Local Exponential Stabilization Strategies of the Navier-Stokes Equations, d = 2, 3, via Feedback Stabilization of its Linearization

A.V. Balakrishnan, Applied Functional Analysis, Springer-Verlag, 1981.Google Scholar

[B-D-D-M.1] A. Bensoussan, G. Da Prato, M. Delfour, S. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol. 1, Birkhäuser, 1992.Google Scholar

V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control Optim. Calc. Var. 9 (2003), 197–206 (electronic).MATHGoogle Scholar

V. Barbu, Stabilization of a plane channel flow by wall-normal controllers, to appear.Google Scholar

V. Barbu, I. Lasiecka, and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoir AMS, Number 852, Volume 181, ISSN 0065-9266, 128 pp., May 2006.Google Scholar

V. Barbu, I. Lasiecka, and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high-and low-gain feedback controllers, J. Nonlinear Anal., to appear.Google Scholar

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana University Mathematics Journal 53(5) (2004), 1443–1494.MATHCrossRefGoogle Scholar

P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, Chicago-London, 1989.Google Scholar

F. Flandoli, I. Lasiecka, and R. Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli equations, Ann. di Matematica Pura e Appl. CLII (1988), 307–382.CrossRefGoogle Scholar

A.V. Fursikov, Real processes of the 3D Navier-Stokes systems and its feedback stabilization from the boundary, in AMS Translations, Series 2, Vol. 206, Partial Differential Equations, Mark Vishik’s Seminar; M.S. Agranovich, M.A. Shubin, editors (2002), 95–123.Google Scholar

A.V. Fursikov, Feedback stabilization for the 2D Oseen equations: Additional remarks, Proc. 8^th Conference on Control of Distributed Parameter Systems, Int. Ser. Numerical Math., Birkhäuser-Verlag 143 (2002), 169–187.Google Scholar

A.V. Fursikov, Stabilization for the 3D Navier-Stokes system of feedback boundary control, Disc. Cont. Dynam. Systems 10(182) (2004)m 289–314.MATHGoogle Scholar

D. Henry, Perturbation of the Boundary in Boundary Value Problems, London Mathematical Society Lecture Notes 318, Cambridge University Press, 2005.Google Scholar

T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, New York-Berlin, 1966.Google Scholar

I. Lasiecka, Boundary control of parabolic systems: Regularity of optimal solutions, Appl. Math. & Optimiz. 4 (1978), 301–327.MATHCrossRefGoogle Scholar

I. Lasiecka, Unified theory for abstract parabolic boundary problems: A semigroup approach, Appl. Math. & Optimiz. 6 (1980), 287–333.MATHCrossRefGoogle Scholar

I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations: A Riccati operator approach, Appl. Anal. 42 (1991), 2434–261.CrossRefGoogle Scholar

I. Lasiecka and R. Triggiani, Dirichlet boundary control problems for parabolic equations with quadratic cost: Analyticity and Riccati feedback synthesis, SIAM J. Control & Optimiz. 21(1) (1983), 41–67.MATHCrossRefGoogle Scholar

I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. Part I: Riccati’s feedback synthesis and regularity of optimal solution, Appl. Math. & Optimiz. 16 (1986), 147–168.CrossRefGoogle Scholar

I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Springer Verlag Lecture Notes, LNCIS, Vol. 164 (1991), 160 pp.Google Scholar

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I: Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications 74, Cambridge University Press (2002), 648 pp.Google Scholar

J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1 (1970); Vol. 2 (1972), Springer-Verlag.Google Scholar

J.P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, preprint, 2005.Google Scholar

R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications, Vol. 1, North Holland Publishing Co., Amsterdam, 1979.Google Scholar

R. Triggiani, On the stabilizability problem in Banach spaces, J. Math. Anal. & Appl. (1975), 383–403.Google Scholar

R. Triggiani, Boundary feedback stabilizability of parabolic equations, Appl. Math. Optimiz. 6 (1980), 201–220.MATHCrossRefGoogle Scholar

R. Triggiani, Exponential feedback stabilization of a 2-D linearized Navier-Stokes channel flow by finite-dimensional wall normal boundary controllers, with arbitrarily small support, Discrete and Continuous Dynamical Systems, B, to appear.Google Scholar

W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg & Sohn, Braunschweig, 1985.Google Scholar