Many systems combine finite-dimensional dynamics, described by ordinary differential equations (ODEs), with phenomena distributed across space, described by partial differential equations (PDEs), such as heat propagation or communication delays. This thesis studies the stability and performance of such interconnected PDE-ODE systems using computationally tractable tools based on Integral Quadratic Constraints (IQCs) and Linear Matrix Inequalities (LMIs).

The first contribution develops a projection-based method for analysing heat equations coupled with ODEs at their boundaries. Polynomial approximations and energy bounds generate a hierarchy of increasingly accurate stability conditions. The second introduces a flexible descriptor framework for finite-dimensional systems affected by multiple uncertainties, together with lifting techniques that reduce conservatism while keeping the computational problem manageable. Finally, this framework is extended to coupled PDE-ODE systems and validated on a transport equation interconnected with an ODE, recovering a known stability result.