Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

T. Tachim Medjo; R. Temam; M. Ziane

doi:10.1115/1.2830523

Applied Mechanics Reviews | Volume 61 | Issue 1 | Review Article

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Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

T. Tachim Medjo, R. Temam and M. Ziane

[+] Author and Article Information

T. Tachim Medjo

Department of Mathematics, Florida International University, DM413B University Park, Miami, FL 33199

R. Temam

The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405

M. Ziane

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

In the space periodic case over $Ω = {(0, 2 π)}^{3}$ , $H^{- 1} (Ω)$ , $s > 0$ is the space of function $u = \sum {\hat{u}}_{k} e^{i k \cdot x}$ , and $k = (k_{1}, k_{2}, k_{3})$ , and $x = (x_{1}, x_{2}, x_{3})$ , such that the norm $\sum_{k} {∣ k ∣}^{- 2 s} {∣ {\hat{u}}_{k} ∣}^{2} < 0$ , thus giving less predominance to the high modes. On the contrary the norm of $H^{s} (Ω)$ , $s > 0$ , $\sum_{k} {∣ k ∣}^{2 s} {∣ {\hat{u}}_{k} ∣}^{2}$ gives enhanced weight to the height modes.

Otherwise, the magnetohydrodynamics (MHD) or radiation equations must be included and the control problem must be fully redefined (objectives, cost functional, etc.).

Robust control is addressed in Sec. 4.

Appl. Mech. Rev 61(1), 010802 (Feb 20, 2008) (23 pages) doi:10.1115/1.2830523 History: Published February 20, 2008

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In this article, we review from the mathematical and numerical points of view some of the recent progresses in the area of flow control.

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References

Alekseev, V. M., Tikhomirov, V. M., and Fomin, S. V., 1987, "Optimal Control", Consultants Bureau, New York.

Gad-El-Hak, M., 1989, “Flow control,” Appl. Mech. Rev., 42 , pp. 262–292.

Gunzburger, M., 1995, “A Prehistory of Flow Control and Optimization,” "Flow Control", Springer, New York, pp. 185–195.

Gad-El-Hak, M., 2000, "Flow Control Passive, Active, and Reactive Flow Management", Cambridge University Press, Cambridge.

Gad-El-Hak, M., 2007, "Large-Scale Disasters: Prediction, Control and Mitigation", Cambridge University Press, Cambridge.

Gad-El-Hak, M., 2006, "MEMS: Introduction and Fundamentals", Taylor & Francis, Boca Roton, FL.

Gad-El-Hak, M., 2006, "MEMS: Applications", Taylor & Francis, Boca Roton, FL.

Gad-El-Hak, M., 2006, "MEMS: Design and Fabrication", Taylor & Francis, Boca Roton, FL.

Sritharan, S., 1998, "Optimal Control of Viscous Flow", SIAM, Philadelphia.

Bewley, T., 2001, “Flow Control: New Challenges for a New Renaissance,” Prog. Aerosp. Sci. [CrossRef], 37 , pp. 21–58.

1995, "Flow Control", Gunzburger, M., ed., Springer, New York.

Gunzburger, M., 1995, “Flow Control and Optimization,” "Computational Fluid Dynamics Review", Wiley, West Sussex, pp. 548–566.

Gunzburger, M., 1998, “Sensitivities in Computational Methods for Optimal Flow Control,” "Computational Methods for Optimal Design and Control", Birkhauser, Boston, pp. 197–236.

Gunzburger, M., 1999, “Sensitivities, Adjoints, and Flow Optimization,” Int. J. Numer. Methods Fluids [CrossRef], 31 , pp. 53–78.

Gunzburger, M., 2000, “Adjoint Equation-Based Methods for Control Problems in Viscous Incompressible Flows,” Flow, Turbul. Combust. [CrossRef], 65 , pp. 249–272.

Menaldi, J. L., and Sritharan, S., 2003, “Impulse Control of Stochastic Navier-Stokes Equations,” Nonlinear Anal. Theory, Methods Appl., 52 (2), pp. 357–381.

Menaldi, J. L., and Sritharan, S., 2002, “Optimal Stopping Time and Impulse Control Problems for the Stochastic Navier-Stokes Equations,” "Stochastic Partial Problems for the Stochastic Navier-Stokes Equations", Dekker, New York, pp. 389–404.

Gozzi, F., Sritharan, S., and Swiech, A., 2005, “Bellman Equations Associated to the Optimal Feedback Control of Stochastic Navier-Stokes Equations,” Commun. Pure Appl. Math. [CrossRef], 58 (5), pp. 671–700.

Gozzi, F., Sritharan, S., and Swiech, A., 2002, “Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations,” Arch. Ration. Mech. Anal. [CrossRef], 163 (4), pp. 295–327.

Temam, R., 1988, "Infinite Dimensional Systems in Mechanics and Physics", Applied Mathematical Sciences , Vol. 68 , 2nd. ed., Springer-Verlag, New York.

Temam, R., 2001, "Navier-Stokes Equations", AMS-Chelsea Series, AMS, Providence.

Lions, J. L., Temam, R., and Wang, S., 1992, “On the Equations of Large-Scale Ocean,” Nonlinearity [CrossRef], 5 , pp. 1007–1053.

Pedlosky, J., 1987, "Geophysical Fluid Dynamics", 2nd. ed., Springer-Verlag, New York.

Peixoto, J. P., and Oort, A. H., 1992, "Physics of Climate", American Institute of Physics, New York.

Lions, J. L, Temam, R., and Wang, S., 1993, “Models of the Coupled Atmosphere and Ocean., (CAO I),” Acta Tech. Acad. Sci. Hung., 1 , pp. 3–54.

Lions, J. L., Temam, R., and Wang, S., 1993, “Numerical Analysis of the Coupled Atmosphere and Ocean Models, (CAO. II),” Acta Tech. Acad. Sci. Hung., 1 , pp. 55–120.

Lions, J. L., Temam, R., and Wang, S., 1995, “Mathematical Study of the Coupled Models of Atmosphere and Ocean (CAO III),” J. Math. Pures Appl., 73 , pp. 105–163.

