}$ (\ref{['y']}) of the linear target system \ref{['lintag']}. The right column compares the transformed state trajectories $\Phi(\boldsymbol{\hat{\varphi}_2}[x])$ (\ref{['r']}) of the closed-loop PDE system \ref{['plant']}, \ref{['sf']} for the IC $x_0(z) = 3.1g(z)$ and the corresponding solution of the linear target system \ref{['lintag']}.
• Figure 3: State profile $x(z,t)$ of the closed-loop PDE system \ref{['plant']}, \ref{['sf']} for the IC $x_0(z) = 3.1g(z)$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Theorem 1: Optimal bilinearization
• Theorem 2: Feedback linearization
• Remark 3
• Theorem 3: Closed-loop eigenfunctionals
• proof
• Lemma 1: Change of coordinates
• proof
• Theorem 4: Closed-loop stability