July  2020, 40(7): 4197-4229. doi: 10.3934/dcds.2020178

Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations

1. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria

2. 

RICAM Institute, Austrian Academy of Sciences, Altenberger Straße 69, 4040 Linz, Austria

* Corresponding author: tobias.breiten@uni-graz.at

Received  May 2019 Revised  January 2020 Published  April 2020

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order.

Citation: Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178
References:
[1]

C. O. Aguilar and A. J. Krener, Numerical solutions to the Bellman equation of optimal control, J. Optim. Theory Appl., 160 (2014), 527-552.  doi: 10.1007/s10957-013-0403-8.  Google Scholar

[2]

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M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.  Google Scholar

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M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463.  doi: 10.1137/090778146.  Google Scholar

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V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series, Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

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V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc., 181 (2006). doi: 10.1090/memo/0852.  Google Scholar

[7]

V. BarbuS. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.  Google Scholar

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V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

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S. BeelerH. Tran and H. Banks, Feedback control methodologies for nonlinear systems, J. Optim. Theory Appl., 107 (2000), 1-33.  doi: 10.1023/A:1004607114958.  Google Scholar

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A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

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F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[12]

T. BreitenK. Kunisch and L. Pfeiffer, Infinite-horizon bilinear optimal control problems: Sensitivity analysis and polynomial feedback laws, SIAM J. Control Optim., 56 (2018), 3184-3214.  doi: 10.1137/18M1173952.  Google Scholar

[13]

T. Breiten, K. Kunisch and L. Pfeiffer, Feedback stabilization of the two-dimensional Navier-Stokes equations by value function approximation, preprint, arXiv: 1902.00394. Google Scholar

[14]

T. Breiten and L. Pfeiffer, Taylor expansions of the value function associated with a bilinear optimal control problem, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 36 (2019), 1361-1399.  doi: 10.1016/j.anihpc.2019.01.001.  Google Scholar

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R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

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E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.  Google Scholar

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A. V. Fursikov, Stabilizability of two-dimensional Navier–Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.  Google Scholar

[18]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 289-314.  doi: 10.3934/dcds.2004.10.289.  Google Scholar

[19]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM. Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.  Google Scholar

[20]

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, Cambridge, 2000.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I/II, Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[22]

D. L. Lukes, Optimal regulation of nonlinear dynamical systems, SIAM J. Control, 7 (1969), 75-100.  doi: 10.1137/0307007.  Google Scholar

[23]

C. Navasca and A. Krener, Patchy solutions of Hamilton-Jacobi-Bellman Partial differential equations, in Modeling, Estimation and Control, Lect. Notes Control Inf. Sci., 364, Springer, Berlin, 2007,251–270. doi: 10.1007/978-3-540-73570-0_20.  Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[25]

M. Pošta and T. Roubíček, Optimal control of Navier-Stokes equations by Oseen approximation, Comput. Math. Appl., 53 (2007), 569-581.  doi: 10.1016/j.camwa.2006.02.034.  Google Scholar

[26]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.  doi: 10.1137/050628726.  Google Scholar

[27]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[29]

L. ThevenetJ.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation, ESAIM Control Optim. Calc. Var., 16 (2010), 929-955.  doi: 10.1051/cocv/2009028.  Google Scholar

[30]

F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 12 (2006), 93-119.  doi: 10.1051/cocv:2005029.  Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar

show all references

References:
[1]

C. O. Aguilar and A. J. Krener, Numerical solutions to the Bellman equation of optimal control, J. Optim. Theory Appl., 160 (2014), 527-552.  doi: 10.1007/s10957-013-0403-8.  Google Scholar

[2]

E. Al'brekht, On the optimal stabilization of nonlinear systems, J. Appl. Math. Mech., 25 (1961), 1254-1266.  doi: 10.1016/0021-8928(61)90005-3.  Google Scholar

[3]

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.  Google Scholar

[4]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463.  doi: 10.1137/090778146.  Google Scholar

[5]

V. Barbu, Stabilization of Navier-Stokes Flows, Communications and Control Engineering Series, Springer, London, 2011. doi: 10.1007/978-0-85729-043-4.  Google Scholar

[6]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc., 181 (2006). doi: 10.1090/memo/0852.  Google Scholar

[7]

V. BarbuS. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478.  doi: 10.1137/100785739.  Google Scholar

[8]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

[9]

S. BeelerH. Tran and H. Banks, Feedback control methodologies for nonlinear systems, J. Optim. Theory Appl., 107 (2000), 1-33.  doi: 10.1023/A:1004607114958.  Google Scholar

[10]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[11]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[12]

T. BreitenK. Kunisch and L. Pfeiffer, Infinite-horizon bilinear optimal control problems: Sensitivity analysis and polynomial feedback laws, SIAM J. Control Optim., 56 (2018), 3184-3214.  doi: 10.1137/18M1173952.  Google Scholar

[13]

T. Breiten, K. Kunisch and L. Pfeiffer, Feedback stabilization of the two-dimensional Navier-Stokes equations by value function approximation, preprint, arXiv: 1902.00394. Google Scholar

[14]

T. Breiten and L. Pfeiffer, Taylor expansions of the value function associated with a bilinear optimal control problem, Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire, 36 (2019), 1361-1399.  doi: 10.1016/j.anihpc.2019.01.001.  Google Scholar

[15]

R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[16]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542.  doi: 10.1016/j.matpur.2004.02.010.  Google Scholar

[17]

A. V. Fursikov, Stabilizability of two-dimensional Navier–Stokes equations with help of a boundary feedback control, J. Math. Fluid Mech., 3 (2001), 259-301.  doi: 10.1007/PL00000972.  Google Scholar

[18]

A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 289-314.  doi: 10.3934/dcds.2004.10.289.  Google Scholar

[19]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM. Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.  Google Scholar

[20]

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, Cambridge, 2000.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I/II, Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, Berlin, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[22]

D. L. Lukes, Optimal regulation of nonlinear dynamical systems, SIAM J. Control, 7 (1969), 75-100.  doi: 10.1137/0307007.  Google Scholar

[23]

C. Navasca and A. Krener, Patchy solutions of Hamilton-Jacobi-Bellman Partial differential equations, in Modeling, Estimation and Control, Lect. Notes Control Inf. Sci., 364, Springer, Berlin, 2007,251–270. doi: 10.1007/978-3-540-73570-0_20.  Google Scholar

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[25]

M. Pošta and T. Roubíček, Optimal control of Navier-Stokes equations by Oseen approximation, Comput. Math. Appl., 53 (2007), 569-581.  doi: 10.1016/j.camwa.2006.02.034.  Google Scholar

[26]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier–Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.  doi: 10.1137/050628726.  Google Scholar

[27]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.  Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[29]

L. ThevenetJ.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation, ESAIM Control Optim. Calc. Var., 16 (2010), 929-955.  doi: 10.1051/cocv/2009028.  Google Scholar

[30]

F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 12 (2006), 93-119.  doi: 10.1051/cocv:2005029.  Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I. Fixed-Point Theorems, Springer-Verlag, New York, 1986.  Google Scholar

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