Temam, R., and Ziane, R., 2004, “Some Mathematical Problems in Geophysical Fluid Dynamics,” "Handbook of Mathematical Fluid Dynamics", North-Holland, Amsterdam, Vol. 3 , pp. 535–657.

Talagrand, O., 1981, “On the Mathematics of Data Assimilation,” Tellus, 33 (4), pp. 321–339.

Talagrand, O., and Courtier, P., 1987, “Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equations I: Theory,” Q. J. R. Meteorol. Soc. [CrossRef], 113 , pp. 1311–1328.

Moin, P., and Bewley, T., 1994, “Feedback Control of Turbulence,” Appl. Mech. Rev., 47 , pp. S3–S13.

Bewley, T., Moin, P., and Temam, R., 2001, “DNS-Based Predictive Control of Turbulence: An Optimal Benchmark of Feedback Algorithms,” J. Fluid Mech., 477 , pp. 179–225.

Fursikov, A., 1982, “Control Problems and Theorems Concerning the Unique Solvability of a Mixed Boundary Value Problem for the Three-Dimensional Navier-Stokes and Euler Systems,” Math. USSR. Sb., 43 , pp. 251–273.

Gunzburger, M., Hou, L., and Svobodny, T., 1990, “A Numerical Method for Drag Minimization Via Suction and Injection of Mass Through the Boundary,” "Stabilization of Flexible Structures", Springer, Berlin, pp. 312–321.

Sritharan, S., 1992, “An Optimal Control Problem in Exterior Hydrodynamics,” Proc. R. Soc. London, Ser. A, 12 (1–2), pp. 5–33.

Aamo, O., Krstic, M., and Bewley, T., 2003, “Control of Mixing by Boundary Feedback in 2D Flow,” Automatica [CrossRef], 39 , pp. 1597–1606.

Lions, J. L., 1971, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer-Verlag, New York.

LeDimet, F. X., and Shutyaev, V., 2001, “On Data Assimilation for Quasilinear Parabolic Problems,” Russian J. Numer. Anal. Math. Modeling, 16 (3), pp. 247–259.

Lee, K., Cortelezzi, L., Kim, J., and Speyer, J., 2001, “Application of Robust Reduced-Order Controller of Wall-Bounded Turbulent Flow,” Phys. Fluids [CrossRef], 13 , pp. 1321–1330.

LeDimet, F. X., Ngnepieba, P., and Shutyaev, V., 2002, “On Error Analysis in Data Assimilation Problems,” Russian J. Numer. Anal. Math. Modeling, 17 (1), pp. 71–97.

Glowinski, R., and He, J., 1998, “On Shape Optimization and Relates Issues,” "Computational Methods for Optimal Design and Control", Birkhauser, Basel, pp. 151–180.

Abergel, F., and Temam, R., 1990, “On Some Control Problems in Fluid Mechanics,” Theor. Comput. Fluid Dyn. [CrossRef], 1 , pp. 303–325.

Bewley, T., and Moin, P., 1994, “Optimal Control of Turbulent Channel Flows,” "Active Control of Vibration and Noise", K.W.Wang, A.H.Von Flotow, R.Shoureshi, E.W.Hendricks, and T.W.Farabee, eds., ASME, New York, DE-Vol. 75 , pp. 221–227.

Bewley, T., Moin, P., and Temam, R., 1997, “Optimal and Robust Control Approaches for Linear and Nonlinear Regulation Problems in Fluid Mechanics,” AIAA Paper No. 97-1872.

Chang, Y., and Collis, S., 1999, “Active Control of Turbulent Channel Flows Based on Large Eddy Simulation,” ASME Paper No. FEDSM-99-6929, pp. 1–8.

Choi, H., Temam, R., and Moin, P., 1993, “Feedback Control for Unsteady Flow and Its Application to the Stochastic Burgers Equation,” J. Fluid Mech. [CrossRef], 253 , pp. 509–543.

Bewley, T., and Liu, S., 1998, “Optimal and Robust Control and Estimation of Linear Paths to Transition,” J. Fluid Mech. [CrossRef], 365 , pp. 305–349.

Bewley, T., Temam, R., and Moin, P., 1997, “Control of Turbulent Flows,” "in Proceedings of the 18th IFIP TC7 Conference on System Modeling and Optimization", Detroit.

Bilić, N., 1988, “Necessary Conditions for an Optimal Distributed Control Problem Governed by Generalized Nonstationary Navier-Stokes Equations,” Glas. Mat. Serija III, 23 (1), pp. 119–133.

Fursikov, A., 1998, “Optimal Control Problems for the Navier-Stokes System With Distributed Control Function,” "Optimal Control of Viscous Flow", S.Sritharan, ed., SIAM, Philadelphia, pp. 109–150.

Gunzburger, M., Hou, L., and Svobodny, T., 1991, “Analysis and Finite Element Approximations of Optimal Control Problems for the Stationary Navier-Stokes Equations With Distributed and Neumann Controls,” Math. Comput. [CrossRef], 57 , pp. 123–151.

Hou, L., and Yan, Y., 1997, “Dynamics for Controlled Navier-Stokes Systems With Distributed Controls,” SIAM J. Control Optim. [CrossRef], 35 , pp. 654–677.

Okumara, H., and Kawahara, M., 2000, “Shape Optimization of Body Located in Incompressible Navier-Stokes Flow Based on Optimal Control Theory,” Comput. Model. Eng. Sci., 1 (2), pp. 71–77.

Gunzburger, M., and Manservisi, S., 1999, “The Velocity Tracking Problem for the Navier-Stokes Flow With Bounded Distributed Controls,” SIAM J. Control Optim. [CrossRef], 37 , pp. 1913–1945.

Gunzburger, M., and Manservisi, S., 2000, “Analysis and Approximation of the Velocity Tracking Problem for the Navier-Stokes Flows With Distributed Controls,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. [CrossRef], 37 , pp. 1481–1512.

