Constructive boundary observer-based control of high-dimensional semilinear heat equations

TL;DR

The paper addresses stabilization of high-dimensional semilinear parabolic PDEs with boundary actuation and sensing under stochastic perturbations. It proposes a constructive finite-dimensional observer-based control using modal decomposition and a lifting transformation to enlarge admissible Lipschitz constants and to allow shape functions distributed on boundary subsets. Stability is established via mean-square exponential results for multiplicative noise and NSS for additive noise, with LMIs determining the observer size and controller gains. Numerical tests in 2D and 3D illustrate improved robustness and practical viability over prior 1D/2D approaches.

Abstract

This paper presents a constructive finite-dimensional output-feedback design for semilinear $M$-dimensional ($M\geq 2$) heat equations with boundary actuation and sensing. A key challenge in high dimensions is the slower growth rate of the Laplacian eigenvalues. The novel features of our modal-decomposition-based design, which allows to enlarge Lipschitz constants, include a larger class of shape functions that may be distributed over a part of the boundary only, the corresponding lifting transformation and the full-order controller gain found from the design LMIs. We further analyze the robustness of the closed-loop system with respect to either multiplicative noise (vanishing at the origin) or additive noise (persistent). Effective LMI conditions are provided for specifying the minimal observer dimension and maximal Lipschitz constants that preserve the stability (mean-square exponential stability for multiplicative noise and noise-to-state stability for additive noise). Numerical examples for 2D and 3D cases demonstrate the efficacy and advantages of our method.

Figures (3)

• Figure 1: (a) and (c) show the distribution of $b_i$ over $\Gamma_1$ and $\psi_i$ over $\mathcal{O}$; (b) and (d) show the distribution of $b_i$ over $\tilde{\Gamma}_{i,1}\subseteq \Gamma_1$ and $\psi_i$ over $\mathcal{O}_i\subseteq \mathcal{O}$.
• Figure 2: The evolution of $\mathbb{E}[\|w(\cdot, t)\|_{L^2}^2+|{\bf u}(t)|^2]$ (The expectation is competed averaging over 20 sample trajectories).
• Figure 3: The evolution of $\mathbb{E}[\|w(\cdot, t)\|_{L^2}^2+|e^N(t)|^2+|{\bf u}(t)|^2]$ (The expectation is competed averaging over 20 sample trajectories).

Theorems & Definitions (11)

• remark 1
• remark 2
• remark 3
• remark 4
• remark 5
• Proposition 1
• proof
• Theorem 1
• remark 6
• Theorem 2