Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T17:58:15.891Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  13 June 2014

Iasson Karafyllis  and
Miroslav Krstic
Iasson Karafyllis
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece. [email protected]
Miroslav Krstic
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA; [email protected]
Get access

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.

Keywords

Integral delay equationsfirst-order hyperbolic partial differential equationsnonlinear systems
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 894 - 923
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aamo, O., Disturbance Rejection in 2 × 2 Linear Hyperbolic Systems. IEEE Trans. Autom. Control 58 (2013) 1095–1106. Google Scholar
G. Bastin and J.-M. Coron, Further Results on Boundary Feedback Stabilization of 2 × 2 Hyperbolic Systems Over a Bounded Interval. In Proc. of IFAC Nolcos 2010, Bologna, Italy (2010).
Bastin, G. and Coron, J.-M., On Boundary Feedback Stabilization of Non-Uniform Linear 2 × 2 Hyperbolic Systems Over a Bounded Interval. Syst. Control Lett. 60 (2011) 900–906. Google Scholar
Cooke, K.L. and Krumme, D.W., Differential-Difference Equations and Nonlinear Initial-Boundary Value Problems for Linear Hyperbolic Partial Differential Equations. J. Math. Anal. Appl. 24 (1968) 372–387. Google Scholar
Coron, J.-M., Bastin, G. and d’Andrea-Novel, B., Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems. SIAM J. Control Optim. 47 (2008) 1460–1498. Google Scholar
Coron, J.-M., Vazquez, R., Krstic, M., and Bastin, G., Local Exponential H2 Stabilization of a 2 × 2 Quasilinear Hyperbolic System Using Backstepping. SIAM J. Control Optim. 51 (2013) 2005–2035. Google Scholar
Diagne, A., Bastin, G. and Coron, J.-M., Lyapunov Exponential Stability of 1-d Linear Hyperbolic Systems of Balance Laws. Automatica 48 (2012) 109–114. Google Scholar
A.V. Fillipov, Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht (1988).
Ha, S.-Y. and Tzavaras, A., Lyapunov Functionals and L1-Stability for Discrete Velocity Boltzmann Equations. Commun. Math. Phys. 239 (2003) 65–92. Google Scholar
J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York (1993).
Karafyllis, I., Pepe, P. and Jiang, Z.-P., Stability Results for Systems Described by Coupled Retarded Functional Differential Equations and Functional Difference Equations. Nonlinear Anal., Theory Methods Appl. 71 (2009) 3339–3362. Google Scholar
I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems. Commun. Control Eng. Springer-Verlag London (2011).
Karafyllis, I. and Krstic, M., Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and Zero-Order Hold. IEEE Trans. Autom. Control 57 (2012) 1141–1154. Google Scholar
Krstic, M. and Smyshlyaev, A., Backstepping Boundary Control for First-Order Hyperbolic PDEs and Application to Systems With Actuator and Sensor Delays. Syst. Control Lett. 57 (2008) 750–758. Google Scholar
M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhuser Boston (2009).
Krstic, M., Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems. IEEE Trans. Autom. Control 55 (2010) 287–303. Google Scholar
T.T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, vol. 3. Higher Education Press, Beijing (2009).
Melchor-Aguilar, D., Kharitonov, V. and Lozano, R., Stability Conditions for Integral Delay Systems. Int. J. Robust Nonlinear Control 20 2010 1–15.
Melchor-Aguilar, D., On Stability of Integral Delay Systems. Appl. Math. Comput. 217 (2010) 3578–3584. Google Scholar
Melchor-Aguilar, D., Exponential Stability of Some Linear Continuous Time Difference Systems. Syst. Control Lett. 61 (2012) 62–68. Google Scholar
Pavel, L. and Chang, L., Lyapunov-Based Boundary Control for a Class of Hyperbolic Lotka-Volterra Systems. IEEE Trans. Autom. Control 57 (2012) 701–714. Google Scholar
Pepe, P., The Lyapunov’s Second Method for Continuous Time Difference Equations. Int. J. Robust Nonlinear Control 13 (2003) 1389–1405. Google Scholar
Prieur, C. and Mazenc, F., ISS-Lyapunov Functions for Time-Varying Hyperbolic Systems of Balance Laws. Math. Control, Signals Syst. 24 (2012) 111–134. Google Scholar
Prieur, C., Winkin, J. and Bastin, G., Robust Boundary Control of Systems of Conservation Laws. Math. Control Signals Syst. 20 (2008) 173–197. Google Scholar
Rasvan, V. and Niculescu, S.I., Oscillations in Lossless Propagation Models: a Liapunov-Krasovskii Approach. IMA J. Math. Control Inf. 19 (2002) 157–172. Google Scholar
J. Rauch and M. Taylor, Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains. Indiana University Math. J. 24 (1975).
Russell, D.L., Canonical Forms and Spectral Determination for a Class of Hyperbolic Distributed Parameter Control Systems. J. Math. Anal. Appl. 62 (1978) 186–225. Google Scholar
Russell, D.L., Neutral FDE Canonical Representations of Hyperbolic Systems. J. Int. Eqs. Appl. 3 (1991) 129–166. Google Scholar
Sontag, E.D., Smooth Stabilization Implies Coprime Factorization. IEEE Trans. Autom. Control 34 (1989) 435–443. Google Scholar
R. Vazquez, M. Krstic and J.-M. Coron, Backstepping Boundary Stabilization and State Estimation of a 2 × 2 Linear Hyperbolic System, in Proc. of 50th Conf. Decision and Control, Orlando (2011).
Xu, C.-Z. and Sallet, G., Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems. ESAIM: COCV 7 (2002) 421–442. Google Scholar