A Distributed SOS Program For Local Stability Analysis of Polynomial PDEs in the PIE Representation

Abstract

It has recently been shown that the evolution of a state, described by a Partial Differential Equation (PDE), can be more conveniently represented as the evolution of the state's highest spatial derivative (the ``fundamental state''), which lies in $L_2$ and has no boundary conditions (BCs) or continuity constraints. For linear PDEs, this yields a Partial Integral Equation (PIE) parametrized by Partial Integral (PI) operators mapping the fundamental state to the PDE state. In this paper, we show that for polynomial PDEs, the dynamics of the fundamental state can instead be compactly expressed as a distributed polynomial in the fundamental state, parametrized by a new tensor algebra of PI operators acting on the tensor product of the fundamental state. We further define a SOS parametrization of the distributed polynomial and use this to construct a distributed SOS program, for testing local stability of polynomial PDEs.