Gunzburger, M., and Manservisi, S., 2000, “The Velocity Tracking Problem for the Navier-Stokes Flow With Boundary Control,” SIAM J. Control Optim. [CrossRef], 39 , pp. 594–634.

Gunzburger, M., Hou, L., and Svobodny, T., 1993, “Heating and Cooling Control of Temperature Distributions Along Boundaries of Flow Domain,” J. Math. Systems Estim. Control, 3 , pp. 147–172.

Gunzburger, M., Hou, L., and Svobodny, T., 1999, “Control of Temperature Distributions Along Boundary of Engine Components,” "Numerical Methods in Laminar and Turbulent Flow VII", Pineridge, Swansea, UK, pp. 765–773.

Ito, K., 1994, “Boundary Temperature Control for Thermally Coupled Navier-Stokes Equations,” "Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena (Vorau, 1993)", Birkhauser, Basel, Vol. 118 , pp. 211–230.

Ito, K., and Ravindran, S. S., 1998, “Optimal Control of Thermally Convected Flows,” SIAM J. Sci. Comput. (USA) [CrossRef], 19 , pp. 1847–1869.

Gunzburger, M., Hou, L., and Svobodny, T., 1992, “Boundary Velocity Control of Incompressible Flow With an Application to Viscous Drag Reduction,” SIAM J. Control Optim. [CrossRef], 30 , pp. 167–181.

Gunzburger, M., and Kim, H., 1998, “Existence of an Optimal Solution of a Shape Control Problem for the Stationary Navier-Stokes,” SIAM J. Control Optim. [CrossRef], 36 , pp. 895–909.

Arian, E., and Ta’asan, S., 1995, “Shape Optimization in One Shot,” "Optimal Design and Control", Birkhauser, Boston, pp. 23–40.

Armugan, G., and Pironneau, O., 1989, “On the Problem of Riblets as a Drag Reduction Device,” Opt. Control Appl. Methods [CrossRef], 10 , pp. 93–112.

Borggaard, J., and Burns, J., 1995, “A Sensitivity Equation Approach to Shape Optimization in Fluid Flows,” "Flow Control", Springer, New York, pp. 49–78.

Burkardt, J., Gunzburger, M., and Peterson, J., 1995, “Discretization of Cost Functionals and Sensitivities in Shape Optimization,” "Computation and Control IV", Birkhauser, Basel, pp. 43–56.

Dziri, R., and Zolesio, J. P., 1999, “Dynamical Shape Control in Non-Cylindrical Navier-Stokes Equations,” J. Convex Anal., 6 , pp. 293–318.

Glowinski, R., Kearsley, A., Pan, T., and Periaux, J., 1995, “Numerical Simulation and Optimal Shape for Viscous Flow by a Fictitious Domain Methods,” Int. J. Numer. Methods Fluids [CrossRef], 20 , pp. 695–711.

Glowinski, R., and Pironneau, O., 1976, “Toward the Computation of Minimum Drag Profile in Viscous Laminar Flow,” Appl. Math. Model. [CrossRef], 1 , pp. 58–66.

Gunzburger, M., Kim, H., and Manservisi, S., 2000, “On a Shape Control Problem for the Stationary Navier-Stokes Equations,” Math. Modell. Numer. Anal. [CrossRef], 34 , pp. 1233–1258.

Gunzburger, M., and Peterson, J., 1992, “Issues in Shape Optimization for the Navier-Stokes Equations,” "Proceedings of 31st IEEE Conference Decision and Control", pp. 3390–3392.

Kim, H. C., 2001, “Penalized Approach and Analysis of an Optimal Shape Control Problem for the Stationary Navier-Stokes Equations,” J. Korean Math. Soc., 38 , pp. 1–23.

Mohammadi, B., 1998, “Mesh Adaptation and Automatic Differentiation for Optimal Shape Design,” Int. J. Comput. Fluid Dyn. [CrossRef], 10 , pp. 199–211.

Mohammadi, B., 1998, “Shape Optimization for the 3D Turbulent Flows Using Automatic Differentiation,” Int. J. Comput. Fluid Dyn. [CrossRef], 11 , pp. 27–50.

Pironneau, O., 1974, “On Optimal Design in Fluid Mechanics,” J. Fluid Mech. [CrossRef], 64 , pp. 97–110.

Svobodny, T., 1995, “Shape Optimization and Control of Separating Flow in Hydrodynamics,” "Flow Control", Springer, New York, pp. 325–339.

Fursikov, A. V., Gunzburger, M. D., and Hou, L. S., 1998, “Boundary Value Problems and Optimal Boundary Control for the Navier-Stokes System: The Two-Dimensional Case,” SIAM J. Control Optim. [CrossRef], 36 , pp. 852–894.

Ekeland, I., and Temam, R., 1999, "Convex Analysis and Variational Problems", Series Classics in Applied Mathematics , Society for Industrial and Applied Mathematics, Philadelphia, PA.

Abergel, F., and Casas, E., 1993, “Some Optimal Control Problems Associated to the Stationary Navier-Stokes Equations,” "Mathematics, Climate and Environment", Masson, Paris, France, pp. 213–220.

Fursikov, A., Gunzburger, M., and Hou, L., 2005, “Optimal Boundary Control for the Evolutionary Navier-Stokes System: The Three-Dimensional Case,” SIAM J. Control Optim. [CrossRef], 43 (6), pp. 2191–2232.

Marchuk, G., Agoshkov, V. I., and Shutyaev, V. P., 1996, "Adjoint Equations and Perturbation Algorithms in Nonlinear Problems", CRC, Boca Roton, FL.

Fursikov, A., Gunzburger, M., and Hou, L. S., 2000, “Optimal Control Problems for the Navier-Stokes Equations,” "Lectures on Applied Mathematics (Munich, 1999)", Springer, Berlin, pp. 143–155.

He, J., Chevalier, M. P., Glowinski, R., Metacalfe, R., Nordlander, A., and Periaux, J., 2000, “Drag Reduction by Active Control for Flow Past Cylinders,” "Computational Mathematics Driven by Industrial Problems (Martina Franca, 1999)", Lecture Notes in Mathematics Vol. 1739 , pp. 287–363.

He, J., Glowinski, R., Metacalfe, R., Nordlander, A., and Periaux, J., 2000, “Active Control and Drag Optimization for Flow Past a Circular Cylinder,” J. Comput. Phys. [CrossRef], 163 (1), pp. 83–117.

He, J., Glowinski, R., Metacalfe, R., and Periaux, J., 1998, “A Numerical Approach to the Control and Stabilization of Advection-Diffusion Systems: Application to Viscous Drag Reduction,” Int. J. Comput. Fluid Dyn. [CrossRef], 11 (1–2), pp. 131–156.

Fursikov, A., Gunzburger, M., and Hou, L., 1998, “Optimal Dirichlet Control and Inhomogeneous Boundary Value Problems for the Unsteady Navier-Stokes,” "Control of Partial Differential Differential Equations", ESAIM, Paris, pp. 97–116.

Fursikov, A., Gunzburger, M., and Hou, L., 1999, “Optimal Boundary Control of the Navier-Stokes With Bounds on the Control,” "Proceedings of the Korean Advanced Institute for Science and Technology Mathematics Workshop on Finite Elements", pp. 41–60.

Gunzburger, M., Hou, L., and Svobodny, T., 1990, “Optimal Boundary Control for the Nonsteady Incompressible Flow: An Application to Viscous Drag Reduction,” "Proceedings of the 29th IEEE Conference Decision and Control, IEEE", pp. 377–378.

Gunzburger, M., Hou, L., and Svobodny, T., 1994, “Boundary Control of Incompressible Flows,” "Computational Fluid Dynamics Techniques", Harwood, Harrow, UK, pp. 839–846.

Hou, L., and Meir, A., 1995, “Boundary Optimal Control for MHD Flows,” Appl. Math. Optim. [CrossRef], 32 , pp. 143–162.

Hou, L., and Ravindran, S., 1996, “Computations of Boundary Optimal Control Problems for Electrically Conducting Flows,” J. Comput. Phys. [CrossRef], 128 , pp. 319–330.

Hou, L., and Ravindran, S., 1998, “Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems,” Int. J. Comput. Fluid Dyn. [CrossRef], 11 , pp. 157–167.

Hou, L., and Ravindran, S., 1999, “Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results,” SIAM J. Sci. Comput. (USA) [CrossRef], 20 , pp. 1753–1777.

Hou, L., and Svobodny, T., 1993, “Optimization Problems for the Navier-Stokes Equations With Regular Boundary Controls,” J. Math. Anal. Appl. [CrossRef], 177 , pp. 342–367.

Lee, H. C., and Imanuvilov, O., 2000, “Analysis of Neuman Boundary Optimal Control Problems for the Stationary Boussinesq Equations Including Solid Media,” SIAM J. Control Optim. [CrossRef], 39 , pp. 457–477.

Sahrapour, A., and Ahmed, N., 1994, “Boundary Control of Navier-Stokes Equations With Potential Applications to Artificial Hearts,” "Proceedings of the CFD Society of Canada", pp. 387–393.

Vedantham, R., 2000, “Optimal Control of the Viscous Burgers Equation Using an Equivalent Index Method,” J. Global Optim. [CrossRef], 18 (3), pp. 255–263.

Volkwein, S., 2001, “Distributed Control Problems for the Burgers Equation,” Comput. Optim. Appl. [CrossRef], 18 (2), pp. 115–140.

Volkwein, S., 2002, “Boundary Control for the Burgers Equation: Optimality Condition and Reduced-Order Approach,” "Optimal Control of Complex Structures (Oberwolfach, 2000)", International Series of Numerical Mathematics Vol. 139 , Birkhauser, Basel, pp. 267–278.

100

Abergel, F., and Casas, E., 1993, “Some Optimal Control Problems of Multiscale Equations Appearing in Fluid Mechanics,” Model. Math. Anal. Numer., 27 , pp. 223–247.

101

Casas, E., 1998, “An Optimal Control Problem Governed by the Evolution Navier-Stokes Equations,” "Optimal Control of Viscous Flow", S.Sritharan, ed., 79–107, SIAM, Philadelphia.

102

Desai, M., and Ito, K., 1994, “Optimal Control of Navier-Stokes Equations,” SIAM J. Control Optim. [CrossRef], 32 , pp. 1428–1446.

103

Fattorini, H., and Sritharan, S., 1992, “Existence of Optimal Controls for Viscous Flow Problems,” Proc. R. Soc. London, Ser. A, 39 , pp. 81–102.

104

Fattorini, H., and Sritharan, S., 1994, “Necessary and Sufficient Conditions for Optimal Control in Viscous Flow Problems,” Proc. - R. Soc. Edinburgh, Sect. A: Math., 124A , pp. 211–251.

105

Fursikov, A., 1980, “On Some Optimal Control Problems and Results Concerning the Unique Solvability of a Mixed Boundary Value Problem for the Three-Dimensional Navier-Stokes and Euler Systems,” Sov. Math. Dokl., 21 , pp. 889–893.

106

Fursikov, A., 1982, “Properties of Solutions of Controls that are Connected With the Navier-Stokes Equations,” Sov. Math. Dokl., 25 , pp. 40–45.

107

Fursikov, A., 1983, “Properties of Solutions of Some Extremal Problems Connected With the Navier-Stokes Equations,” Math. USSR. Sb., 46 , pp. 323–351.

108

Ghattas, O., and Bark, J., 1997, “Optimal Control of the Two- and the Three-Dimensional Incompressible Navier-Stokes Flow,” J. Comput. Phys. [CrossRef], 136 , pp. 231–244.

109

Hou, L. S., and Ravindran, S. S., 1998, “A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier-Stokes Equations,” SIAM J. Control Optim. [CrossRef], 36 , pp. 1795–1814.

110

Ito, K., Scroggs, J., and Tran, H., 1995, “Optimal Control of Thermally Coupled Navier-Stokes Equations,” "Optimal Design and Control", Birkhauser, Boston, pp. 199–214.

