On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients
Coron, Jean-Michel1  ; Nguyen, Hoai-Minh1
1Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, équipe Cage, Paris (France)
Annales de l'Institut Fourier, Online first, 70 p.

The optimal time for the controllability of linear hyperbolic systems in one-dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.

Le temps optimal pour la contrôlabilité des systèmes hyperboliques linéaires sur un espace unidimensionnel avec des contrôles unilatéraux a été obtenu récemment pour des coefficients indépendants du temps dans nos travaux antérieurs. Dans cet article, nous considérons des systèmes hyperboliques linéaires avec des termes d’ordre zéro variables dans le temps. Nous montrons la possibilité que le temps optimal pour la contrôlabilité nulle devienne significativement plus grand que celui du cadre invariant dans le temps, même lorsque le terme d’ordre zéro est indéfiniment différentiable. Lorsque l’analyticité par rapport au temps est imposée pour le terme d’ordre zéro, nous établissons également que le temps optimal est le même que dans le cadre indépendant du temps.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3745
Classification : 93C20, 35Q93, 35L50, 47A55
Keywords: hyperbolic systems, controllability, optimal time, time-varying coefficients, analytic coefficients in time, unique continuation principle, well-posedness of hyperbolic systems
Mots-clés : systèmes hyperboliques, contrôlabilité, temps optimal, coefficients dépendant du temps, coefficients analytiques en temps, principe de prolongement unique, caractère bien posé des systèmes hyperboliques

Coron, Jean-Michel  1   ; Nguyen, Hoai-Minh  1

1 Sorbonne Université, Université Paris-Diderot SPC, CNRS, INRIA, Laboratoire Jacques-Louis Lions, équipe Cage, Paris (France) 
@unpublished{AIF_0__0_0_A33_0,
     author = {Coron, Jean-Michel and Nguyen, Hoai-Minh},
     title = {On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients},
     journal = {Annales de l'Institut Fourier},
     year = {2026},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     doi = {10.5802/aif.3745},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Coron, Jean-Michel
AU  - Nguyen, Hoai-Minh
TI  - On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients
JO  - Annales de l'Institut Fourier
PY  - 2026
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3745
LA  - en
ID  - AIF_0__0_0_A33_0
ER  - 
%0 Unpublished Work
%A Coron, Jean-Michel
%A Nguyen, Hoai-Minh
%T On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients
%J Annales de l'Institut Fourier
%D 2026
%V 0
%N 0
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3745
%G en
%F AIF_0__0_0_A33_0
Coron, Jean-Michel; Nguyen, Hoai-Minh. On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients. Annales de l'Institut Fourier, Online first, 70 p.

[1] Amin, Saurabh; Hante, Falk M.; Bayen, Alexandre M. On stability of switched linear hyperbolic conservation laws with reflecting boundaries, Hybrid systems: computation and control (Lecture Notes in Computer Science), Volume 4981, Springer, 2008, pp. 602-605 | DOI | MR | Zbl

[2] Anantharaman, Nalini; Léautaud, Matthieu; Macià, Fabricio Wigner measures and observability for the Schrödinger equation on the disk, Invent. Math., Volume 206 (2016) no. 2, pp. 485-599 | DOI | MR | Zbl

[3] Auriol, Jean; Di Meglio, Florent Minimum time control of heterodirectional linear coupled hyperbolic PDEs, Automatica, Volume 71 (2016), pp. 300-307 | DOI | MR | Zbl

[4] Bardos, Claude; Lebeau, Gilles; Rauch, Jeffrey Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992) no. 5, pp. 1024-1065 | MR | Zbl | DOI

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

[6] Bressan, Alberto Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, 2000, xii+250 pages | MR | DOI | Zbl

[7] Cerpa, Eduardo; Coron, Jean-Michel Rapid stabilization for a Korteweg–de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Autom. Control, Volume 58 (2013) no. 7, pp. 1688-1695 | DOI | MR | Zbl

[8] Coron, Jean-Michel On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain, SIAM J. Control Optim., Volume 37 (1999) no. 6, pp. 1874-1896 | DOI | MR | Zbl

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

[10] Coron, Jean-Michel; Bastin, Georges; d’Andréa-Novel, Brigitte Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., Volume 47 (2008) no. 3, pp. 1460-1498 | DOI | MR | Zbl

