On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. Systemtheoretic 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. Mathematics Subject Classification. 34K20, 35L04, 35L60, 93D20, 34K05, 93C23. Received February 5, 2013. Revised July 14, 2013. Published online June 13, 2014.