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

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

Tobias Breiten 1,, and  Karl Kunisch 1,2,

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.

Keywords: 3-D Navier-Stokes equations, generalized Lyapunov equations, optimal control, value function, Taylor expansion, feedback control, stabilization.
Mathematics Subject Classification: Primary: 35Q35, 49J20, 49N35, 93D05, 93D15.
Citation: Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

V. Barbu, S. 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.

[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.

[9]

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

[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.

[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.

[12]

T. Breiten, K. 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.

[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.

[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.

[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.

[16]

E. Fernández-Cara, S. Guerrero, O. 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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

L. Thevenet, J.-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.

[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.

[31]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[7]

V. Barbu, S. 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.

[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.

[9]

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

[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.

[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.

[12]

T. Breiten, K. 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.

[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.

[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.

[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.

[16]

E. Fernández-Cara, S. Guerrero, O. 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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[29]

L. Thevenet, J.-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.

[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.

[31]

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

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