  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.