[11] Coron, Jean-Michel; Gagnon, Ludovick; Morancey, Morgan Rapid stabilization of a linearized bilinear 1-D Schrödinger equation, J. Math. Pures Appl. (9), Volume 115 (2018), pp. 24-73 | DOI | MR | Zbl

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

[13] Coron, Jean-Michel; Hu, Long; Olive, Guillaume; Shang, Peipei Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space, J. Differ. Equations, Volume 271 (2021), pp. 1109-1170 | DOI | MR | Zbl

[14] Coron, Jean-Michel; Lü, Qi Local rapid stabilization for a Korteweg–de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), Volume 102 (2014) no. 6, pp. 1080-1120 | DOI | MR | Zbl

[15] Coron, Jean-Michel; Nguyen, Hoai-Minh Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach, Arch. Ration. Mech. Anal., Volume 225 (2017), pp. 993-1023 | DOI | Zbl | MR

[16] Coron, Jean-Michel; Nguyen, Hoai-Minh Optimal time for the controllability of linear hyperbolic systems in one-dimensional space, SIAM J. Control Optim., Volume 57 (2019) no. 2, pp. 1127-1156 | DOI | MR | Zbl

[17] Coron, Jean-Michel; Nguyen, Hoai-Minh Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 119, 24 pages | DOI | MR | Numdam | Zbl

[18] Coron, Jean-Michel; Nguyen, Hoai-Minh Null-controllability of linear hyperbolic systems in one dimensional space, Syst. Control Lett., Volume 148 (2021), 104851, 8 pages | DOI | MR | Zbl

[19] Coron, Jean-Michel; Nguyen, Hoai-Minh Lyapunov functions and finite-time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 39 (2022) no. 5, pp. 1235-1260 | DOI | MR | Numdam | Zbl

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

[21] Di Meglio, Florent; Vazquez, Rafael; Krstic, Miroslav Stabilization of a system of n+1 coupled first-order hyperbolic linear PDEs with a single boundary input, IEEE Trans. Autom. Control, Volume 58 (2013) no. 12, pp. 3097-3111 | DOI | MR | Zbl

[22] Dos Santos, Valérie; Prieur, Christophe Boundary control of open channels with numerical and experimental validations, IEEE Trans. Control Sys. Technol., Volume 16 (2008) no. 6, pp. 1252-1264 | DOI

[23] Goatin, Paola The Aw–Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, Volume 44 (2006) no. 3-4, pp. 287-303 | DOI | MR | Zbl

[24] Gohberg, Israel; Lancaster, Peter; Rodman, Leiba Matrix polynomials, Classics in Applied Mathematics, 58, Society for Industrial and Applied Mathematics, 2009, xxiv+409 pages | DOI | MR

[25] Greenberg, James M.; Li, Tatsien The effect of boundary damping for the quasilinear wave equation, J. Differ. Equations, Volume 52 (1984) no. 1, pp. 66-75 | DOI | MR | Zbl

[26] Gugat, Martin; Leugering, Günter Global boundary controllability of the de St. Venant equations between steady states, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003) no. 1, pp. 1-11 | DOI | MR | Numdam | Zbl

[27] Gugat, Martin; Leugering, Günter; Georg Schmidt, E. J. P. Global controllability between steady supercritical flows in channel networks, Math. Methods Appl. Sci., Volume 27 (2004) no. 7, pp. 781-802 | DOI | MR | Zbl

[28] de Halleux, Jonathan; Prieur, Christophe; Coron, Jean-Michel; d’Andréa-Novel, Brigitte; Bastin, Georges Boundary feedback control in networks of open channels, Automatica, Volume 39 (2003) no. 8, pp. 1365-1376 | DOI | MR | Zbl

[29] Hörmander, Lars On the uniqueness of the Cauchy problem under partial analyticity assumptions, Geometrical optics and related topics (Cortona, 1996) (Progress in Nonlinear Differential Equations and their Applications), Volume 32, Birkhäuser, 1997, pp. 179-219 | MR | DOI | Zbl

[30] Hu, Long; Di Meglio, Florent; Vazquez, Rafael; Krstic, Miroslav Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs, IEEE Trans. Autom. Control, Volume 61 (2016) no. 11, pp. 3301-3314 | DOI | MR | Zbl

