Abstract
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).
Bibliography
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)
Auriol J, Di Meglio F (2016) Minimum time control of heterodirectional linear coupled hyperbolic PDEs. Automatica J IFAC 71:300–307
Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60(3): 916–938 (electronic)
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 International
Bastin G, Coron J-M, Tamasoiu SO (2015) Stability of linear density-flow hyperbolic systems under PI boundary control. Automatica J IFAC 53:37–42
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 problem
Bressan A, Coclite GM (2002) On the boundary control of systems of conservation laws. SIAM J Control Optim 41(2):607–622 (electronic)
Coron J-M, Hayat A (2018) PI controllers for 1-D nonlinear transport equation. IEEE Trans Automat Control 64(11):4570–4582
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):1127–1156
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):2005–2035
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):1–30
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 84:95–100
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):3097–3111
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):1310–1318
Glass O (2007) On the controllability of the 1-D isentropic Euler equation. J Eur Math Soc (JEMS) 9(3):427–486
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 Control
Hayat A (2019) PI controller for the general Saint-Venant equations. Preprint, hal-01827988
Horsin T (1998) On the controllability of the Burgers equation. ESAIM Control Optim Calc Var 3:83–95 (electronic)
Hu L (2015) Sharp time estimates for exact boundary controllability of quasilinear hyperbolic systems. SIAM J Control Optim 53(6):3383–3410
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-01982662
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):3301–3314
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):963–998
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.
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 86:24–33
Li T (2010) Controllability and observability for quasilinear hyperbolic systems, volume 3 of AIMS series on applied mathematics. American Institute of Mathematical Sciences (AIMS), Springfield
Li T, Rao B-P (2003) Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J Control Optim 41(6):1748–1755 (electronic)
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):4527–4536
Wang Z (2006) Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Ann Math Ser B 27(6):643–656
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Bastin, G., Coron, JM. (2020). Boundary Control of 1-D Hyperbolic Systems. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_11-2
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DOI: https://doi.org/10.1007/978-1-4471-5102-9_11-2
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Latest
Boundary Control of 1-D Hyperbolic Systems- Published:
- 21 November 2019
DOI: https://doi.org/10.1007/978-1-4471-5102-9_11-2
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Original
Boundary Control of 1-D Hyperbolic Systems- Published:
- 17 February 2014
DOI: https://doi.org/10.1007/978-1-4471-5102-9_11-1