×

zbMATH — the first resource for mathematics

Stabilizability of infinite-dimensional systems by finite-dimensional controls. (English) Zbl 1432.93263
Comput. Methods Appl. Math. 19, No. 4, 797-811 (2019); retraction ibid. 20, No. 2, 395 (2020).
Summary: 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.
Editorial remark: From the text of the retraction notice: “The Editors and the Publisher of Computational Methods in Applied Mathematics retract from publication this article. Due to technical reasons and the erroneous assignment of different DOI numbersan identical version of this paper was published in an earlier issue of the journal [J.-P. Raymond, Comput. Methods Appl. Math. 19, No. 2, 267–282 (2019; Zbl 1422.93158)]. Hence the version of thearticle that was published later is herewith retracted.”

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C25 Control/observation systems in abstract spaces
93B52 Feedback control
93C20 Control/observation systems governed by partial differential equations
76D55 Flow control and optimization for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
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.; Airiau, C.; Buchot, J.-M.; Dubey, R. K.; Fournié, M.; Raymond, J.-P.; Weller-Calvo, J., Stabilization and best actuator location for the Navier-Stokes equations, SIAM J. Sci. Comput., 39, 5, B993-B1020 (2017) · Zbl 1373.93129
[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.; Amann, H., Feedback stabilization of linear and semilinear parabolic systems, Semigroup Theory and Applications, 21-57 (1989) · Zbl 0576.35067
[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.; Amodei, L.; Buchot, J.-M., A stabilization algorithm of the Navier-Stokes equations based on algebraic Bernoulli equation, Numer. Linear Algebra Appl., 19, 4, 700-727 (2012) · Zbl 1274.76186
[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.; Badra, M., Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15, 4, 934-968 (2009) · Zbl 1183.35216
[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.; Badra, M., Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48, 3, 1797-1830 (2009) · Zbl 1405.93178
[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.; Badra, M., 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, 4, 1169-1208 (2012) · Zbl 1235.93200
[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.; Badra, M.; Takahashi, T., Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49, 2, 420-463 (2011) · Zbl 1217.93137
[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.; Badra, M.; Takahashi, T., On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20, 3, 924-956 (2014) · Zbl 1292.93022
[9] V. Barbu, Stabilization of Navier-Stokes Flows, Comm. Control Engrg. Ser., Springer, London, 2011.; Barbu, V., Stabilization of Navier-Stokes Flows (2011) · Zbl 1213.76001
[10] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc. 181 (2006), Paper No. 852.; Barbu, V.; Lasiecka, I.; Triggiani, R., Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006) · Zbl 1098.35026
[11] C. D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), no. 3, 373-379.; Benchimol, C. D., A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim., 16, 3, 373-379 (1978) · Zbl 0384.93035
[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.; Bensoussan, A.; Da Prato, G.; Delfour, M. C.; Mitter, S. K., Representation and Control of Infinite-Dimensional Systems. Vol. 2 (1993) · Zbl 0790.93016
[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.; Fabre, C.; Lebeau, G., Prolongement unique des solutions de l’equation de Stokes, Comm. Partial Differential Equations, 21, 3-4, 573-596 (1996) · Zbl 0849.35098
[14] H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control 4 (1966), 686-694.; Fattorini, H. O., Some remarks on complete controllability, SIAM J. Control, 4, 686-694 (1966) · Zbl 0168.34906
[15] H. O. Fattorini, On complete controllability of linear systems, J. Differential Equations 3 (1967), 391-402.; Fattorini, H. O., On complete controllability of linear systems, J. Differential Equations, 3, 391-402 (1967) · Zbl 0155.15903
