Abstract
In this paper, we consider control systems for which the underlying semigroup is analytic, and the resolvent of its generator is compact. In that case we give a characterization of the stabilizability of such control systems. When the stabilizability condition is satisfied the system is also stabilizable by finite-dimensional controls. We end the paper by giving an application of this result to the stabilizability of the Oseen equations with mixed boundary conditions.
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR 15-CE40.0010
Funding statement: The author is partially supported by the ANR-Project IFSMACS (ANR 15-CE40.0010).
References
[1] C. Airiau, J.-M. Buchot, R. K. Dubey, M. Fournié, J.-P. Raymond and J. Weller-Calvo, Stabilization and best actuator location for the Navier–Stokes equations, SIAM J. Sci. Comput. 39 (2017), no. 5, B993–B1020. 10.1137/16M107503XSearch in Google Scholar
[2] H. Amann, Feedback stabilization of linear and semilinear parabolic systems, Semigroup Theory and Applications (Trieste 1987), Lecture Notes Pure Appl. Math. 116, Dekker, New York (1989), 21–57. Search in Google Scholar
[3] L. Amodei and J.-M. Buchot, A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation, Numer. Linear Algebra Appl. 19 (2012), no. 4, 700–727. 10.1002/nla.799Search in Google Scholar
[4] M. Badra, Feedback stabilization of the 2-D and 3-D Navier–Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var. 15 (2009), no. 4, 934–968. 10.1007/0-387-33882-9_2Search in Google Scholar
[5] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier–Stokes equations, SIAM J. Control Optim. 48 (2009), no. 3, 1797–1830. 10.1137/070682630Search in Google Scholar
[6] M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier–Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1169–1208. 10.3934/dcds.2012.32.1169Search in Google Scholar
[7] M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier–Stokes system, SIAM J. Control Optim. 49 (2011), no. 2, 420–463. 10.1137/090778146Search in Google Scholar
[8] M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var. 20 (2014), no. 3, 924–956. 10.1051/cocv/2014002Search in Google Scholar
[9] V. Barbu, Stabilization of Navier–Stokes Flows, Comm. Control Engrg. Ser., Springer, London, 2011. 10.1007/978-0-85729-043-4Search in Google Scholar
[10] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 181 (2006), Paper No. 852. 10.1090/memo/0852Search in Google Scholar
[11] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), no. 3, 373–379. 10.21236/ADA043852Search in Google Scholar
[12] A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems. Vol. 2, Systems Control Found. Appl., Birkhäuser, Boston, 1993. 10.1007/978-1-4612-2750-2Search in Google Scholar
[13] C. Fabre and G. Lebeau, Prolongement unique des solutions de l’equation de Stokes, Comm. Partial Differential Equations 21 (1996), no. 3–4, 573–596. 10.1080/03605309608821198Search in Google Scholar
[14] H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control 4 (1966), 686–694. 10.1137/0304048Search in Google Scholar
[15] H. O. Fattorini, On complete controllability of linear systems, J. Differential Equations 3 (1967), 391–402. 10.1016/0022-0396(67)90039-3Search in Google Scholar
[16] M. Fournié, M. Ndiaye and J.-P. Raymond, Feedback stabilization of a two-dimensional fluid-structure intercation system with mixed boundary conditions, preprint (2018), https://hal.archives-ouvertes.fr/hal-01743783. Search in Google Scholar
[17] A. V. Fursikov, Stabilizability of a quasilinear parabolic equation by means of boundary feedback control, Mat. Sb. 192 (2001), no. 4, 115–160. 10.1070/SM2001v192n04ABEH000560Search in Google Scholar
[18] A. V. Fursikov, Stabilizability of two-dimensional Navier–Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech. 3 (2001), no. 3, 259–301. 10.1007/PL00000972Search in Google Scholar
[19] A. V. Fursikov, Stabilization for the 3D Navier–Stokes system by feedback boundary control, Discrete Contin. Dyn. Syst. 10 (2004), no. 1–2, 289–314. 10.3934/dcds.2004.10.289Search in Google Scholar
[20] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995. 10.1007/978-3-642-66282-9Search in Google Scholar
[21] I. Lasiecka and R. Triggiani, Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim. 21 (1983), no. 5, 766–803. 10.1137/0321047Search in Google Scholar
[22] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. II. Galerkin approximation, Appl. Math. Optim. 16 (1987), no. 3, 187–216. 10.1007/BF01442191Search in Google Scholar
[23] I. Lasiecka and R. Triggiani, Stability and Stabilizability of Infinite Dimensional Systems. Vol. 1, Cambridge University Press, Cambridge, 2000. Search in Google Scholar
[24] D. Maity, J.-P. Raymond and A. Roy, Local-in-time existence of strong solutions to a 3D fluid-structure intercation model, preprint (2018). Search in Google Scholar
[25] V. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr. 162, American Mathematical Society, Providence, 2010. 10.1090/surv/162Search in Google Scholar
[26] T. Nambu, On the stabilization of diffusion equations: boundary observation and feedback, J. Differential Equations 52 (1984), no. 2, 204–233. 10.1016/0022-0396(84)90177-3Search in Google Scholar
[27] P. A. Nguyen and J.-P. Raymond, Boundary stabilization of the Navier–Stokes equations in the case of mixed boundary conditions, SIAM J. Control Optim. 53 (2015), no. 5, 3006–3039. 10.1137/13091364XSearch in Google Scholar
[28] A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite-dimensional systems, SIAM Rev. 23 (1981), no. 1, 25–52. 10.1137/1023003Search in Google Scholar
[29] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations, SIAM J. Control Optim. 45 (2006), no. 3, 790–828. 10.1137/050628726Search in Google Scholar
[30] J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations, J. Math. Pures Appl. (9) 87 (2007), no. 6, 627–669. 10.1016/j.matpur.2007.04.002Search in Google Scholar
[31] J.-P. Raymond, Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 6, 921–951. 10.1016/j.anihpc.2006.06.008Search in Google Scholar
[32] J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier–Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst. 27 (2010), no. 3, 1159–1187. 10.3934/dcds.2010.27.1159Search in Google Scholar
[33] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), no. 3, 383–403. 10.1016/0022-247X(75)90067-0Search in Google Scholar
[34] M. Tucsnak and G. Weiss, Mathematical Control Theory. An Introduction, Mod. Birkhäuser Class., Birkhäuser, Basel, 2009. Search in Google Scholar
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