A Parameterization of Polynomials on Distributed States and a PIE Representation of Nonlinear PDEs

Partial Integral Equations (PIEs) have previously been used to represent systems of linear (1D) Partial Differential Equations (PDEs) with homogeneous Boundary Conditions (BCs), facilitating analysis and simulation of such distributed-state systems. In this paper, we extend these result to derive an equivalent PIE representation of scalar-valued, 1D, polynomial PDEs, with linear, homogoneous BCs. To derive this PIE representation of polynomial PDEs, we first propose a new definition of polynomials on distributed states $\mathbf{u}\in L_2[a,b]$, that naturally generalizes the concept of polynomials on finite-dimensional states to infinite dimensions. We then define a subclass of distributed polynomials that is parameterized by Partial Integral (PI) operators. We prove that this subclass of polynomials is closed under addition and multiplication, providing formulae for computing the sums and products of such polynomials. Applying these results, we then show how a large class of polynomial PDEs can be represented in terms of distributed PI polynomials, proving equivalence of solutions of the resulting PIE representation to those of the original PDE. Finally, parameterizing quadratic Lyapunov functions by PI operators as well, we formulate a stability test for quadratic PDEs as a linear operator inequality optimization problem, which can be solved using the PIETOOLS software suite. We illustrate how this framework can be used to test stability of several common nonlinear PDEs.