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A PIE Representation of Delayed Coupled Linear ODE-PDE Systems and Stability Analysis using Convex Optimization

Declan S. JagtMatthew M. Peet
Nov 2022
摘要
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, proving that a PIE representation of boththe 1D and 2D subsystems can be derived. Taking the feedback interconnection ofthese PIE representations, we then obtain a 2D PIE representation of the 1D PDEwith delay. We show that this PIE representation can be coupled to that of anOrdinary Differential Equation (ODE) with delay, to obtain a PIE representationof delayed linear ODE-PDE systems. Next, based on the PIE representation, weformulate the problem of stability analysis as a Linear Operator Inequality(LPI) optimization problem which can be solved using the PIETOOLS softwaresuite. We apply the result to several examples from the existing literatureinvolving delay in the dynamics as well as the boundary conditions of the PDE.
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