In this paper, we first introduce an abstract viscous hyperbolic problem for which we prove exponential decay under appropriated assumptions. We then give some illustrative examples, like the linearized viscous Saint-Venant system. In order to achieve the optimal decay rate, we also perform a detailed spectral analysis of our abstract problem under a natural assumption satisfied by various examples. We finally consider the boundary stabilizability of the linearized viscous Saint-Venant system on trees.
Accepté le :
DOI : 10.1051/cocv/2018020
Keywords: Hyperbolic systems, viscosity, stabilization
Nicaise, Serge 1
@article{COCV_2019__25__A33_0,
author = {Nicaise, Serge},
title = {Stability results of some first order viscous hyperbolic systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018020},
zbl = {1441.35055},
mrnumber = {4001033},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018020/}
}
TY - JOUR AU - Nicaise, Serge TI - Stability results of some first order viscous hyperbolic systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018020/ DO - 10.1051/cocv/2018020 LA - en ID - COCV_2019__25__A33_0 ER -
%0 Journal Article %A Nicaise, Serge %T Stability results of some first order viscous hyperbolic systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018020/ %R 10.1051/cocv/2018020 %G en %F COCV_2019__25__A33_0
Nicaise, Serge. Stability results of some first order viscous hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 33. doi: 10.1051/cocv/2018020
[1] , A characterisation of generalized c∞ notion on nets. Integr. Equ. Oper. Theory 9 (1986) 753–766. | Zbl | MR | DOI
[2] , Nonlinear Wave in Networks. Vol. 80 of Math. Res. Akademie Verlag (1994). | Zbl | MR
[3] and , Stabilization of Elastic Systems by Collocated Feedback. Vol. 2124 of Lecture Notes in Mathematics. Springer, Cham (2015). | Zbl | MR | DOI
[4] , , and , Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | Zbl | MR | DOI
[5] , , and , Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete Contin. Dyn. Syst. Ser. B 15 (2011) 491–511. | Zbl | MR
[6] and , Stability and Boundary Stabilization of 1-D Hyperbolic Systems. PNLDE Subseries in Control. Birkhäuser, Basel (2016). | MR | DOI
[7] , and , On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Netw. Heterog. Media 4 (2009) 177–187. | Zbl | MR | DOI
[8] , A characteristic equation associated to an eigenvalue problem on c2-networks. Linear Algebra Appl. 71 (1985) 309–325. | Zbl | MR | DOI
[9] , Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72 (1988) 316–337. | Zbl | MR | DOI
[10] , Sturm-Liouville eigenvalue problems on networks. Math. Methods Appl. Sci. 10 (1988) 383–395. | Zbl | MR | DOI
[11] and , Introduction to Quantum Graphs. Vol. 186 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2013). | Zbl | MR
[12] , Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). | Zbl | MR | DOI
[13] , and , Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. | Zbl | MR | DOI
[14] , Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | Zbl | MR
[15] and , The rate at which energy decays in a damped string. Commun. Partial Differ. Equ. 19 (1994) 213–243. | Zbl | MR | DOI
[16] and , Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures. Vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006). | Zbl | MR | DOI
[17] and , Mathematical Analysis and Numerical Methods for Science and Technology. In Vol. 3 of Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson. Springer-Verlag, Berlin (1990). | Zbl | MR
[18] and , Introduction à quelques problèmes d’EDP. Notes du groupe de travail de l’équipe EDP du Laboratoire de Mathématiques et leurs Applications de Valenciennes. Editions universitaires européennes (2014).
[19] , , , and , Boundary feedback control in networks of open channels. Autom. J. IFAC 39 (2003) 1365–1376. | Zbl | MR | DOI
[20] , Etude des équations de la magnéto-hydrodynamique stationnaire et de leur approximation par éléments finis. Thèse de 3eme cycle, Université Pierre et Marie Curie (1982).
[21] and , Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Math. Acad. Sci. Paris 345 (2007) 113–118. | Zbl | MR | DOI
[22] and , Les inéquations en mécanique et en physique. Travaux et Recherches Mathématiques, No. 21, Dunod, Paris (1972). | Zbl | MR
[23] and , Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ. 245 (2008) 2660–2703. | Zbl | MR | DOI
[24] and , Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978) 28–63. | Zbl | MR | Numdam
[25] and , Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Vol. 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986). | Zbl | MR | DOI
[26] and , Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44 (1995) 603–676. | Zbl | MR | DOI
[27] , Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56. | Zbl | MR
[28] , Boundary stabilization, observation and control of Maxwell’s equations. PanAm. Math. J. 4 (1994) 47–61. | Zbl | MR
[29] , and , Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Birkhäuser, Boston (1994). | Zbl | MR | DOI
[30] , Derivation of a viscous Boussinesq system for surface water waves. Asymptot. Anal. 94 (2015) 309–345. | Zbl | MR
[31] and , On the modelling and stabilization of flows in networks of open canals. SIAM J. Control Optim. 41 (2002) 164–180. | Zbl | MR | DOI
[32] , Connecting of local operators and evolution equations on networks, in Potential Theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979). Vol. 787 of Lecture Notes in Math. Springer, Berlin (1980) 219–234. | Zbl | MR | DOI
[33] , Finite element methods for Maxwell’s equations. Numerical Analysis and Scientific Computation Series Oxford Univ. Press, New York (2003). | Zbl | MR
[34] and , On a Lp-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186 (2002) 377–393. | Zbl | MR | DOI
[35] , Spectre des réseaux topologiques finis. Bull. Sc. Math., 2ème série 111 (1987) 401–413. | Zbl | MR
[36] , Exact boundary controllability of Maxwell’s equations in heterogeneous media and an application to an inverse source problem. SIAM J. Control Optim. 38 (2000) 1145–1170. | Zbl | MR | DOI
[37] and , Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks. SIAM J. Control Optim. 52 (2014) 143–163. | Zbl | MR | DOI
[38] , Contrôle et stabilisation d’ondes électromagnétiques. ESAIM: COCV 5 (2000) 87–137. | Zbl | MR | Numdam
[39] , Observability and controllability of Maxwell’s equations. Rend. Mat. Appl. 19 (1999) 523–546. | Zbl | MR
[40] , On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847–857. | Zbl | MR
[41] , About the resolvent of an operator from fluid dynamics. Math. Z. 194 (1987) 183–191. | Zbl | MR | DOI
Cité par Sources :