111

Li, Z., Navon, M., Hussaini, M., and Dimet, F., 2002, “Optimal Control of the Unsteady Navier-Stokes Equations,” Comput. Fluids [CrossRef], 32 , pp. 149–171.

112

Imanuvilov, O. Y., 1992, “On Some Optimal Control Problems Connected With the Navier-Stokes System,” J. Sov. Math. [CrossRef], 60 (6), pp. 1725–1741.

113

Lee, H. C., and Imanuvilov, O., 2000, “Analysis of Optimal Control Problems for the 2-D Stationary Boussinesq Equations,” J. Math. Anal. Appl. [CrossRef], 242 , pp. 191–211.

114

Lee, H. C., 1997, “Existence of Boundary Optimal Control for the Boussinesq Equations,” "Proceedings of the Korea-Japan Partial Differential Equations Conference (Taejon, 1996)", Lecture Notes Series , 39, Seoul National University, Seoul.

115

Lee, H., 2002, “Dynamics for the Controlled 2-D Boussinesq Systems With Distributed Controls,” J. Math. Anal. Appl. [CrossRef], 273 (2), pp. 457–479.

116

LeDimet, F. X., and Shutyaev, V., 2000, “On Newton Method in Data Assimilation,” Russian J. Numer. Anal. Math. Modeling, 15 (5), pp. 419–434.

117

Tachim Medjo, T., 2003, “On the Newton Method in Robust Control of Fluid Flow,” Discrete Contin. Dyn. Syst., 9 (5), pp. 1201–1222.

118

Parmuzin, E., and Shutyaev, V., 1999, “Numerical Analysis of Iterative Methods for Solving Evolution Data Assimilation Problems,” Russ. J. Numer. Anal. Math. Modelling, 14 (3), pp. 275–289.

119

Shutyaev, V., 1991, “An Algorithm for Computing Functionals for a Class of Nonlinear Problems Using the Conjugate Equation,” Russ. J. Numer. Anal. Math. Modelling, 6 (2), pp. 169–178.

120

Shutyaev, V., 1995, “Some Properties of the Control Operator in the Problem of Data Assimilation and Iterative Algorithms,” Russ. J. Numer. Anal. Math. Modelling, 10 (4), pp. 357–371.

121

Shutyaev, V., 1999, “Control Operators and Iterative Algorithms in Problems of Reconstructing Source Functions and Initial Data,” Russ. J. Numer. Anal. Math. Modelling, 14 (2), pp. 137–176.

122

Marchuck, G., and Shutyaev, V., 1994, “Iterative Methods for Solving a Data Assimilation Problem,” Russ. J. Numer. Anal. Math. Modelling, 9 (3), pp. 265–279.

123

Marchuk, G., 1975, "Methods of Numerical Mathematics", Springer-Verlag, New York.

124

Tachim Medjo, T., 2002, “Iterative Methods for Robust Control Problems in Fluid Mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. [CrossRef], 39 (5), pp. 1625–1645.

125

Lebedev, V., and Zabelin, V., 1990, “On One Class of Three-Term Iterative Methods With Tchebyshev,” Russ. J. Numer. Anal. Math. Modelling, 5 (4–5), pp. 275–298.

126

Bewley, T., Temam, R., and Ziane, M., 2000, “A General Framework for Robust Control in Fluid Mechanics,” Physica D [CrossRef], 138 , pp. 360–392.

127

Afanasiev, K., and Hinze, M., 1999, “Adaptive Control of a Wake Flow Using Proper Orthogonal Decomposition,” Tecnische Universitat Berlin, Report No. 648.

128

Cortelezzi, L., and Speyer, J., 1998, “Robust Reduced-Order Controller of Transitional Layer Transition,” Phys. Rev. E [CrossRef], 58 , pp. 1906–1910.

129

Ito, K., and Ravindran, S., 1998, “A Reduced-Order Method for Simulation and Control of Fluid Flows,” J. Comput. Phys. [CrossRef], 143 , pp. 1403–1425.

130

Ito, K., and Ravindran, S., 2001, “Reduced Basis Method for Optimal Control of Unsteady Viscous Flows,” Int. J. Comput. Fluid Dyn. [CrossRef], 15 , pp. 97–113.

131

Berggren, M., and Heikenschloss, M., 1997, “Parallel Solution of Optimal-Control Problems by Time-Domain Decomposition,” "Computational Science for the 21st Century", Wiley, Chichester, pp. 102–112.

132

Choi, H., 1995, “Suboptimal Control of Turbulent Flow Using Control Theory,” "Proceedings of the International Symposium on Mathematical Modelling of Turbulent Flows", Tokyo, Japan.

133

Hinze, M., and Kunisch, K., 1997, “Suboptimal Control Strategies for Backward Facing Step Flows,” "Proceedings of the 15th IMACS World Congress on Scientific Computation, Modeling and Applied Mathematics", A.Sydow, ed., Berlin, Vol. 3 , pp. 53–58.

134

Hinze, M., and Kunisch, K., 1998, “On Suboptimal Control Strategies for the Navier-Stokes Equations,” "Control and Partial Differential Equations", ESAIM, Paris, 181–198.

135

Hinze, M., and Kunisch, K., 2000, “Three Control Methods for Time-Dependent Fluid Flow,” Flow, Turbul. Combust. [CrossRef], 65 , pp. 273–298.

136

Hou, L. S., and Yan, Y., 1997, “Dynamics and Approximations of a Velocity Tracking Problem for the Navier-Stokes Equations With Piecewise Distributed Controls,” SIAM J. Control Optim. [CrossRef], 35 , pp. 1847–1885.

137

Lee, C., Kim, J., and Choi, H., 1998, “Suboptimal Control of Turbulent Channel Flow for Drag Reduction,” J. Fluid Mech. [CrossRef], 358 , pp. 245–258.

138

Min, C., and Choi, H., 1999, “Suboptimal Feedback Control of Vortex Shedding at Low Reynolds Numbers,” J. Fluid Mech. [CrossRef], 401 , pp. 123–156.

