Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Boundary Control of 1-D Hyperbolic Systems

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DOI: https://doi.org/10.1007/978-1-4471-5102-9_11-2


One-dimensional hyperbolic systems are commonly used to describe the evolution of various physical systems. For many of these systems, controls are available on the boundary. There are then two natural questions: controllability (steer the system from a given state to a desired target) and stabilization (construct feedback laws leading to a good behavior of the closed-loop system around a given set point).


Hyperbolic systems Controllability Stabilization Electrical lines Open channels Road traffic Chromatography 
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  1. Ancona F, Marson A (1998) On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J Control Optim 36(1):290–312 (electronic)Google Scholar
  2. Auriol J, Di Meglio F (2016) Minimum time control of heterodirectional linear coupled hyperbolic PDEs. Automatica J IFAC 71:300–307MathSciNetCrossRefGoogle Scholar
  3. Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60(3): 916–938 (electronic)Google Scholar
  4. Bastin G, Coron J-M (2016) Stability and boundary stabilisation of 1-d hyperbolic systems. Number 88 in progress in nonlinear differential equations and their applications. Springer InternationalGoogle Scholar
  5. Bastin G, Coron J-M, Tamasoiu SO (2015) Stability of linear density-flow hyperbolic systems under PI boundary control. Automatica J IFAC 53:37–42MathSciNetCrossRefGoogle Scholar
  6. Bressan A (2000) Hyperbolic systems of conservation laws, volume 20 of Oxford lecture series in mathematics and its applications. Oxford University Press, Oxford. The one-dimensional Cauchy problemGoogle Scholar
  7. Bressan A, Coclite GM (2002) On the boundary control of systems of conservation laws. SIAM J Control Optim 41(2):607–622 (electronic)Google Scholar
  8. Coron J-M, Hayat A (2018) PI controllers for 1-D nonlinear transport equation. IEEE Trans Automat Control 64(11):4570–4582CrossRefGoogle Scholar
  9. Coron J-M, Nguyen H-M (2019) Optimal time for the controllability of linear hyperbolic systems in one-dimensional space. SIAM J Control Optim 57(2):1127–1156MathSciNetCrossRefGoogle Scholar
  10. Coron J-M, Vazquez R, Krstic M, Bastin G (2013) Local exponential H2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J Control Optim 51(3):2005–2035MathSciNetCrossRefGoogle Scholar
  11. Coron J-M, Ervedoza S, Ghoshal SS, Glass O, Perrollaz V (2017) Dissipative boundary conditions for 2 × 2 hyperbolic systems of conservation laws for entropy solutions in BV. J Differ Equ 262(1):1–30MathSciNetCrossRefGoogle Scholar
  12. Coron J-M, Hu L, Olive G (2017) Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation. Automatica J IFAC 84:95–100MathSciNetCrossRefGoogle Scholar
  13. Di Meglio F, Vazquez R, Krstic M (2013) Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans Automat Control 58(12):3097–3111MathSciNetCrossRefGoogle Scholar
  14. Dos Santos V, Bastin G, Coron JM, d’Andréa Novel B (2008) Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments. Automatica J IFAC 44(5):1310–1318MathSciNetCrossRefGoogle Scholar
  15. Glass O (2007) On the controllability of the 1-D isentropic Euler equation. J Eur Math Soc (JEMS) 9(3):427–486MathSciNetCrossRefGoogle Scholar
  16. Hayat A (2017) Exponential stability of general 1-D quasilinear systems with source terms for the C1 norm under boundary conditions. Accepted for publication in Siam J ControlGoogle Scholar
  17. Hayat A (2019) PI controller for the general Saint-Venant equations. Preprint, hal-01827988Google Scholar
  18. Horsin T (1998) On the controllability of the Burgers equation. ESAIM Control Optim Calc Var 3:83–95 (electronic)Google Scholar
  19. Hu L (2015) Sharp time estimates for exact boundary controllability of quasilinear hyperbolic systems. SIAM J Control Optim 53(6):3383–3410MathSciNetCrossRefGoogle Scholar
  20. Hu L, Olive G (2019) Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Preprint, hal-01982662Google Scholar
  21. Hu L, Di Meglio F, Vazquez R, Krstic M (2016) Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs. IEEE Trans Automat Control 61(11):3301–3314MathSciNetCrossRefGoogle Scholar
  22. Hu L, Vazquez R, Di Meglio F, Krstic M (2019) Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems. SIAM J Control Optim 57(2):963–998MathSciNetCrossRefGoogle Scholar
  23. Krstic M, Smyshlyaev A (2008) Boundary control of PDEs, volume 16 of advances in design and control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. A course on backstepping designs.Google Scholar
  24. Lamare P-O, Bekiaris-Liberis N (2015) Control of 2 × 2 linear hyperbolic systems: backstepping-based trajectory generation and PI-based tracking. Syst Control Lett 86:24–33MathSciNetCrossRefGoogle Scholar
  25. Li T (2010) Controllability and observability for quasilinear hyperbolic systems, volume 3 of AIMS series on applied mathematics. American Institute of Mathematical Sciences (AIMS), SpringfieldGoogle Scholar
  26. Li T, Rao B-P (2003) Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J Control Optim 41(6):1748–1755 (electronic)Google Scholar
  27. Trinh N-T, Andrieu V, Xu C-Z (2017) Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations. IEEE Trans Automat Control 62(9):4527–4536MathSciNetCrossRefGoogle Scholar
  28. Wang Z (2006) Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Ann Math Ser B 27(6):643–656MathSciNetCrossRefGoogle Scholar

Authors and Affiliations

  1. 1.Department of Mathematical EngineeringICTEAM, UCLouvainLouvain-La-NeuveBelgium
  2. 2.Laboratoire Jacques-Louis LionsSorbonne UniversitéParisFrance

Section editors and affiliations

  • Miroslav Krstic
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA