Comptes Rendus
Control theory
Spectral stabilization of linear transport equations with boundary and in-domain couplings
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 219-240.

In this work, the problem of stabilization of general systems of linear transport equations with in-domain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite-dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.

Published online:
DOI: 10.5802/crmath.288
Classification: 93D05, 93D15, 93D20
Mathias Dus 1; Francesco Ferrante 2; Christophe Prieur 3

1 IMT Toulouse, 118 Route de Narbonne, 31400 Toulouse, France
2 Department of Engineering, University of Perugia, Via G. Duranti, 67, 06125 Perugia, Italy
3 Univ. Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, 38000, Grenoble, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Mathias Dus and Francesco Ferrante and Christophe Prieur},
     title = {Spectral stabilization of linear transport equations with boundary and in-domain couplings},
     journal = {Comptes Rendus. Math\'ematique},
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     publisher = {Acad\'emie des sciences, Paris},
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     year = {2022},
     doi = {10.5802/crmath.288},
     language = {en},
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DO  - 10.5802/crmath.288
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%T Spectral stabilization of linear transport equations with boundary and in-domain couplings
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Mathias Dus; Francesco Ferrante; Christophe Prieur. Spectral stabilization of linear transport equations with boundary and in-domain couplings. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 219-240. doi : 10.5802/crmath.288.

[1] Alassane Aw; Michel Rascle Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math., Volume 60 (2000) no. 3, pp. 916-938 | DOI | MR | Zbl

[2] Georges Bastin; Jean-Michel Coron On boundary feedback stabilization of non-uniform linear 2×2 hyperbolic systems over a bounded interval, Systems Control Lett., Volume 60 (2011) no. 11, pp. 900-906 | DOI | MR | Zbl

[3] Georges Bastin; Jean-Michel Coron Stability and boundary stabilization of 1-D hyperbolic systems, Progress in Nonlinear Differential Equations and their Applications, 88, Birkhäuser/Springer, 2016, xiv+307 pages | DOI | MR

[4] Georges Bastin; Jean-Michel Coron; Brigitte d’Andréa-Novel On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Netw. Heterog. Media, Volume 4 (2009) no. 2, pp. 177-187 | DOI | MR | Zbl

[5] Felipe Castillo; Emmanuel Witrant; Christophe Prieur; Luc Dugard Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control, Automatica, Volume 49 (2013) no. 11, pp. 3180-3188 | DOI | MR | Zbl

[6] C. Cheverry; N. Raymond Handbook of Spectral Theory (2019) (Lecture)

[7] Jean-Michel Coron Control and nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, 2007, xiv+426 pages | DOI | MR

[8] Jean-Michel Coron; Long Hu; Guillaume Olive Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation, Automatica, Volume 84 (2017), pp. 95-100 | DOI | MR | Zbl

[9] Jean-Michel Coron; Rafael Vazquez; Miroslav Krstic; Georges Bastin Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping, SIAM J. Control Optimization, Volume 51 (2013) no. 3, pp. 2005-2035 | DOI | MR | Zbl

[10] Florent Di Meglio Dynamic and control of slugging in oil production, Ph. D. Thesis, Mines ParisTech (2011)

[11] Florent Di Meglio; Delphine Bresch-Pietri; Ulf Jakob F. Aarsnes An adaptive observer for hyperbolic systems with application to UnderBalanced Drilling, IFAC Proceedings Volumes, Volume 47 (2014) no. 3, pp. 11391-11397 (19th IFAC World Congress) | DOI

[12] Florent Di Meglio; Glenn Kaasa; N. Petit; Vidar Alstad Slugging in multiphase flow as a mixed initial-boundary value problem for a quasilinear hyperbolic system, Proceedings of the 2011 American Control Conference (2011), pp. 3589-3596 | DOI

[13] Ababacar Diagne; Georges Bastin; Jean-Michel Coron Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, Volume 48 (2012) no. 1, pp. 109-114 | DOI | MR | Zbl