139

Hinze, M., and Volkwein, S., 2001, “Analysis of Instantaneous Control for the Burgers Equation,” Nonlinear Anal. Theory, Methods Appl. [CrossRef], 50 (1), pp. 1–26.

140

Hinze, M., and Kunisch, K., 1998, “On Suboptimal Control Strategies for the Navier-Stokes Equations,” "Control and Partial Differential Equations (Marseille-Luminy, 1997)", ESAIM, Paris, Vol. 4 , pp. 181–198.

141

Gunzburger, M., and Manservisi, S., 2000, “Analysis and Approximation for Linear Feedback Control for Tracking the Velocity in Navier-Stokes Flows,” Comput. Methods Appl. Mech. Eng. [CrossRef], 189 , pp. 803–823.

142

Ito, K., and Kang, S., 1994, “A Dissipative Feedback Control Synthesis for Systems Arising in Fluid Dynamics,” SIAM J. Control Optim. [CrossRef], 32 , pp. 831–854.

143

Ravindran, S., 2000, “Proper Orthogonal Decomposition in Optimal Control of Fluids,” Int. J. Numer. Methods Fluids [CrossRef], 34 , pp. 425–448.

144

Ravindran, S., 2002, “Adaptive Reduced-Order Controllers for a Thermal Flow System Using Proper Orthogonal Decomposition,” SIAM J. Sci. Comput. (USA) [CrossRef], 23 , pp. 1924–1942.

145

Cortelezzi, L., Kim, J., and Speyer, J., 1998, “Skin-Friction Drag Reduction Via Robust Reduction-Order Linear Feedback Control,” Int. J. Comput. Fluid Dyn. [CrossRef], 11 , pp. 79–92.

146

Ito, K., and Schroeter, J., 2001, “Reduced Order Feedback Synthesis for Viscous Incompressible Flows,” Math. Comput. Modell. [CrossRef], 33 , pp. 173–192.

147

Ravindran, S., 2000, “A Reduced Order Approach for Optimal Control of Fluid Flow Using Proper Orthogonal Decomposition,” Int. J. Numer. Methods Fluids [CrossRef], 34 , pp. 425–448.

148

Hinze, M., and Kunisch, K., 2000, “Three Control Methods for Time-Dependent Fluid Flow,” Flow, Turbul. Combust. [CrossRef], 65 (3–4), pp. 273–298.

149

Hinze, M., and Kunisch, K., 2001, “Second Methods for Optimal Control of Time-Dependent Fluid Flow,” SIAM J. Control Optim. [CrossRef], 40 (3), pp. 925–946.

150

Bischof, C., Carle, A., Corliss, G., Griewnak, A., and Hoveland, P., 1992, “ADIFOR: Generating Derivative Codes From Fortran Programs,” Sci. Prog., 1 , pp. 11–29.

151

Green, M., and Limebeer, D. J. N., 1995, "Linear Robust Control", Prentice-Hall, Englewood Cliffs, NJ.

152

Lagnese, J. E., Russell, D. L., and White, L. W., 1995, "Control and Optimal Design of Distributed Parameter Systems", The Institute for Mathematics and Its Application Vol. 70 , Springer, Berlin.

153

Tachim Medjo, T., 2002, “Fixed-Point Iteration Method for Nonlinear Robust Control Problems in Fluid Mechanics,” Numer. Funct. Anal. Optim. [CrossRef], 23 (7–8), pp. 849–876.

154

Tachim Medjo, T., and Tébou, L. R. T., 2003, “Adjoint-Based Iterative Method for Robust Control Problems in Fluid Mechanics,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. [CrossRef], 42 (1), pp. 302–325.

155

Hu, C., and Temam, R., 2001, “Robust Control of the Kuramoto-Sivashinsky Equation,” Dyn. Contin. Discrete Impulsive Syst. Ser. B Appl. Algorithms, 8 (3), pp. 315–338.

156

Barbu, V., and Sritharan, S., 1998, “H∞—Control Theory of Fluid Dynamics,” Proc. R. Soc. London, Ser. A [CrossRef], 454 (1979), pp. 3009–3033.

157

Tachim Medjo, T., 2001, “Robust Control Problems in Fluid Mechanics,” Physica D [CrossRef], 149 (4), pp. 278–292.

158

Sritharan, S., 1995, “Optimal Feedback Control of Hydrodynamics: A Progress Report,” "Flow Control", Springer, New York, pp. 257–274.

159

Ou, Y. H., 1998, “Design of Feedback Compensators for Viscous Flow,” "Optimal Control of Viscous Flow", S.Sritharan, ed., SIAM, Philadelphia, pp. 151–180.

160

Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A., 1989, “State-Space Solution to H2 and H∞ Control Problems,” IEEE Trans. Autom. Control [CrossRef], 34 (8), pp. 831–847.

161

Başar, T., and Bernhard, P., 1991, "H∞—Optimal Control and Related Minmax Design Problems", Birkhäuser, Boston.

162

Keulen, B. V., 1993, "H∞—Control for Distributed Parameter Systems: A State Space Approach", Birkhäuser, Boston.

163

Armaou, A., and Christofides, P. D., 2000, “Feedback Control of the Kuramoto-Sivashinsky Equation,” Physica D [CrossRef], 137 (1–2), pp. 49–61.

164

Fursikov, A. V., 2002, “Feedback Stabilization for the 2D Navier-Stokes Equations,” "The Navier-Stokes Equations: Theory and Numerical Methods (Varenna, 2000)", Lecture Notes in Pure and Applied Math , Dekker, New York, Vol. 223 , pp. 179–196.

165

Barbu, V., 1998, “Feedback Control of Time Dependent Stokes Flows,” "Optimal Control of Viscous Flow", S.Sritharan, ed., SIAM, Philadelphia, pp. 63–77.

166

Barbu, V., and Sritharan, S., 2000, “Flow Invariance Preserving Feedback Controllers for the Navier-Stokes Equations,” J. Math. Anal. Appl., 255 , pp. 281–307.

