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Partial Integral Equations (PIEs) have been used to represent both systemswith delay and systems of Partial Differential Equations (PDEs) in one or twospatial dimensions. In this paper, we show that these results can be combinedto obtain a PIE representation of any suitably well-posed 1D PDE model withconstant delay. In particular, we represent these delayed PDE systems ascoupled systems of 1D and 2D PDEs, obtaining a PIE representation of bothsubsystems. Taking the feedback interconnection of these PIE subsystems, wethen obtain a 2D PIE representation of the 1D PDE with delay. Next, based onthe PIE representation, we formulate the problem of stability analysis asconvex optimization of positive operators which can be solved using thePIETOOLS software suite. We apply the result to PDE examples with delay in thestate and boundary conditions.

Representation of PDE Systems with Delay and Stability Analysis using Convex Optimization -- Extended Version