On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems

Badra, Mehdi; Takahashi, Takéo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 924-956.

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.

Classification : 93B05, 93D15, 35Q30, 76D05, 76D07, 76D55, 93B52, 93C20
Mots clés : approximate controllability, stabilizability, parabolic equation, finite dimensional control, coupled−Stokes and mhd system

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

[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 | Zbl