A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs
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A distributional solution framework for linear hyperbolic PDEs coupled to switched DAEs R. Borsche1 · D. Kocoglu1 · S. Trenn2 Received: 28 November 2019 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract A distributional solution framework is developed for systems consisting of linear hyperbolic partial differential equations and switched differential-algebraic equations (DAEs) which are coupled via boundary conditions. The unique solvability is then characterize in terms of a switched delay DAE. The theory is illustrated with an example of electric power lines modelled by the telegraph equations which are coupled via a switching transformer where simulations confirm the predicted impulsive solutions. Keywords Solution theory · Coupled systems · Switched systems · Partial differential equations · Differential algebraic equations
1 Introduction In this paper, we develop a rigorous solution theory for systems where a linear hyperbolic partial differential equation (PDE) is coupled with a switched differentialalgebraic equation (DAE) via boundary conditions (BC), see Fig. 1 as an overview. Such systems occur, for example, when modelling power grids using the telegraph equation [8] including switches (e.g. induced by disconnecting lines), water flow
This work was supported by DFG-Grants BO 4768/1-1 and TR 1223/4-1 as well as NWO Vidi Grant 639.032.733.
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S. Trenn [email protected] R. Borsche [email protected] D. Kocoglu [email protected]
1
Department of Mathematics, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany
2
Bernoulli Institute for Mathematics, CS and AI, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands
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Mathematics of Control, Signals, and Systems Fig. 1 Coupling of a PDE with a switched DAE via boundary condition
networks with valves [13,14], supply chain models including processor breakdown [1,7], district heating systems with rapid consumption changes [5] and blood flow with simplified valve models in the heart [15]. Similar to [3,11] the closed-loop setting illustrated in Fig. 1 can include general network structures. In this coupled system, the output of the switched DAE provides the boundary condition for the PDE and the boundary values of the PDE serve as input to the DAE. Solutions of switched DAEs in general contain jumps and derivatives thereof, e.g. derivatives of Dirac impulses [17,19]; hence, the solution concept of the PDE has to be extended to allow for jumps, Dirac impulses and their derivatives at the boundary. In particular, this is a wider class compared to the solutions of small bounded variation, e.g. used in [4] where a nonlinear hyperbolic PDE is coupled to an ODE. Similarly, in [2,12], the investigations of switched linear PDEs with source terms are restricted to solutions with bounded variation. In [16], Dirac impulses are introduced at the position of an interface of nonlinear PDEs. A more general appearance of Dirac impulses is allowed in [6,23] for a partially linear syst
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