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

A PIE Representation of Coupled Linear 2D PDEs and Stability Analysis using LPIs

We introduce a Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D PDEs and time-delay systems without continuity constraints or boundary conditions -- making these PIE representations amenable to stability analysis using convex optimization. To extend the PIE framework to 2D PDEs, we first construct an algebra of Partial Integral (PI) operators on the function space L_2[x,y], providing formulae for composition, adjoint, and inversion. We then extend this algebra to R^n x L_2[x] x L_2[y] x L_2[x,y] and demonstrate that, for any suitable coupled, linear PDE in 2 spatial variables, there exists an associated PIE whose solutions bijectively map to solutions of the original PDE -- providing conversion formulae between these representations. Next, we use positive matrices to parameterize the convex cone of 2D PI operators -- allowing us to optimize PI operators and solve Linear PI Inequality (LPI) feasibility problems. Finally, we use the 2D LPI framework to provide conditions for stability of 2D linear PDEs. We test these conditions on 2D heat and wave equations and demonstrate that the stability condition has little to no conservatism

A PIE Representation of Delayed Coupled Linear ODE-PDE Systems and Stability Analysis using Convex Optimization

Partial Integral Equations (PIEs) have been used to represent both systems with delay and systems of Partial Differential Equations (PDEs) in one or two spatial dimensions. In this paper, we show that these results can be combined to obtain a PIE representation of any suitably well-posed 1D PDE model with constant delay. In particular, we represent these delayed PDE systems as coupled systems of 1D and 2D PDEs, proving that a PIE representation of both the 1D and 2D subsystems can be derived. Taking the feedback interconnection of these PIE representations, we then obtain a 2D PIE representation of the 1D PDE with delay. We show that this PIE representation can be coupled to that of an Ordinary Differential Equation (ODE) with delay, to obtain a PIE representation of delayed linear ODE-PDE systems. Next, based on the PIE representation, we formulate the problem of stability analysis as a Linear Operator Inequality (LPI) optimization problem which can be solved using the PIETOOLS software suite. We apply the result to several examples from the existing literature involving delay in the dynamics as well as the boundary conditions of the PDE