On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations
For control theorists, it offers a new framework to analyze and stabilize hyperbolic PDEs using delay system tools, but the results are limited to a specific class of PDEs.
This paper proves the equivalence between systems described by a single first-order hyperbolic PDE and integral delay equations, providing system-theoretic results including converse Lyapunov theorems. An example demonstrates that converting a hyperbolic PDE to a delay system simplifies robust feedback stabilization.
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.