On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, p. 924-956
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In this paper, we consider the well-known Fattorini's criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686-694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini's criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini's criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier-Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini's criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.

DOI : https://doi.org/10.1051/cocv/2014002
Classification:  93B05,  93D15,  35Q30,  76D05,  76D07,  76D55,  93B52,  93C20
Keywords: approximate controllability, stabilizability, parabolic equation, finite dimensional control, coupled−Stokes and mhd system
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     author = {Badra, Mehdi and Takahashi, Tak\'eo},
     title = {On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {3},
     year = {2014},
     pages = {924-956},
     doi = {10.1051/cocv/2014002},
     zbl = {1292.93022},
     mrnumber = {3264229},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2014__20_3_924_0}
}
Badra, Mehdi; Takahashi, Takéo. On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, pp. 924-956. doi : 10.1051/cocv/2014002. http://www.numdam.org/item/COCV_2014__20_3_924_0/

[1] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system. ESAIM: COCV 15 (2009) 934-968. | Numdam | MR 2567253 | Zbl 1183.35216

[2] M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Control Optim. 48 (2009) 1797-1830. | MR 2516189 | Zbl pre05719723

[3] 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. - Series A 32 (2011) 1169-1208. | MR 2851895 | Zbl 1235.93200

[4] M. Badra, Local controllability to trajectories of the magnetohydrodynamic equations. J. Math. Fluid Mech. (Sumitted). | MR 3267539

[5] 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) 420-463. | MR 2784695 | Zbl 1217.93137

[6] M. Badra and T. Takahashi, Feedback stabilization of a fluid-rigid body interaction system. preprint. | MR 3261920 | Zbl pre06356246

[7] M. Badra and T. Takahashi, Feedback stabilization of a simplified 1d fluid - particle system. Ann. Inst. Henri Poincaré Anal. Non Linéaire (Sumitted). | Zbl pre06347281

[8] V. Barbu and R.L. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53 (2004) 1443-1494. | MR 2104285 | Zbl 1073.76017

[9] V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers. Nonlinear Anal. 64 (2006) 2704-2746. | MR 2218543 | Zbl 1098.35025

[10] V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations. Mem. Amer. Math. Soc. 181 (2006) 128. | MR 2215059 | Zbl 1098.35026

[11] V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, d = 2,3, via feedback stabilization of its linearization, in Control of coupled partial differential equations, vol. 155. Int. Ser. Numer. Math. Birkhäuser, Basel (2007) 13-46. | MR 2328600 | Zbl 1239.93094

[12] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite dimensional systems. Systems & Control: Foundations & Applications. 2nd edition. Birkhäuser Boston Inc., Boston, MA (2007). | MR 2273323 | Zbl 1117.93002

[13] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim. 43 (2004) 549-569. | MR 2086173 | Zbl 1101.93011

[14] R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, vol. 21. Texts Appl. Math.. Springer-Verlag, New York (1995). | MR 1351248 | Zbl 0839.93001

[15] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 5. Spectre des opérateurs. [The operator spectrum], With the collaboration of Michel Artola, Michel Cessenat, Jean Michel Combes and Bruno Scheurer, Reprinted from the 1984 edition. INSTN: Collection Enseignement. [INSTN: Teaching Collection]. Masson, Paris (1988). | MR 944303 | Zbl 0749.35004

[16] E.B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances. R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) 585-599. | MR 1700903 | Zbl 0931.70016

[17] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes. Commun. Partial Differ. Eqs. 21 (1996) 573-596. | MR 1387461 | Zbl 0849.35098

[18] H.O. Fattorini, Some remarks on complete controllability. SIAM J. Control 4 (1966) 686-694. | MR 202484 | Zbl 0168.34906

[19] H.O. Fattorini, On complete controllability of linear systems. J. Differ. Eqs. 3 (1967) 391-402. | MR 212322 | Zbl 0155.15903

[20] E. Fernández-Cara and S. Guerrero, Local exact controllability of micropolar fluids. J. Math. Fluid Mech. 9 (2007) 419-453. | MR 2336077 | Zbl 1133.35423

[21] E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR 2103189 | Zbl 1267.93020

[22] A.V. Fursikov, Stabilizability of a quasilinear parabolic equation by means of boundary feedback control. Mat. Sb. 192 (2001) 115-160. | MR 1834095 | Zbl 1019.93047

[23] A.V. Fursikov, Stabilizability of two-dimensional Navier-Stokes equations with help of a boundary feedback control. J. Math. Fluid Mech. 3 (2001) 259-301. | MR 1860125 | Zbl 0983.93021

