A Convex Approach for Stability Analysis of Coupled PDEs with Spatially Dependent Coefficients

TL;DR

This paper introduces a convex LMI-based method for stability analysis of coupled PDEs with spatially varying coefficients, unifying various PDE types and boundary conditions, and enabling computational verification.

Contribution

It develops a novel SOS Lyapunov functional framework with positive matrices and projection operators, allowing stability conditions to be checked via LMIs and SDP solvers.

Findings

Method successfully applied to several PDE examples

Provides a computational tool for stability verification

Outperforms traditional discretization-based approaches

Abstract

In this paper, we present a methodology for stability analysis of a general class of systems defined by coupled Partial Differential Equations (PDEs) with spatially dependent coefficients and a general class of boundary conditions. This class includes PDEs of the parabolic, elliptic and hyperbolic type as well as coupled systems without boundary feedback. Our approach uses positive matrices to parameterize a new class of SOS Lyapunov functionals and combines these with a parametrization of projection operators which allow us to enforce positivity and negativity on subspaces of L_2. The result allows us to express Lyapunov stability conditions as a set of Linear Matrix Inequality (LMI) constraints which can be constructed using SOSTOOLS and tested using Semi-Definite Programming (SDP) solvers such as SeDuMi or Mosek. The methodology is tested using several simple numerical examples and compared with results obtained from simulation using a standard form of numerical discretization.