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We present a new Partial Integral Equation (PIE) representation of PartialDifferential Equations (PDEs) in which it is possible to use convexoptimization to perform stability analysis with little or no conservatism. Thefirst result gives a standardized representation for coupled linear PDEs in asingle spatial variable and shows that any such PDE, suitably well-posed,admits an equivalent PIE representation, defined by the given conversionformulae. This leads to a new prima facie representation of the dynamicswithout the implicit constraints on system state imposed by boundaryconditions. The second result is to show that for systems in this PIErepresentation, convex optimization may be used to verify stability withoutdiscretization. The resulting algorithms are implemented in the Matlab toolboxPIETOOLS, tested on several illustrative examples, compared with previousresults, and the code has been posted on Code Ocean. Scalability testingindicates the algorithm can analyze systems of up to 40 coupled PDEs on adesktop computer.

A Partial Integral Equation (PIE) Representation of Coupled Linear PDEs and Scalable Stability Analysis using LMIs