167

Bewley, T., Moin, P., and Temam, R., 1999, “A Method for Optimizing Feedback Control Laws for Wall-Bounded Flows Based on Control Theory,” "Proceedings of Engineering Division Conference, Part 2", ASME, New York, FED-237.

168

Burns, J., King, B., and Rubio, D., 1998, “Feedback Control of a Thermal Fluid Using State Estimation,” Int. J. Comput. Fluid Dyn. [CrossRef], 11 , pp. 93–112.

169

Cortelezzi, L., 1996, “Nonlinear Feedback Control of the Wake Past a Plate With Suction Point on the Downstream Wall,” J. Fluid Mech. [CrossRef], 327 , pp. 303–324.

170

Cortelezzi, L., Chen, Y., and Chang, H., 1997, “Nonlinear Feedback Control of the Wake Past a Plate: From a Low Order Model to a Higher Order Model,” Phys. Fluids [CrossRef], 9 , pp. 2009–2022.

171

Gunzburger, M., and Lee, H. C., 1994, “Feedback Control of Fluid Flows,” "14th IMACS World Congress on Computational and Applied Mathematics", Georgia Tech., Atlanta, pp. 716–719.

172

Gunzburger, M., and Lee, H. C., 1996, “Feedback Control of Karman Vortex Shedding,” ASME J. Appl. Mech. [CrossRef], 63 , pp. 828–835.

173

Hu, H., and Bau, H., 1994, “Feedback Control to Delay or Advance Linear Loss of Stability in Planar Poiseuille Flow,” Proc. R. Soc. London, Ser. A, 447 , pp. 229–312.

174

Ito, K., and Yan, Y., 1998, “Viscous Scalar Conservation Law With Nonlinear Flux Feedback and Global Attractors,” J. Math. Anal. Appl. [CrossRef], 227 , pp. 271–299.

175

Joshi, S., Speyer, J., and Kim, J., 1997, “A System Theory Approach to the Feedback Stabilization of Infinitesimal and Finite-Amplitude Disturbances in Plane Poiseuille Flow,” J. Fluid Mech., 332 , pp. 157–184.

176

Lee, H. C., and Shin, B., 2001, “Dynamics for Linear Feedback Controller for the Two Dimensional Bernard Equations With Distributed Control,” Appl. Math. Optim. [CrossRef], 44 , pp. 163–175.

177

Park, D., 1994, “Theoretical Analysis of Feedback Control of Karman Vortex Shedding at Slightly Supercritical Reynolds Numbers,” Eur. J. Mech. B/Fluids, 13 , pp. 387–399.

178

Park, D., Ladd, D., and Hendricks, E., 1994, “Feedback Control of Karman Vortex Shedding Behind a Circular Cylinder at Low Reynolds Numbers,” Phys. Fluids [CrossRef], 6 , pp. 2390–2405.

179

Roussopoulos, K., 1993, “Feedback Control of Vortex Shedding at Low Reynolds Numbers,” J. Fluid Mech. [CrossRef], 248 , pp. 267–296.

180

Leith, D. J., and Leithhead, W. E., 2000, “Survey of Gain-Scheduling Analysis and Design,” Int. J. Control [CrossRef], 73 (11), pp. 1001–1025.

181

Fernández-Cara, E., and Zuazua, E., 2004, “On the History and Perspectives of Control Theory,” Sb. Russ. Acad. Sci., 74 , pp. 47–73.

182

Coron, J. M., 1996, “On the Controllability of the 2D Incompressible Navier-Stokes Equations With Navier-Slip Boundary Conditions,” COCV, 1 , pp. 35–75.

183

Coron, J. M., and Fursikov, A. V., 1996, “Global Exact Controllability of the 2D Navier-Stokes Equations on a Manifold Without Boundary,” Russ. J. Math. Phys., 4 (4), pp. 1–19.

184

Fernández-Cara, E., Guerrero, S., Imanuvilov, O. Y., and Puel, J. P., 2004, “Local Exact Controllability of the Navier-Stokes System,” J. Math. Pures Appl. [CrossRef], 83 , pp. 1501–1542.

185

Fursikov, A. V., 1995, “Exact Boundary Zero Controllability of Three-Dimensional Navier-stokes Equations,” J. Dyn. Control Syst. [CrossRef], 1 (3), pp. 325–350.

186

Fursikov, A. V., and Imanuvilov, O. Y., 1994, “On Exact Boundary Zero-Controllability of Two-Dimensional Navier-Stokes Equations: Mathematical Problems for Navier-Stokes Equations (Centro, 1993),” Acta Appl. Math. [CrossRef], 37 (1–2), pp. 67–76.

187

Fernández-Cara, E., and Guerrero, S., 2006, “Global Carleman Inequalities for Parabolic Systems and Applications to Controllability,” SIAM J. Control Optim., 45 (4), pp. 1399–1446.

188

Havârneanu, T., Popa, C., and Sritharan, S., 2005, “Exact Controllability for the Three-Dimensional Navier-stokes Equations With the Navier Slip Boundary Conditions,” Indiana Univ. Math. J. [CrossRef], 54 (5), pp. 1303–1350.

189

Imanuvilov, O. Y., 1997, “Local Exact Controllability for the 2-D Navier-Stokes Equations With the Navier Slip Boundary Conditions,” "Turbulence Modeling and Vortex Dynamics (Istanbul, 1996)", Lecture Notes in Physics Vol. 491 , Springer, Berlin, pp. 148–168.

190

Imanuvilov, O. Y., 1993, “On Exact Controllability for the Navier-Stokes Equations,” COCV, 3 , pp. 97–131.

191

Imanuvilov, O. Y., 2001, “Remark on Exact Controllability for the Navier-Stokes Equations,” COCV, 6 , pp. 39–72.

192

Barbu, V., Havârneanu, T., Popa, C., and Sritharan, S., 2005, “Local Exact Controllability for the Magnetohydrodynamic Equations Revisited,” Adv. Differ. Equ., 10 (5), pp. 481–504.

193

Barbu, V., Havârneanu, T., Popa, C., and Sritharan, S., 2003, “Exact Controllability for the Magnetohydrodynamic Equations,” Commun. Pure Appl. Math. [CrossRef], 56 (6), pp. 732–783.

