A Convex Approach for Stability Analysis of Coupled PDEs using Lyapunov Functionals

TL;DR

This paper introduces a convex method utilizing Lyapunov functionals and LMIs for stability analysis of coupled linear PDEs with mixed boundary conditions, enabling efficient computational verification.

Contribution

It proposes a novel convex framework using positive matrices to parameterize Lyapunov functionals, facilitating stability analysis via LMIs for coupled PDE systems.

Findings

Successfully applied to a numerical example

Stability conditions verified through SDP solvers

Compared results with PDE simulations

Abstract

In this paper, we present an algorithm for stability analysis of systems described by coupled linear Partial Differential Equations (PDEs) with constant coefficients and mixed boundary conditions. Our approach uses positive matrices to parameterize functionals which are positive or negative on certain function spaces. Applying this parameterization to construct Lyapunov functionals with negative derivative allows us to express stability conditions as a set of LMI constraints which can be constructed using SOSTOOLS and tested using standard SDP solvers such as SeDuMi. The results are tested using a simple numerical example and compared results obtained from simulation using a standard form of discretization.