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

TL;DR

This paper develops a novel algebraic representation called PIE for 2D PDEs, enabling stability analysis via convex optimization with minimal conservatism, demonstrated on heat and wave equations.

Contribution

It extends the PIE framework to 2D PDEs, providing conversion formulas, an algebra of PI operators, and a convex optimization approach for stability analysis.

Findings

PIE representation bijectively maps to 2D PDE solutions

Convex optimization via LPIs effectively tests stability

Minimal conservatism in stability conditions for heat and wave equations

Abstract

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.