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

TL;DR

This paper demonstrates the equivalence between first-order hyperbolic PDEs and integral delay equations, providing system-theoretic insights and simplifying the analysis of nonlinear systems with boundary inputs.

Contribution

It introduces a framework linking hyperbolic PDEs to delay equations, enabling new analysis and control strategies for such systems.

Findings

Equivalence between hyperbolic PDEs and delay equations established

Lyapunov-based stability results derived for both system classes

Conversion simplifies robust stabilization of hyperbolic PDE systems

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. An illustrative example shows 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 solution of the corresponding robust feedback stabilization problem.