On the relation of delay equations to first-order hyperbolic partial differential equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, p. 894-923
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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.

DOI : https://doi.org/10.1051/cocv/2014001
Classification:  34K20,  35L04,  35L60,  93D20,  34K05,  93C23
Keywords: integral delay equations, first-order hyperbolic partial differential equations, nonlinear systems
@article{COCV_2014__20_3_894_0,
author = {Karafyllis, Iasson and Krstic, Miroslav},
title = {On the relation of delay equations to first-order hyperbolic partial differential equations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
pages = {894-923},
doi = {10.1051/cocv/2014001},
zbl = {1295.35299},
mrnumber = {3264228},
language = {en},
url = {http://www.numdam.org/item/COCV_2014__20_3_894_0}
}

Karafyllis, Iasson; Krstic, Miroslav. On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 894-923. doi : 10.1051/cocv/2014001. http://www.numdam.org/item/COCV_2014__20_3_894_0/

[1] O. Aamo, Disturbance Rejection in 2 × 2 Linear Hyperbolic Systems. IEEE Trans. Autom. Control 58 (2013) 1095-1106. | MR 3047914

[2] 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).

[3] G. Bastin and J.-M. Coron, On Boundary Feedback Stabilization of Non-Uniform Linear 2 × 2 Hyperbolic Systems Over a Bounded Interval. Syst. Control Lett. 60 (2011) 900-906. | MR 2895133 | Zbl 1229.93130

[4] K.L. Cooke and D.W. Krumme, Differential-Difference Equations and Nonlinear Initial-Boundary Value Problems for Linear Hyperbolic Partial Differential Equations. J. Math. Anal. Appl. 24 (1968) 372-387. | MR 232089 | Zbl 0186.16902

[5] J.-M. Coron, G. Bastin and B. D'Andrea-Novel, Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems. SIAM J. Control Optim. 47 (2008) 1460-1498. | MR 2407024 | Zbl 1172.35008

[6] J.-M. Coron, R. Vazquez, M. Krstic, and G. Bastin, Local Exponential H2 Stabilization of a 2 × 2 Quasilinear Hyperbolic System Using Backstepping. SIAM J. Control Optim. 51 (2013) 2005-2035. | MR 3049647 | Zbl pre06212040

[7] A. Diagne, G. Bastin and J.-M. Coron, Lyapunov Exponential Stability of 1-d Linear Hyperbolic Systems of Balance Laws. Automatica 48 (2012) 109-114. | MR 2879417 | Zbl 1244.93143

[8] A.V. Fillipov, Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht (1988). | Zbl 0664.34001

[9] S.-Y. Ha and A. Tzavaras, Lyapunov Functionals and L1-Stability for Discrete Velocity Boltzmann Equations. Commun. Math. Phys. 239 (2003) 65-92. | MR 1997116 | Zbl 1024.35067

[10] J.K. Hale and S.M.V. Lunel, Introduction to Functional Differential Equations. Springer-Verlag, New York (1993). | MR 1243878 | Zbl 0787.34002

[11] I. Karafyllis, P. Pepe and Z.-P. Jiang, Stability Results for Systems Described by Coupled Retarded Functional Differential Equations and Functional Difference Equations. Nonlinear Anal., Theory Methods Appl. 71 (2009) 3339-3362. | MR 2532756 | Zbl 1188.34097

[12] I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems. Commun. Control Eng. Springer-Verlag London (2011). | MR 2791585 | Zbl 1243.93004

[13] I. Karafyllis and M. Krstic, 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. | MR 2923875

[14] M. Krstic and A. Smyshlyaev, Backstepping Boundary Control for First-Order Hyperbolic PDEs and Application to Systems With Actuator and Sensor Delays. Syst. Control Lett. 57 (2008) 750-758. | MR 2446460 | Zbl 1153.93022

[15] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkhuser Boston (2009). | MR 2553294 | Zbl 1181.93003

[16] M. Krstic, Input Delay Compensation for Forward Complete and Strict-Feedforward Nonlinear Systems. IEEE Trans. Autom. Control 55 (2010) 287-303. | MR 2604408

[17] T.T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, vol. 3. Higher Education Press, Beijing (2009). | MR 2655971 | Zbl 1198.93003

[18] D. Melchor-Aguilar, V. Kharitonov and R. Lozano, Stability Conditions for Integral Delay Systems. Int. J. Robust Nonlinear Control 20 2010 1-15. | MR 2589723 | Zbl 1192.93096

[19] D. Melchor-Aguilar, On Stability of Integral Delay Systems. Appl. Math. Comput. 217 (2010) 3578-3584. | MR 2733802 | Zbl 1221.45012

[20] D. Melchor-Aguilar, Exponential Stability of Some Linear Continuous Time Difference Systems. Syst. Control Lett. 61 (2012) 62-68. | MR 2878688 | Zbl 1250.93107

[21] L. Pavel and L. Chang, Lyapunov-Based Boundary Control for a Class of Hyperbolic Lotka-Volterra Systems. IEEE Trans. Autom. Control 57 (2012) 701-714. | MR 2932826

[22] P. Pepe, The Lyapunov's Second Method for Continuous Time Difference Equations. Int. J. Robust Nonlinear Control 13 (2003) 1389-1405. | MR 2027371 | Zbl 1116.93383

[23] C. Prieur and F. Mazenc, ISS-Lyapunov Functions for Time-Varying Hyperbolic Systems of Balance Laws. Math. Control, Signals Syst. 24 (2012) 111-134. | MR 2899713 | Zbl 1238.93089

[24] C. Prieur, J. Winkin and G. Bastin, Robust Boundary Control of Systems of Conservation Laws. Math. Control Signals Syst. 20 (2008) 173-197. | MR 2411418 | Zbl 1147.93036

[25] V. Rasvan and S.I. Niculescu, Oscillations in Lossless Propagation Models: a Liapunov-Krasovskii Approach. IMA J. Math. Control Inf. 19 (2002) 157-172. | MR 1899011 | Zbl 1020.93010

[26] J. Rauch and M. Taylor, Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains. Indiana University Math. J. 24 (1975). | MR 361461 | Zbl 0281.35012

[27] D.L. Russell, Canonical Forms and Spectral Determination for a Class of Hyperbolic Distributed Parameter Control Systems. J. Math. Anal. Appl. 62 (1978) 186-225. | MR 472175 | Zbl 0371.93010

[28] D.L. Russell, Neutral FDE Canonical Representations of Hyperbolic Systems. J. Int. Eqs. Appl. 3 (1991) 129-166. | MR 1094933 | Zbl 0767.93045

[29] E.D. Sontag, Smooth Stabilization Implies Coprime Factorization. IEEE Trans. Autom. Control 34 (1989) 435-443. | MR 987806 | Zbl 0682.93045

[30] 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).

[31] C.-Z. Xu and G. Sallet, Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems. ESAIM: COCV 7 (2002) 421-442. | Numdam | MR 1925036 | Zbl 1040.93031