Representation and Stability Analysis of PDE-ODE Coupled Systems

In this work, we present a scalable Linear Matrix Inequality (LMI) based framework to verify the stability of a set of linear Partial Differential Equations (PDEs) in one spatial dimension coupled with a set of Ordinary Differential Equations (ODEs) via input-output based interconnection. Our approach extends the newly developed state space representation and stability analysis of coupled PDEs that allows parametrizing the Lyapunov function on $L_2$ with multipliers and integral operators using polynomial kernels of semi-separable class. In particular, under arbitrary well-posed boundary conditions, we define the linear operator inequalities on $\mathbb{R}^n \times L_2$ and cast the stability condition as a feasibility problem constrained by LMIs. In this framework, no discretization or approximation is required to verify the stability conditions of PDE-ODE coupled systems. The developed algorithm has been implemented in MATLAB where the stability of example PDE-ODE coupled systems are verified.