Stability results of some first order viscous hyperbolic systems

Nicaise, Serge

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 33.

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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.

[1] F. Ali Mehmeti, A characterisation of generalized c^∞ notion on nets. Integr. Equ. Oper. Theory 9 (1986) 753–766. | DOI | MR | Zbl

[2] F. Ali Mehmeti, Nonlinear Wave in Networks. Vol. 80 of Math. Res. Akademie Verlag (1994). | MR | Zbl

[3] K. Ammari and S. Nicaise, Stabilization of Elastic Systems by Collocated Feedback. Vol. 2124 of Lecture Notes in Mathematics. Springer, Cham (2015). | DOI | MR | Zbl

[4] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998) 823–864. | DOI | MR | Zbl

[5] H. Arfaoui, F. Ben Belgacem, H. El Fekih and J.-P. Raymond, Boundary stabilizability of the linearized viscous Saint-Venant system. Discrete Contin. Dyn. Syst. Ser. B 15 (2011) 491–511. | MR | Zbl

[6] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems. PNLDE Subseries in Control. Birkhäuser, Basel (2016). | DOI | MR

[7] G. Bastin, J.-M. Coron and B. D’Andréa Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Netw. Heterog. Media 4 (2009) 177–187. | DOI | MR | Zbl

[8] J. Von Below, A characteristic equation associated to an eigenvalue problem on c²-networks. Linear Algebra Appl. 71 (1985) 309–325. | DOI | MR | Zbl

[9] J. Von Below, Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72 (1988) 316–337. | DOI | MR | Zbl

[10] J. Von Below, Sturm-Liouville eigenvalue problems on networks. Math. Methods Appl. Sci. 10 (1988) 383–395. | DOI | MR | Zbl

[11] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs. Vol. 186 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2013). | MR | Zbl

[12] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). | DOI | MR | Zbl

[13] S. Chowdhury, M. Ramaswamy and J.-P. Raymond, Controllability and stabilizability of the linearized compressible Navier-Stokes system in one dimension. SIAM J. Control Optim. 50 (2012) 2959–2987. | DOI | MR | Zbl

[14] J.-M. Coron, Control and Nonlinearity. Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007). | MR | Zbl

[15] S. Cox and E. Zuazua, The rate at which energy decays in a damped string. Commun. Partial Differ. Equ. 19 (1994) 213–243. | DOI | MR | Zbl

[16] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures. Vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006). | DOI | MR | Zbl

[17] R. Dautray and J.-L. Lions, 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). | MR | Zbl

[18] C. De Coster and S. Nicaise, 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] J. De Halleux, C. Prieur, J.-M. Coron, B. D’Andréa Novel and G. Bastin, Boundary feedback control in networks of open channels. Autom. J. IFAC 39 (2003) 1365–1376. | DOI | MR | Zbl

[20] J.-M. Dominguez, 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] D. Dutykh and F. Dias, Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Math. Acad. Sci. Paris 345 (2007) 113–118. | DOI | MR | Zbl

[22] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Travaux et Recherches Mathématiques, No. 21, Dunod, Paris (1972). | MR | Zbl

[23] S. Evje and K.H. Karlsen, Global existence of weak solutions for a viscous two-phase model. J. Differ. Equ. 245 (2008) 2660–2703. | DOI | MR | Zbl

[24] C. Foiaş and R. Temam, 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. | Numdam | MR | Zbl

[25] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Vol. 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986). | DOI | MR | Zbl

[26] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44 (1995) 603–676. | DOI | MR | Zbl

[27] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56. | MR | Zbl

[28] V. Komornik, Boundary stabilization, observation and control of Maxwell’s equations. PanAm. Math. J. 4 (1994) 47–61. | MR | Zbl

[29] J.E. Lagnese, G. Leugering and E. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures. Birkhäuser, Boston (1994). | DOI | MR | Zbl

[30] H.V.J. Le Meur, Derivation of a viscous Boussinesq system for surface water waves. Asymptot. Anal. 94 (2015) 309–345. | MR | Zbl

[31] G. Leugering and E.J.P.G. Schmidt, On the modelling and stabilization of flows in networks of open canals. SIAM J. Control Optim. 41 (2002) 164–180. | DOI | MR | Zbl

[32] G. Lumer, 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. | DOI | MR | Zbl

[33] P. Monk, Finite element methods for Maxwell’s equations. Numerical Analysis and Scientific Computation Series Oxford Univ. Press, New York (2003). | MR | Zbl

[34] P.B. Mucha and W.M. Zajaczkowski, On a L_p-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186 (2002) 377–393. | DOI | MR | Zbl

[35] S. Nicaise, Spectre des réseaux topologiques finis. Bull. Sc. Math., 2ème série 111 (1987) 401–413. | MR | Zbl

[36] S. Nicaise, 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. | DOI | MR | Zbl

[37] V. Perrollaz and L. Rosier, Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks. SIAM J. Control Optim. 52 (2014) 143–163. | DOI | MR | Zbl

[38] K.D. Phung, Contrôle et stabilisation d’ondes électromagnétiques. ESAIM: COCV 5 (2000) 87–137. | Numdam | MR | Zbl

[39] C. Pignotti, Observability and controllability of Maxwell’s equations. Rend. Mat. Appl. 19 (1999) 523–546. | MR | Zbl

[40] J. Prüss, On the spectrum of C₀-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847–857. | MR | Zbl

[41] G. Ströhmer, About the resolvent of an operator from fluid dynamics. Math. Z. 194 (1987) 183–191. | DOI | MR | Zbl