On the relation of delay equations to first-order hyperbolic partial differential equations

¹ Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece
iasonkar@central.ntua.gr
² Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA
krstic@ucsd.edu

Received: 5 February 2013
Revised: 14 July 2013

Abstract

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. Illustrative examples show that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the stability analysis and the solution of robust feedback stabilization problems.