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Backstepping Control Laws for Higher-Dimensional PDEs: Spatial Invariance and Domain Extension Methods

Rafael Vazquez

TL;DR

The paper addresses boundary stabilization of higher-dimensional PDEs by exploiting spatial invariance and geometry to decompose the problem into tractable lower-dimensional modules. It develops three complementary routes: (i) Fourier-based modal analysis for rectangular domains with finite-dimensional actuation, (ii) angular-eigenfunction/sectors yielding explicit kernels via modified Bessel functions, and (iii) a domain-extension technique that maps irregular domains to regular target domains while preserving stability margins. Key results include explicit backstepping kernels, mode-wise stability proofs, and precise conditions on the number of controlled modes or actuators needed to achieve a prescribed decay rate $c$; for sector domains, the stability condition scales with domain geometry as $N > rac{ ext{}} ext{ }}{ ext{}}$ and domain size parameters. This framework unifies prior higher-dimensional backstepping results and provides constructive, geometry-aware guidelines for actuator placement and kernel design across complex domains. The approach has practical implications for distributed control of reaction-diffusion processes in physics, chemistry, and biology, where higher-dimensional domains and irregular geometries are common.

Abstract

This paper extends backstepping to higher-dimensional PDEs by leveraging domain symmetries and structural properties. We systematically address three increasingly complex scenarios. First, for rectangular domains, we characterize boundary stabilization with finite-dimensional actuation by combining backstepping with Fourier analysis, deriving explicit necessary conditions. Second, for reaction-diffusion equations on sector domains, we use angular eigenfunction expansions to obtain kernel solutions in terms of modified Bessel functions. Finally, we outline a domain extension method for irregular domains, transforming the boundary control problem into an equivalent one on a target domain. This framework unifies and extends previous backstepping results, offering new tools for higher-dimensional domains where classical separation of variables is inapplicable.

Backstepping Control Laws for Higher-Dimensional PDEs: Spatial Invariance and Domain Extension Methods

TL;DR

The paper addresses boundary stabilization of higher-dimensional PDEs by exploiting spatial invariance and geometry to decompose the problem into tractable lower-dimensional modules. It develops three complementary routes: (i) Fourier-based modal analysis for rectangular domains with finite-dimensional actuation, (ii) angular-eigenfunction/sectors yielding explicit kernels via modified Bessel functions, and (iii) a domain-extension technique that maps irregular domains to regular target domains while preserving stability margins. Key results include explicit backstepping kernels, mode-wise stability proofs, and precise conditions on the number of controlled modes or actuators needed to achieve a prescribed decay rate ; for sector domains, the stability condition scales with domain geometry as and domain size parameters. This framework unifies prior higher-dimensional backstepping results and provides constructive, geometry-aware guidelines for actuator placement and kernel design across complex domains. The approach has practical implications for distributed control of reaction-diffusion processes in physics, chemistry, and biology, where higher-dimensional domains and irregular geometries are common.

Abstract

This paper extends backstepping to higher-dimensional PDEs by leveraging domain symmetries and structural properties. We systematically address three increasingly complex scenarios. First, for rectangular domains, we characterize boundary stabilization with finite-dimensional actuation by combining backstepping with Fourier analysis, deriving explicit necessary conditions. Second, for reaction-diffusion equations on sector domains, we use angular eigenfunction expansions to obtain kernel solutions in terms of modified Bessel functions. Finally, we outline a domain extension method for irregular domains, transforming the boundary control problem into an equivalent one on a target domain. This framework unifies and extends previous backstepping results, offering new tools for higher-dimensional domains where classical separation of variables is inapplicable.

Paper Structure

This paper contains 10 sections, 6 theorems, 94 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Higher-dimensional backstepping control of PDEs
  3. Contribution and paper structure
  4. Spatial Invariance and Backstepping Control in Higher Dimension
  5. Example: Heat Equation in Semi-Infinite Strip
  6. Backstepping Control of Reaction-Diffusion Systems in Square Domains
  7. Extension to Finite-Dimensional Control
  8. Control of Reaction-Diffusion Equations on Sector Domains
  9. Domain Extension Method for Irregular Domains
  10. Conclusions

Key Result

Theorem 1

For any given decay rate $c > 0$, set $N_0 = \sqrt{\frac{c+\lambda}{4\pi^2\epsilon}}$ and let $N\in \mathbb{N}$ such that $N \geq N_0$. For the closed-loop system (eq:heat_strip)--(eq:heat_strip_control) with the truncated feedback control law $U_N$ in (eqn-Un), the equilibrium $u(t,x,y)\equiv 0$ ac for some $M\geq1$ that does not depend on $N$.

Figures (6)

  • Figure 1: Semi-infinite strip domain with distributed boundary control
  • Figure 2: Square domain with boundary control on the right edge
  • Figure 3: Sector domain with boundary control at the outer radius. The control $U(t,\theta)$ is applied along the curved boundary (shown in red), while homogeneous Dirichlet conditions are imposed on the straight edges.
  • Figure 4: Piano-shaped domain with control at the "back" boundaries
  • Figure 5: General domain extension concept: irregular domain (shaded) extended to regular domain (square) with extension region (colored)
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1: Spatial Invariance
  • Theorem 1: Stability with Spectral Truncation
  • proof
  • Theorem 2: Stability with Finite-Dimensional Control
  • proof
  • Proposition 3: Angular Eigenfunctions
  • Proposition 4: Explicit Kernel Solution
  • Theorem 5: Exponential Stability of the Closed-Loop System
  • Theorem 6
  • proof