Dedicated to Jean Lévine for his 50 th birthday Due to recent progress in advanced technologies in many fields of engineering sciences, applications of flow control are developing very quickly. In this paper we survey only a tiny and theoretical part of the recent results obtained in flow control, namely some results on the controllability and on the stabilizability of the equations of incompressible fluids which have been obtained by means of the return method. 1
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597
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Sobolev Spaces
– Adams
- 1975
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218
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Mathematical Control Theory, Deterministic Finite Dimensional Systems, Second Edition
– Sontag
- 1998
|
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96
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A General Theorem on Local Controllability
– Sussmann
- 1987
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90
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Flatness and defect of non-linear systems: Introductory theory and examples
– Fliess, Levine, et al.
- 1995
|
|
63
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Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory (R. W. Brockett et al
– Brockett
- 1983
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46
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Controllability of nonlinear systems
– Sussmann, Jurdjevic
- 1972
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44
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Global asymptotic stabilization for controllable systems without drift
– CORON
- 1991
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40
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Compressible fluid flows and systems of conservation laws in several space variables
– Majda
- 1984
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39
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Controle optimal de syst`emes gouvern'es par des 'equations aux d'eriv'ees partielles. Dunod
– Lions
- 1968
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35
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Lie brackets and local controllability: a sufficient condition for scalar-input systems
– Sussmann
- 1983
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30
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Systeme von linearen partiellen Differentialgleichungen erster Ordnung
– Chow
- 1939
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23
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Remarks on continuous feedback
– Sontag, Sussmann
- 1980
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20
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Exact controllability, stabilization and perturbations for distributed systems
– Lions
- 1988
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18
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Vorticity and the mathematical theory of incompressible fluid flow
– Majda
- 1986
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15
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Subanalytic sets and feedback control
– Sussmann
- 1979
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14
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Control of systems without drift via generic loops
– Sontag
- 1995
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12
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Control systems which generate decomposable Lie algebras
– Hermes
- 1982
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10
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Controllability along a trajectory: a variational approach
– Bianchini, Stefani
- 1993
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10
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Motion planning and nonlinear simulations for a tank containing a fluid
– Dubois, Petit, et al.
- 1999
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10
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Dynamics and solutions to some control problems for water-tank systems
– Petit, Rouchon
- 2000
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9
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Mathematical Models of Turbulence
– Launder, Spalding
- 1972
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9
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Finite dimensional open-loop control generators for nonlinear systems
– Sontag
- 1988
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8
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Lectures on nonlinear hyperbolic differential equations
– Hormander
- 1997
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7
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On the attainable set for scalar nonlinear conservation laws with boundary control
– Ancona, Marson
- 1998
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7
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Fluid dynamic limits of kinetic equations: Formal derivations
– Bardos, Golse, et al.
- 1991
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7
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Linearized control systems and applications to smooth stabilization
– Coron
- 1994
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7
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Nonlinear Water Waves
– Debnath
- 1994
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6
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On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions
– Coron
- 1996
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6
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On the inversion of Lyapunov's second theorem on stability of motion
– Kurzweil
- 1956
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6
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Exact boundary value controlability theorems for wave and heat processes in star-complemented regions
– Russell
- 1974
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5
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On the controllability of 2-D incompressible perfect fluids
– Coron
- 1996
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5
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Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
– Coron
- 2002
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5
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Universal nonsingular controls
– Sontag
- 1992
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4
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Sufficient conditions for local controllability, in
– Bianchini, Stefani
- 1986
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4
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Exact boundary controllability of 3-D Euler equation
– Glass
- 2000
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4
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On the controllability of the Burgers equation
– Horsin
- 1998
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4
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Exact controllability for the semilinear wave equation
– Zuazua
- 1990
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3
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diagrams and tangent cones to attainable sets, in Analysis of Controlled Dynamical Systems, Progr. Systems Control Theory 8
– Agrachev, Newton
- 1991
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3
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Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation
– Coron
- 1989
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3
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Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels
– Coron
- 1993
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3
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Approximate controllability for semilinear heat equation
– Fabre, Puel, et al.
- 1995
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3
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Exact boundary zero controllability of three-dimensional Navier-Stokes equations
– Fursikov, V
- 1995
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3
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Local exact controllability of the Navier-Stokes equations
– Fursikov, Yu
- 1996
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3
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Exact controllability for distributed systems. Some trends and some problems
– Lions
- 1991
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2
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Imanuvilov, On controllability of certain systems simulating a fluid flow
– Fursikov, Yu
- 1994
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2
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Remarks on exact controllability for the Navier-Stokes equations
– Imanuvilov
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2
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Tsien and Yu Wen-Ci, Boundary value problems for quasilinear hyperbolic systems, Mathematics series V
– Ta
- 1985
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2
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Are there connections between turbulence and controllability
– Lions
- 1990
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2
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Systemes de lois de conservation I et II. Diderot Editeur, Arts et Sciences
– Serre
- 1996
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2
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A note on complete controllability and stabilizability for linear control systems in Hilbert space
– Slemrod
- 1974
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