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We introduce a Partial Integral Equation (PIE) representation of PartialDifferential Equations (PDEs) in two spatial variables. PIEs are an algebraicstate-space representation of infinite-dimensional systems and have been usedto model 1D PDEs and time-delay systems without continuity constraints orboundary conditions -- making these PIE representations amenable to stabilityanalysis using convex optimization. To extend the PIE framework to 2D PDEs, wefirst construct an algebra of Partial Integral (PI) operators on the functionspace L_2[x,y], providing formulae for composition, adjoint, and inversion. Wethen extend this algebra to R^n x L_2[x] x L_2[y] x L_2[x,y] and demonstratethat, for any suitable coupled, linear PDE in 2 spatial variables, there existsan associated PIE whose solutions bijectively map to solutions of the originalPDE -- providing conversion formulae between these representations. Next, weuse 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 provideconditions for stability of 2D linear PDEs. We test these conditions on 2D heatand wave equations and demonstrate that the stability condition has little tono conservatism.

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