194

Havârneanu, T., Popa, C., and Sritharan, S., 2005, “Exact Controllability for the Three-Dimensional Navier-Stokes Equations With the Navier Slip Boundary Conditions,” Indiana Univ. Math. J. [CrossRef], 54 (5), pp. 1303–1350.

195

Fursikov, A. V., and Imanuvilov, O. Y., 1997, “Local Exact Boundary Controllability of the Navier-Stokes System,” Contemp. Math., 209 , pp. 115–129.

196

Coron, J. M., 1996, “Global Exact Controllability of the 2D Navier-Stokes Equations on a Manifold Without Boundary,” Russ. J. Math. Phys., 4 , pp. 429–448.

197

Coron, J. M., 1996, “On the Controllability of the 2-D Incompressible Navier-Stokes Equations With the Navier Slip Condition,” COCV, 1 , pp. 35–75.

198

Fursikov, A., 1995, “Exact Boundary Zero Controllability of Three-Dimensional Navier-Stokes Equations,” J. Dyn. Control Syst. [CrossRef], 1 , pp. 325–350.

199

Fursikov, A., and Imanuvilov, O., 1994, “On Exact Boundary Zero-Controllability of Two-Dimensional Navier-Stokes Equations,” Acta Appl. Math. [CrossRef], 36 , pp. 1–10.

200

Fursikov, A., and Imanuvilov, O. Y., 1995, “On Controllability of Certain Systems Simulating a Fluid Flow,” "Flow Control", Springer, New York, pp. 149–184.

201

Fursikov, A., and Imanuvilov, O. Y., 1996, “Exact Local Controllability for 2-D Navier-Stokes Equations,” Sbornik Math., 187 , pp. 1355–1390.

202

Fursikov, A., and Imanuvilov, O. Y., 1996, “Local Exact Controllability of the Navier-Stokes Equations,” C. R. Acad. Sci., Ser. I: Math., 323 , pp. 275–280.

203

Fursikov, A., and Imanuvilov, O. Y., 1997, “Local Exact Controllability of the Navier-Stokes Equations,” Contemp. Math., 209 , pp. 115–129.

204

Imanuvilov, O. Y., 1997, “On Exact Controllability for the Navier-Stokes Equations,” COCV, 3 , pp. 97–131.

205

Lions, J. L., 1988, "Contrôlabilité exacte, perturbations et stabilisation de systémes distribués", Masson, Paris, Tome 1.

206

Fursikov, A., and Imanuvilov, O. Y., 1998, “Local Exact Boundary Controllability of the Boussinesq Equations,” SIAM J. Control Optim. [CrossRef], 36 , pp. 391–421.

207

Fursikov, A. V., and Imanuvilov, O. Y., 1999, “Exact Controllability for the Navier-Stokes and Boussinesq Equations,” Russ. Math. Surveys [CrossRef], 54 (3), pp. 565–618.

208

Fursikov, A. V., and Imnuvilov, O. Y., 1996, "Controllability of Evolution Equations", Lecture Notes Series 34 , Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul.

209

Barbu, V., Havârneanu, T., Popa, C., and Sritharan, S., 2003, “Exact Controllability for the Magnetohydrodynamic Equations,” Commun. Pure Appl. Math. [CrossRef], 56 (6), pp. 732–783.

Figures

Performance of optimized controls for formulations based on Eq. 213 as a function of the optimization horizon t1 as computed in direct numerical simulations at Reτ=100: history of turbulent kinetic energy (see Ref. 32)

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Figure 3

Classification of flow control strategies

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Figure 1

Classification of flow control strategies

Different control loops for active flow control. (a) Predetermined, open-looped control; (b) reactive, feedforward, open-loop control; (c) reactive, feedback, closed-loop control.

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Figure 2

Different control loops for active flow control. (a) Predetermined, open-looped control; (b) reactive, feedforward, open-loop control; (c) reactive, feedback, closed-loop control.

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Figure 4

Performance of optimized controls for formulations based on Eq. 213 as a function of the optimization horizon t1 as computed in direct numerical simulations at Reτ=100: history of drag. For small optimization horizons (t1=O(1)), approximately 20% drag reduction is obtained, a result which can be obtained with a variety of other approaches. For sufficiently large optimization horizons (t1⩾20), the flow is returned to the region of stability of the laminar flow and the flow relaminarizes, resulting in a 57.2% drag reduction with no further effort required (see Ref. 32).

Side view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Large eddies are visualized using a sheet of laser fluorescent dye (see Ref. 4).

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Figure 5

Side view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Large eddies are visualized using a sheet of laser fluorescent dye (see Ref. 4).

Top view of a low Reynolds number turbulent boundary layer, Reθ=742. Wind tunnel experiment. Pockets, believed to be the fingerprints of typical eddies, are visualized using dense smoke illuminated with a sheet of laser (see Ref. 4).

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Figure 6

Top view of a low Reynolds number turbulent boundary layer, Reθ=742. Wind tunnel experiment. Pockets, believed to be the fingerprints of typical eddies, are visualized using dense smoke illuminated with a sheet of laser (see Ref. 4).

Top view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Low-speed streak are visualized using a sheet of laser fluorescent dye (see Ref. 4).

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Figure 7

Top view of a low Reynolds number turbulent boundary layer, Reθ=725. Flat plate towed in a water tank. Low-speed streak are visualized using a sheet of laser fluorescent dye (see Ref. 4).

Top view of an artificially generated spot in a laminar boundary layer. The displacement thickness Reynolds number at the spot’s initiation is Reθ=625 (see Ref. 4).

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Figure 8

Top view of an artificially generated spot in a laminar boundary layer. The displacement thickness Reynolds number at the spot’s initiation is Reθ=625 (see Ref. 4).

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Medjo T, Temam RR, Ziane MM. Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects. ASME. Appl. Mech. Rev. 2008;61(1):010802-010802-23. doi:10.1115/1.2830523.

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Optimal and Robust Control of Fluid Flows: Some Theoretical and Computational Aspects

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