[14] S. Dudret; K. Beauchard; F. Ammouri; P. Rouchon Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models, 2012 American Control Conference (ACC) (2012), pp. 3352-3358 | DOI

[15] Francesco Ferrante; Andrea Cristofaro; Christophe Prieur Boundary observer design for cascaded ODE–hyperbolic PDE systems: a matrix inequalities approach, Automatica, Volume 119 (2020), 109027, 9 pages | DOI | MR | Zbl

[16] Larry Gearhart Spectral theory for contraction semigroups on Hilbert space, Trans. Am. Math. Soc., Volume 236 (1978), pp. 385-394 | DOI | MR | Zbl

[17] Mohammad Ghousein; Emmanuel Witrant; Viren Bhanot; Paolo Petagna Adaptive boundary observer design for linear hyperbolic systems; application to estimation in heat exchangers, Automatica, Volume 114 (2020), 108824, 13 pages | DOI | MR | Zbl

[18] Jonathan de Halleux; Christophe Prieur; Jean-Michel Coron; Brigitte d’Andréa-Novel; Georges Bastin Boundary feedback control in networks of open channels, Automatica, Volume 39 (2003), pp. 1365-1376 | DOI | MR | Zbl

[19] Amaury Hayat Boundary stability of 1-D nonlinear inhomogeneous hyperbolic systems for the C 1 norm, SIAM J. Control Optimization, Volume 57 (2019) no. 6, pp. 3603-3638 | DOI | MR | Zbl

[20] Amaury Hayat On boundary stability of inhomogeneous 2×2 1-D hyperbolic systems for the C 1 norm, ESAIM, Control Optim. Calc. Var., Volume 25 (2019), p. 31 | DOI | MR | Zbl

[21] Long Hu; Rafael Vazquez; Florent Di Meglio; Miroslav Krstic Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems, SIAM J. Control Optimization, Volume 57 (2019) no. 2, pp. 963-998 | MR | Zbl

[22] Tosio Kato Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995, xxii+619 pages (Reprint of the 1980 edition) | DOI | MR

[23] Miroslav Krstic; Andrey Smyshlyaev Boundary control of PDEs. A course on backstepping designs, Advances in Design and Control, 16, Society for Industrial and Applied Mathematics, 2008, x+192 pages | DOI | MR

[24] Mark Lichtner Spectral mapping theorem for linear hyperbolic systems, Proc. Am. Math. Soc., Volume 136 (2008) no. 6, pp. 2091-2101 | DOI | MR | Zbl

[25] Aloisio Freiria Neves; Hermano de Souza Ribeiro; Orlando Lopes On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., Volume 67 (1986) no. 3, pp. 320-344 | DOI | MR | Zbl

[26] Amnon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983 | DOI

[27] Jan Prüss On the spectrum of C 0 -semigroups, Trans. Am. Math. Soc., Volume 284 (1984) no. 2, pp. 847-857 | DOI | MR | Zbl

[28] Michel Renardy On the type of certain C 0 -semigroups, Commun. Partial Differ. Equations, Volume 18 (1993) no. 7-8, pp. 1299-1307 | DOI | MR

[29] Sophie Tarbouriech; Germain Garcia; João Manoel Gomes da Silva; Isabelle Queinnec Stability and stabilization of linear systems with saturating actuators, Springer, 2011, xxii+430 pages (with a foreword by Ian Postlethwaite) | DOI | MR

[30] Marius Tucsnak; George Weiss Observation and control for operator semigroups, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2009, xii+483 pages | DOI | MR

[31] Rafael Vazquez; Miroslav Krstic; Jean-Michel Coron Backstepping boundary stabilization and state estimation of a 2 x 2 linear hyperbolic system, 2011 50th IEEE Conference on Decision and Control and European Control Conference (2011), pp. 4937-4942 | DOI

[32] Cheng-Zhong Xu; Gauthier Sallet Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems, ESAIM, Control Optim. Calc. Var., Volume 7 (2002), pp. 421-442 | DOI | Numdam | MR | Zbl

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