[31] Hu, Long; Olive, Guillaume Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls, J. Math. Pures Appl. (9), Volume 148 (2021), pp. 24-74 | DOI | MR | Zbl

[32] Kato, Tosio Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995, xxii+619 pages | MR | Zbl | DOI

[33] Krstic, Miroslav; Guo, Bao-Zhu; Balogh, Andras; Smyshlyaev, Andrey Output-feedback stabilization of an unstable wave equation, Automatica, Volume 44 (2008) no. 1, pp. 63-74 | DOI | MR | Zbl

[34] Krstic, Miroslav; Smyshlyaev, Andrey Boundary control of PDEs, Advances in Design and Control, 16, Society for Industrial and Applied Mathematics, 2008, x+192 pages | DOI | MR | Zbl

[35] Laurent, Camille; Léautaud, Matthieu Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves, J. Eur. Math. Soc., Volume 21 (2019) no. 4, pp. 957-1069 | DOI | MR | Zbl

[36] Li, Tatsien; Yu, Wen Ci Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series, 5, Duke University Press, 1985, viii+325 pages | MR | Zbl

[37] Li, Tatsien; Yu, Wen Ci; Shen, Wei Xi Second initial-boundary value problems for quasilinear hyperbolic-parabolic coupled systems, Chin. Ann. Math., Volume 2 (1981) no. 1, pp. 65-90 | MR | Zbl

[38] Lions, Jacques-Louis Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, 1988, x+541 pages | MR | Zbl

[39] Rauch, Jeffrey; Taylor, Michael Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., Volume 24 (1974), pp. 79-86 | DOI | MR | Zbl

[40] Robbiano, Luc; Zuily, Claude Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., Volume 131 (1998) no. 3, pp. 493-539 | DOI | MR | Zbl

[41] Russell, David L. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., Volume 20 (1978) no. 4, pp. 639-739 | DOI | MR | Zbl

[42] Slemrod, Marshall Boundary feedback stabilization for a quasilinear wave equation, Control theory for distributed parameter systems and applications (Vorau, 1982) (Lecture Notes in Control and Information Sciences), Volume 54, Springer, 1983, pp. 221-237 | DOI | MR | Zbl

[43] Smyshlyaev, Andrey; Cerpa, Eduardo; Krstic, Miroslav Boundary stabilization of a 1-D wave equation with in-domain antidamping, SIAM J. Control Optim., Volume 48 (2010) no. 6, pp. 4014-4031 | Zbl | DOI | MR

[44] Smyshlyaev, Andrey; Krstic, Miroslav Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. Autom. Control, Volume 49 (2004) no. 12, pp. 2185-2202 | DOI | MR | Zbl

[45] Smyshlyaev, Andrey; Krstic, Miroslav On control design for PDEs with space-dependent diffusivity or time-dependent reactivity, Automatica, Volume 41 (2005) no. 9, pp. 1601-1608 | DOI | MR | Zbl

[46] Smyshlyaev, Andrey; Krstic, Miroslav Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary, Syst. Control Lett., Volume 58 (2009) no. 8, pp. 617-623 | DOI | MR | Zbl

[47] Sontag, Eduardo D. Mathematical control theory, Texts in Applied Mathematics, 6, Springer, 1998, xvi+531 pages | DOI | MR | Zbl

[48] Tataru, Daniel Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem, Commun. Partial Differ. Equations, Volume 20 (1995) no. 5-6, pp. 855-884 | DOI | MR | Zbl

[49] Vazquez, Rafael; Krstic, Miroslav Control of 1-D parabolic PDEs with Volterra nonlinearities. I. Design, Automatica, Volume 44 (2008) no. 11, pp. 2778-2790 | DOI | MR | Zbl

[50] Xu, Cheng-Zhong; Sallet, Gauthier Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM, Control Optim. Calc. Var., Volume 7 (2002), pp. 421-442 | DOI | MR | Numdam | Zbl

[51] Xu, Cheng-Zhong; Sallet, Gauthier Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems, ESAIM, Control Optim. Calc. Var., Volume 7 (2002), pp. 421-442 | DOI | MR | Numdam | Zbl

[52] Zhang, Christophe Finite-time internal stabilization of a linear 1-D transport equation, Syst. Control Lett., Volume 133 (2019), 104529, 8 pages | DOI | MR | Zbl

Cité par Sources :