[24] A.V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications. Discrete Contin. Dyn. Syst. 10 (2004) 289-314. | MR 2026196 | Zbl 1174.93675

[25] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems, vol. 38. Springer Tracts in Natural Philosophy. Springer-Verlag, New York (1994). | MR 1284205 | Zbl 0949.35004

[26] I.C. Gohberg and M.G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators. Translated from the Russian by A. Feinstein. Vol. 18. Translations of Mathematical Monographs. Amer. Math. Soc., Providence, R.I. (1969). | MR 246142 | Zbl 0181.13504

[27] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Oxford University Press, Oxford, 6th edition (2008). Revised by D.R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles. | MR 2445243 | Zbl 1159.11001

[28] M.L.J. Hautus, Controllability and observability conditions of linear autonomous systems. Nederl. Akad. Wetensch. Proc. Ser. A. Vol. 31 of Indag. Math. (1969) 443-448. | MR 250694 | Zbl 0188.46801

[29] A. Henrot and M. Pierre, Variation et optimisation de formes, Une analyse géométrique (A geometric analysis). Vol. 48. Mathématiques & Applications [Mathematics & Applications]. Springer, Berlin (2005). | MR 2512810 | Zbl 1098.49001

[30] L. Hörmander, The analysis of linear partial differential operators I. Classics in Mathematics. Springer-Verlag, Berlin (2003). Distribution theory and Fourier analysis, Reprint of the 2nd edition (1990) [Springer, Berlin; MR1065993 (91m:35001a)]. | MR 1996773 | Zbl 1028.35001

[31] O. Yu. Imanuvilov. Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR 1804497 | Zbl 0961.35104

[32] O.Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Int. Math. Res. Not. 16 (2003) 883-913. | MR 1959940 | Zbl 1146.35340

[33] T. Kato, Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin (1995). Reprint of the 1980 edition. | MR 1335452 | Zbl 0836.47009

[34] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I, Abstract parabolic systems. Vol. 74. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2000). | MR 1745475 | Zbl 0942.93001

[35] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. applications to unique continuation and control of parabolic equations. ESAIM: COCV 18 (2012) 712-747. | Numdam | MR 3041662 | Zbl 1262.35206

[36] C.-G. Lefter, On a unique continuation property related to the boundary stabilization of magnetohydrodynamic equations. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 56 (2010) 1-15. | MR 2605928 | Zbl 1212.93256

[37] G. Łukaszewicz, Micropolar fluids. Theory and applications. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston Inc., Boston, MA (1999). | MR 1711268 | Zbl 0923.76003

[38] A.J. Meir, The equations of stationary, incompressible magnetohydrodynamics with mixed boundary conditions. Comput. Math. Appl. 25 (1993) 13-29. | MR 1216012 | Zbl 0774.35059

[39] A.M. Micheletti, Perturbazione dello spettro dell'operatore di Laplace, in relazione ad una variazione del campo. Ann. Scuola Norm. Sup. Pisa 26 (1972) 151-169. | Numdam | MR 367480 | Zbl 0234.35073

[40] S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 44 (2006) 1950-1972. | MR 2248170 | Zbl 1116.93022

[41] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44. Appl. Math. Sci. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[42] J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J. Control Optim. 45 (2006) 790-828. | MR 2247716 | Zbl 1121.93064

[43] J.-P. Raymond and T. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers. Issue in Discrete and Continuous Dynamical Systems A 27 (2010) 1159-1187. | MR 2629581 | Zbl 1211.93103

[44] D.L. Russell and G. Weiss, A general necessary condition for exact observability. SIAM J. Control Optim. 32 (1994) 1-23. | MR 1255956 | Zbl 0795.93023

[45] H. Triebel, Interpolation theory, function spaces, differential operators, 2nd edition. Johann Ambrosius Barth, Heidelberg (1995). | MR 1328645 | Zbl 0830.46028

[46] R. Triggiani, On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52 (1975) 383-403,. | MR 445388 | Zbl 0326.93023

[47] R. Triggiani, Extensions of rank conditions for controllability and observability to Banach spaces and unbounded operators. SIAM J. Control Optim. 14 (1976) 313-338. | MR 479499 | Zbl 0326.93003

[48] R. Triggiani, Boundary feedback stabilizability of parabolic equations. Appl. Math. Optim. 6 (1980) 201-220. | MR 576260 | Zbl 0434.35016

[49] R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation. Nonlinear Anal. 71 (2009) 4967-4976. | MR 2548728 | Zbl 1181.35034

[50] M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2009). | MR 2502023 | Zbl 1188.93002