[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), .; <element-citation publication-type=”other“> Fournié, M.Ndiaye, M.Raymond, J.-P.Feedback stabilization of a two-dimensional fluid-structure intercation system with mixed boundary conditionsPreprint2018 <ext-link ext-link-type=”uri“ xlink.href=”>https://hal.archives-ouvertes.fr/hal-01743783
[17] A. V. Fursikov, Stabilizability of a quasilinear parabolic equation by means of boundary feedback control, Mat. Sb. 192 (2001), no. 4, 115-160.; Fursikov, A. V., Stabilizability of a quasilinear parabolic equation by means of boundary feedback control, Mat. Sb., 192, 4, 115-160 (2001) · Zbl 1019.93047
[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.; Fursikov, A. V., Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3, 3, 259-301 (2001) · Zbl 0983.93021
[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.; Fursikov, A. V., Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete Contin. Dyn. Syst., 10, 1-2, 289-314 (2004) · Zbl 1174.93675
[20] T. Kato, Perturbation Theory for Linear Operators, Classics Math., Springer, Berlin, 1995.; Kato, T., Perturbation Theory for Linear Operators (1995) · Zbl 0836.47009
[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.; Lasiecka, I.; Triggiani, R., Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations, SIAM J. Control Optim., 21, 5, 766-803 (1983) · Zbl 0518.93046
[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.; Lasiecka, I.; Triggiani, R., The regulator problem for parabolic equations with Dirichlet boundary control. II. Galerkin approximation, Appl. Math. Optim., 16, 3, 187-216 (1987) · Zbl 0652.49026
[23] I. Lasiecka and R. Triggiani, Stability and Stabilizability of Infinite Dimensional Systems. Vol. 1, Cambridge University Press, Cambridge, 2000.; Lasiecka, I.; Triggiani, R., Stability and Stabilizability of Infinite Dimensional Systems. Vol. 1 (2000) · Zbl 0983.35032
[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).; Maity, D.; Raymond, J.-P.; Roy, A., Local-in-time existence of strong solutions to a 3D fluid-structure intercation model, Preprint (2018)
[25] V. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr. 162, American Mathematical Society, Providence, 2010.; Maz’ya, V.; Rossmann, J., Elliptic Equations in Polyhedral Domains (2010) · Zbl 1196.35005
[26] T. Nambu, On the stabilization of diffusion equations: boundary observation and feedback, J. Differential Equations 52 (1984), no. 2, 204-233.; Nambu, T., On the stabilization of diffusion equations: boundary observation and feedback, J. Differential Equations, 52, 2, 204-233 (1984) · Zbl 0545.93053
[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.; Nguyen, P. A.; Raymond, J.-P., Boundary stabilization of the Navier-Stokes equations in the case of mixed boundary conditions, SIAM J. Control Optim., 53, 5, 3006-3039 (2015) · Zbl 1327.93194
[28] A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite-dimensional systems, SIAM Rev. 23 (1981), no. 1, 25-52.; Pritchard, A. J.; Zabczyk, J., Stability and stabilizability of infinite-dimensional systems, SIAM Rev., 23, 1, 25-52 (1981) · Zbl 0452.93029
[29] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim. 45 (2006), no. 3, 790-828.; Raymond, J.-P., Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45, 3, 790-828 (2006) · Zbl 1121.93064
[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.; Raymond, J.-P., Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87, 6, 627-669 (2007) · Zbl 1114.93040
[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.; Raymond, J.-P., Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24, 6, 921-951 (2007) · Zbl 1136.35070
[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.; Raymond, J.-P.; Thevenet, L., Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete Contin. Dyn. Syst., 27, 3, 1159-1187 (2010) · Zbl 1211.93103
[33] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), no. 3, 383-403.; Triggiani, R., On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52, 3, 383-403 (1975) · Zbl 0326.93023
[34] M. Tucsnak and G. Weiss, Mathematical Control Theory. An Introduction, Mod. Birkhäuser Class., Birkhäuser, Basel, 2009.; Tucsnak, M.; Weiss, G., Mathematical Control Theory. An Introduction (2009) · Zbl 1188.93002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.