[Paper Review] A New State-Space Representation of Lyapunov Stability for Coupled PDEs and Scalable Stability Analysis in the SOS Framework

Matthew M. Peet|arXiv (Cornell University)|Mar 20, 2018

Stability and Controllability of Differential Equations12 references3 citations

TL;DR

This paper introduces a novel state-space representation for Lyapunov stability analysis of coupled linear PDEs, including parabolic, elliptic, and hyperbolic systems with various boundary conditions. By expressing the Lyapunov derivative as a linear operator inequality directly on $L_2$, the method eliminates reliance on integration by parts or spacing functions, enabling scalable, conservative-free stability analysis via the sum-of-squares (SOS) framework.

ABSTRACT

In this paper, we present a framework for Stability Analysis of Systems of Coupled Linear Partial-Differential Equations. The class of PDE systems considered in this paper includes parabolic, elliptic and hyperbolic systems with Dirichelet, Neuman and mixed boundary conditions. The results in this paper apply to systems with a single spatial variable and assume existence and continuity of solutions except in such cases when existence and continuity can be inferred from existence of a Lyapunov function. Our approach is based on a new concept of state for PDE systems which allows us to express the derivative of the Lyapunov function as a Linear Operator Inequality directly on $L_2$ and allows for any type of suitably well-posed boundary conditions. This approach obviates the need for integration by parts, spacing functions or similar mathematical encumbrances. The resulting algorithms are implemented in Matlab, tested on several motivating examples, and the codes have been posted online. Numerical testing indicates the approach has little or no conservatism for a large class of systems.

Motivation & Objective

• To develop a scalable and conservative-free stability analysis framework for systems of coupled linear PDEs with diverse boundary conditions.
• To eliminate the need for integration by parts, spacing functions, or other complex mathematical constructs in Lyapunov-based stability proofs.
• To enable direct formulation of Lyapunov derivative conditions as linear operator inequalities in $L_2$ space.
• To support a broad class of PDEs, including parabolic, elliptic, and hyperbolic systems, with Dirichlet, Neumann, and mixed boundary conditions.
• To provide a numerically tractable and implementable method using sum-of-squares optimization, validated through numerical testing.

Proposed method

• Introduce a new state-space representation for PDE systems that facilitates direct expression of the Lyapunov function derivative in $L_2$.
• Formulate the stability condition as a linear operator inequality (LOI) on $L_2$, avoiding integration by parts and auxiliary functions.
• Ensure the framework is applicable to systems with any well-posed boundary conditions, including mixed types.
• Implement the resulting algorithms in MATLAB and release the code publicly for reproducibility and testing.
• Apply the sum-of-squares (SOS) framework to solve the LOI numerically, enabling scalable stability analysis.
• Validate the approach on multiple benchmark examples to demonstrate low conservatism and robustness.

Experimental results

Research questions

• RQ1 Can a new state-space representation be developed that simplifies Lyapunov stability analysis for coupled PDEs without relying on integration by parts?
• RQ2 To what extent can the proposed framework handle diverse PDE types, including parabolic, elliptic, and hyperbolic systems?
• RQ3 How does the absence of spacing functions or auxiliary constructs affect the conservatism of the stability analysis?
• RQ4 Can the resulting linear operator inequality formulation be efficiently solved using sum-of-squares optimization?
• RQ5 What is the numerical performance and scalability of the proposed method across different boundary conditions and PDE classes?

Key findings

• The proposed state-space representation enables direct formulation of the Lyapunov derivative as a linear operator inequality in $L_2$, bypassing traditional analytical burdens.
• The method is applicable to parabolic, elliptic, and hyperbolic PDEs with Dirichlet, Neumann, and mixed boundary conditions.
• Numerical testing shows the approach exhibits little or no conservatism for a wide class of systems.
• The framework avoids the use of integration by parts, spacing functions, or other complex mathematical tools commonly used in PDE stability analysis.
• The implementation in MATLAB and public code release support reproducibility and practical application of the method.
• The sum-of-squares (SOS) framework successfully enables scalable and numerically tractable stability analysis within the proposed framework.