2018/12/14 by Matthew M. Peet, Peet, Matthew M. · 1 citation
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Matrix Theory and Algorithms #Nonlinear Dynamics and Pattern Formation #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1812.06794
openalex publication_date 2018/12/14 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We present a new Partial Integral Equation (PIE) representation of Partial\nDifferential Equations (PDEs) in which it is possible to use convex\noptimization to perform stability analysis with little or no conservatism. The\nfirst result gives a standardized representation for coupled linear PDEs in a\nsingle spatial variable and shows that any such PDE, suitably well-posed,\nadmits an equivalent PIE representation, defined by the given conversion\nformulae. This leads to a new prima facie representation of the dynamics\nwithout the implicit constraints on system state imposed by boundary\nconditions. The second result is to show that for systems in this PIE\nrepresentation, convex optimization may be used to verify stability without\ndiscretization. The resulting algorithms are implemented in the Matlab toolbox\nPIETOOLS, tested on several illustrative examples, compared with previous\nresults, and the code has been posted on Code Ocean. Scalability testing\nindicates the algorithm can analyze systems of up to 40 coupled PDEs on a\ndesktop computer.\n