Index. A Abstract parabolic, 26 Accretive, 15 Algebraic multiplicity, 2 Algebraic Riccati equation, 36 Asymptotically stable, 26

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3 References Da Prato G, Zabczyk J (1996) Ergodicity for Infinite Dimensional Systems. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge 46. Deng H, Krstic M, Williams RJ (2001) Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Autom. Control 46: Doyle J, Glover K, Khargonekar P, Francis B (1989) State space solutions to standard H 2 and H -control problems. IEEE Trans. Autom. Control AC 34: Duan J, Fursikov AV (2005) Feedback stabilization for Oseen Fluid Equations. A stochastic approach. J. Math. Fluids Mech. 7: Edwards RE (1965) Functional Analysis. Holt, Rinehart and Winston, New York 50. Fursikov AV (2000) Optimal Control of Systems Theory and Applications. AMS, Providence RI 51. Fursikov AV (2002) Real processes of the 3-D Navier Stokes systems and its feedback stabilization from the boundary. In: Agranovic MS, Shubin MA (eds) AMS Translations. Partial Differential Equations. M. Vishnik Seminar Fursikov AV (2004) Stabilization for the 3-D Navier Stokes systems by feedback boundary control. Discrete Contin. Dyn. Syst. 10: Fursikov AV, Imanuvilov OY (1998) Local exact controllability of the Boussinesque equation. SIAM J. Control Optim. 36: Henry D (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics 840. Springer, Berlin, Heidelberg, New York 55. Hormander L (1976) Linear Partial Differential Operators. Springer, Berlin, Heidelberg, New York 56. Imanuvilov OY (1998) On exact controllability for Navier Stokes equations. ESAIM COCV 3: Joseph DD (1976) Stability of Fluid Motions. Springer, Berlin, Heidelberg, New York 58. Kuksin S, Shirikyan A (2001) Ergodicity for the randomly forced 2D Navier Stokes equations. Math. Phys. Anal. Geom. 4: Kato T (1966) Perturbation Theory of Linear Operators. Springer, Berlin, Heidelberg, New York 60. Lasiecka I, Triggiani R (2000) Control Theory for Partial Differential Equations: Continuous and Approximation Theory. Cambridge University Press, Cambridge 61. Lefter C (2009) Feedback stabilization of 2-D Navier Stokes equations with Navier slip boundary conditions. Nonlinear Anal. 70: Lions JL (1969) Quelques Méthodes de Resolution des Problèmes aux Limites Nonlinéaires. Dunod-Gauthier Villars, Paris 63. Lipster R, Shiraev AN (1989) Theory of Martingals. Kluwer, Dordrecht 64. Mao XR (2003) Stochastic stabilization and destabilization. Syst. Control Lett. 23: Munteanu I (2010) Normal feedback stabilization of periodic flows in a 2-D channel (to appear) 66. Pazy A (1983) Semigroups of Linear Operators. Springer, Berlin, Heidelberg, New York 67. Pazy A (1985) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin 68. Ravindran SS (2000) Reduced-order adaptive controllers for fluid flows using POD. J. Sci. Comput. 15(4): Smale S (1965) An infinite dimensional version of Sard s theorem. Am. J. Math. 18: Raymond JP (2006) Feedback boundary stabilization of the two dimensional Navier Stokes equations. SIAM J. Control Optim. 45: Raymond JP (2007) Feedback boundary stabilization of the three dimensional incompressible Navier Stokes equations. J. Math. Pures Appl. 87: Shirikyan A (2004) Exponential mixing 2D Navier Stokes equations perturbed by an unbounded noise. J. Math. Fluids Mech. 6: Temam R (1979) Navier Stokes Equations. North-Holland, Amsterdam 74. Temam R (1985) Navier Stokes Equations and Nonlinear Functional Analysis. SIAM, Philadelphia

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5 Index A Abstract parabolic, 26 Accretive, 15 Algebraic multiplicity, 2 Algebraic Riccati equation, 36 Asymptotically stable, 26 B Bilinear, 7 Bochner integrable, 10 Boundary controllers, 28 Boundary stabilization, 90 Brownian motion, 231 C C 0 -semigroup, 12 Closed, 1 Closed and densely defined operator, 3 Closed-loop system, 27, 53 Coercive, 7 Complexified space, 39 Conditional expectation, 230 Continuous, 7 Control system, 27 Controlled evolution system, 66 Controlled Navier Stokes system, 93 D Dirichlet map, 29 Distribution, 3 E Eigenvalue, 1 Eigenvectors, 1 Entry, 1 subentry, 1 F {F t }-stopping time, 231 Feedback controller, 53 Feedback stabilizable, 27 Filtration, 231 Floquet exponents, 74 Floquet s transformation, 159 Fredholm map, 169 Fredholm operator, 170 Function absolutely continuous, 11 finitely-valued, 10 G γ -suboptimal solution, 238 Gâteaux differentiable, 25 Generalized eigenvector, 1 Geometric multiplicity, 1 Growth logarithmic property, 14 H H -control problem, 238 Hamiltonian system, 77 High-gain Riccati-based feedback, 54 I Internal controller, 89 Internal controllers, 28 Invariant manifold, 255 Ito s formula, 49, 234 L Lax Milgram lemma, 7 Local martingale, 231 Low-gain Riccati-based feedback, 54 V. Barbu, Stabilization of Navier Stokes Flows, Communications and Control Engineering, DOI / , Springer-Verlag London Limited

6 276 Index M m-accretive, 15 Martingale, 230 P Periodic solution, 67 Poincaré inequality, 6 R Regular value, 169 Residual, 168 Riccati equation, 75 S Semigroup of class C 0,12 Semilinear parabolic equation, 81 Semimartingale, 231 Semisimple, 2 Sobolev embedding theorem, 5 Sobolev space, 3 Stabilizable controller, 27 Stabilizable feedback controller, 53 Stabilization problem, 27 Stochastic feedback controller, 50 Stokes operator, 18 Stokes Oseen operator, 89, 118, 168 Strong solution, 19 Strongly measurable, 10 Submartingale, 230 Supermartingale, 230 Support, 3 T Trace, 5 Transversality theorem, 169 V Variational solution, 8 W Weak, 8 Weak solution, 18 White noise, 232 Wiener process, 231 Y Yosida approximation, 15