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

• Published in: ESAIM. Control, optimisation and calculus of variations Vol. 20; no. 3; pp. 894 - 923
• Main Authors: Karafyllis, Iasson, Krstic, Miroslav
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.07.2014
• Subjects: 34K05; 34K20; 35L04; 35L60; 93C23; 93D20; Boundary conditions; first-order hyperbolic partial differential equations; Integral delay equations; Integral equations; Mathematical analysis; Nonlinear systems; Partial differential equations; Studies
• ISSN: 1292-8119; 1262-3377
• DOI: 10.1051/cocv/2014001

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.