Return method : some applications to flow control

Abstract 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]  M. Slemrod.  A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space , 1974 .

[2]  J. L. Lions,et al.  Exact Controllability for Distributed Systems. Some Trends and Some Problems , 1991 .

[3]  Jean-Michel Coron,et al.  Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations , 2002 .

[4]  H. Hermes.  Control systems which generate decomposable Lie algebras , 1982 .

[5]  Pierre Rouchon,et al.  Motion planning and nonlinear simulations for a tank containing a fluid , 1999, 1999 European Control Conference (ECC).

[6]  Dr. M. G. Worster.  Methods of Mathematical Physics , 1947, Nature.

[7]  Andrei V. Fursikov,et al.  On exact boundary zero-controlability of two-dimensional Navier-Stokes equations , 1994 .

[8]  Eduardo Sontag.  Universal nonsingular controls , 1993 .

[9]  F. Ancona,et al.  On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control , 1998 .

[10]  J. Lions,et al.  Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles , 1968 .

[11]  Oleg Yu. Imanuvilov,et al.  On Controllability of Certain Systems Simulating a Fluid Flow , 1995 .

[12]  Eduardo D. Sontag,et al.  Finite-dimensional open-loop control generators for non-linear systems , 1988 .

[13]  B. Launder,et al.  Mathematical Models of turbulence , 1972 .

[14]  Wei-Liang Chow.  Über Systeme von linearen partiellen Differential-gleichungen erster Ordnung , 1941 .

[15]  J. Coron.  On the controllability of 2-D incompressible perfect fluids , 1996 .

[16]  A. Fursikov,et al.  Exact boundary zero controllability of three-dimensional Navier-Stokes equations , 1995 .

[17]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[18]  Gianna Stefani,et al.  Controllability along a trajectory: a variational approach , 1993 .

[19]  Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid , 1980 .

[20]  Oleg Yu. Imanuvilov,et al.  Remarks on exact controllability for the Navier-Stokes equations , 2001 .

[21]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[22]  Irena Lasiecka,et al.  Differential and Algebraic Riccati Equations With Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory , 1991 .

[23]  Giuseppe Geymonat,et al.  On the vanishing viscosity limit for acoustic phenomena in a bounded region , 1981 .

[24]  F. Golse,et al.  Fluid dynamic limits of kinetic equations. I. Formal derivations , 1991 .

[25]  François Coron,et al.  Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation , 1989 .

[26]  M. Krstić,et al.  On global stabilization of Burgers' equation by boundary control , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[27]  Olivier Glass.  Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles en dimension 3 , 1997 .

[28]  J. Coron,et al.  On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions , 1996 .

[29]  Eduardo Sontag,et al.  Remarks on continuous feedback , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[30]  S. Bhat.  Controllability of Nonlinear Systems , 2022 .

[31]  L. Hörmander,et al.  Lectures on Nonlinear Hyperbolic Differential Equations , 1997 .

[32]  H. Sussmann.  Subanalytic sets and feedback control , 1979 .

[33]  Olivier Glass,et al.  Exact boundary controllability of 3-D Euler equation , 2000 .

[34]  R. Bianchini,et al.  Sufficient conditions of local controllability , 1986, 1986 25th IEEE Conference on Decision and Control.

[35]  H. Sussmann.  Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems , 1983 .

[36]  Newton Diagrams and Tangent Cones to Attainable Sets , 1991 .

[37]  J. Coron.  Linearized Control Systems and Applications to Smooth Stabilization , 1994 .

[38]  Eduardo D. Sontag,et al.  Universal nonsingular controls: Systems and control letters 19 (1992) 221–224 , 1993 .

[39]  O. Glass.  An addendum to a J.M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids , 2001 .

[40]  H. Sussmann.  A general theorem on local controllability , 1987 .

[41]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[42]  Pierre Rouchon,et al.  Dynamics and solutions to some control problems for water-tank systems , 2002, IEEE Trans. Autom. Control..

[43]  Andrei V. Fursikov,et al.  Exact controllability of the Navier-Stokes and Boussinesq equations , 1999 .

[44]  ON ODD-ORDER NECESSARY CONDITIONS FOR OPTIMALITY IN A TIME-OPTIMAL CONTROL PROBLEM FOR SYSTEMS LINEAR IN THE CONTROL , 1991 .

[45]  C. Fabre,et al.  Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems , 1996 .

[46]  Andrew J. Majda,et al.  Vorticity and the mathematical theory of incompressible fluid flow , 1986 .

[47]  Jean-Michel Coron,et al.  Global asymptotic stabilization for controllable systems without drift , 1992, Math. Control. Signals Syst..

[48]  C. SIAMJ..  RAPID BOUNDARY STABILIZATION OF LINEAR DISTRIBUTED SYSTEMS , 1997 .

[49]  Enrique Zuazua,et al.  Exact controllability for semilinear wave equations in one space dimension , 1993 .

[50]  J. Coron.  Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels , 1993 .

[51]  Eduardo Sontag.  Control of systems without drift via generic loops , 1995, IEEE Trans. Autom. Control..

[52]  Jean-Michel Coron,et al.  Global Exact Controllability of the 2D Navier-Stokes Equations on a Manifold without boundary , 1996 .

[53]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[54]  L. Silverman,et al.  Controllability and Observability in Time-Variable Linear Systems , 1967 .

[55]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .