# A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations

## TL;DR

This paper introduces a semi-definite programming method for stability analysis of linear 1-D PDEs with polynomial data, enabling systematic construction of Lyapunov certificates for various PDE systems.

## Contribution

It develops a novel Lyapunov-based approach that reduces PDE stability verification to solving polynomial problems via semi-definite programming.

## Key findings

- Successfully applied to different PDE types
- Provides a systematic computational framework
- Enables stability certification for complex PDEs

## Abstract

We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs.