January  2017, 16(1): 189-208. doi: 10.3934/cpaa.2017009

Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D

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

Received  March 2016 Revised  July 2016 Published  November 2016

We show the continuity of a specific cost functional $J(\phi) =\mathbb{E} \sup_{ t \in [0, T]}(\varphi(\mathcal{L}[t, u_\phi(t), \phi(t)]))$ of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ with respect to a special set of feedback controls $\{\phi_n\}_{n \geq 0}$, where $\varphi(x) =\log(1 + x)^{1-\epsilon}$ with $0 < \epsilon < 1$.

Citation: Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure & Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput.Fluid Dynamics., 1 (1990), 303-325.   Google Scholar

[2]

A. Bensoussan and R. Temam, Òquations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.   Google Scholar

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H. Breckner, Existence of optimal and epsilon-optimal controls for the stochastic NavierStokes Equation, Nonlinear Analysis, 51 (2002), 95-118.  doi: 10.1016/S0362-546X(01)00814-8.  Google Scholar

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J. BricmontA. Kupiainen and R. Lefevere, Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations, J. Statist. Phys., 100 (2000), 743-756.  doi: 10.1023/A:1018627609718.  Google Scholar

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Z. Brzézniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations, Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , vol. 52, R. Neth. Acad. Arts Sci. , Amsterdam, 2000, 85-98.  Google Scholar

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M. Capinski and NJ. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore, 1995. doi: 10.1142/9789812831958.  Google Scholar

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M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with application to NavierStokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35.  doi: 10.1006/jfan.1994.1140.  Google Scholar

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M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

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H. ChoiR. TemamP. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), 509-543.  doi: 10.1017/S0022112093001880.  Google Scholar

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P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

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A. B. Cruzeiro, Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes, Exposition. Math., 7 (1989), 73-82.   Google Scholar

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G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947.  doi: 10.1016/S0021-7824(03)00025-4.  Google Scholar

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[14]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, (2011), to appear. doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[15]

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I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM Series Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

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F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1942, Springer Berlin / Heidelberg, 2008, 51-150. doi: 10.1007/978-3-540-78493-7_2.  Google Scholar

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[19]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241.  doi: 10.1090/S0002-9947-02-02975-6.  Google Scholar

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C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[21]

A. V. Fursikov and MJ. Vishik, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988. Google Scholar

[22]

D. Gatarek, Existence of optimal controls for stochastic evolution systems, in: G. Da Prato et al. , Control of Partial Differential Equations, IFIP WG 7. 2 Conference, Villa Madruzzo, Trento, Italy, January 4-9,1993, New York, Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. , 165 (1994), 81-86.  Google Scholar

[23]

D. Gatarek and J. Sobczyk, On the existence of optimal controls of Hilbert space-valued diffusions, SIAM Control Optim., 32 (1994), 170-175.  doi: 10.1137/S0363012992226260.  Google Scholar

[24]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability, 42 (2014), 80-145.  doi: 10.1214/12-AOP773.  Google Scholar

[25]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.   Google Scholar

[26]

W. Grecksch, Stochastische Evolutionsgleichungen undderen Steuerung, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987.  Google Scholar

[27]

S. B. Kuksin, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2006. doi: 10.4171/021.  Google Scholar

[28]

I. Kukavica, K. Uğurlu and M. Ziane, On the Galerkin approximation and norm estimates of the stochastic Navier-Stokes equations with multiplicative noise, submitted. Google Scholar

[29]

I. Kukavica and V. Vicol, On moments for strong solutions of the 2D stochastic Navier-Stokes equations in a bounded domain, Asymptotic Analysis, 90 (2014), 189-206.   Google Scholar

[30]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity, J. Statist. Phys. , 108 (2002), 1157-1179, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. doi: 10.1023/A:1019799700126.  Google Scholar

[31]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[32]

R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[33]

J. -L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[34]

C. Odasso, Spatial smoothness of the stationary solutions of the 3D Navier-Stokes equations, Electron. J. Probab., 11 (2006), 686-699.  doi: 10.1214/EJP.v11-336.  Google Scholar

[35]

G. Da Prato and A. Debussche, Control of the stochastic burgers model of turbulence, SIAM J. Control Optim., 37 (1999), 1123-1149.  doi: 10.1137/S0363012996311307.  Google Scholar

[36]

G. Da Prato and A. Ichikawa, Stability and quadratic control for linear stochastic equations with unbounded coefficients, Boll. Unione Mat. Ital., Ⅵ. Ser. B, 4 (1985), 987-1001.   Google Scholar

[37]

C. Prévôt and M. Röckner, A Concise Course On Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.  Google Scholar

[38]

A. Shirikyan, Analyticity of solutions and Kolmogorov's dissipation scale for 2D NavierStokes equations, Evolution equations (Warsaw, 2001), Banach Center Publ. , vol. 60, Polish Acad. Sci. , Warsaw, 2003, 49-53.  Google Scholar

[39]

S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyper viscosities, Appl. Math. Optim., 41 (2000), 255-308.  doi: 10.1007/s0024599110140.  Google Scholar

[40]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979.  Google Scholar

[41]

C. Tudor, Optimal control for semi-linear evolution equations, Appl. Math. Optim., 20 (1989), 319-331.  doi: 10.1007/BF01447659.  Google Scholar

[42]

C. Tudor, Optimal and optimal control for the stochastic linear-quadratic problem, Math. Nachr., 145 (1990), 135-149.  doi: 10.1002/mana.19901450111.  Google Scholar

[43]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thèse, Université Pierre et Marie Curie, Paris, 1976. Google Scholar

[44]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Ⅲ: Variational Methods and Optimization, Springer, New York, 1985. doi: 10.1007/978-1-4612-5020-3.  Google Scholar

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput.Fluid Dynamics., 1 (1990), 303-325.   Google Scholar

[2]

A. Bensoussan and R. Temam, Òquations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.   Google Scholar

[3]

H. Breckner, Existence of optimal and epsilon-optimal controls for the stochastic NavierStokes Equation, Nonlinear Analysis, 51 (2002), 95-118.  doi: 10.1016/S0362-546X(01)00814-8.  Google Scholar

[4]

J. BricmontA. Kupiainen and R. Lefevere, Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations, J. Statist. Phys., 100 (2000), 743-756.  doi: 10.1023/A:1018627609718.  Google Scholar

[5]

Z. Brzézniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations, Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , vol. 52, R. Neth. Acad. Arts Sci. , Amsterdam, 2000, 85-98.  Google Scholar

[6]

M. Capinski and NJ. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore, 1995. doi: 10.1142/9789812831958.  Google Scholar

[7]

M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with application to NavierStokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35.  doi: 10.1006/jfan.1994.1140.  Google Scholar

[8]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200.  doi: 10.1007/PL00001415.  Google Scholar

[9]

H. ChoiR. TemamP. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), 509-543.  doi: 10.1017/S0022112093001880.  Google Scholar

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[11]

A. B. Cruzeiro, Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes, Exposition. Math., 7 (1989), 73-82.   Google Scholar

[12]

G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947.  doi: 10.1016/S0021-7824(03)00025-4.  Google Scholar

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[14]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, (2011), to appear. doi: 10.1016/j.physd.2011.03.009.  Google Scholar

[15]

R. Durrett, Probability: Theory and Examples, Cambridge University Press, Cambridge, 2013.  Google Scholar

[16]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM Series Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[17]

F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1942, Springer Berlin / Heidelberg, 2008, 51-150. doi: 10.1007/978-3-540-78493-7_2.  Google Scholar

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391.  doi: 10.1007/BF01192467.  Google Scholar

[19]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241.  doi: 10.1090/S0002-9947-02-02975-6.  Google Scholar

[20]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.   Google Scholar

[21]

A. V. Fursikov and MJ. Vishik, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988. Google Scholar

[22]

D. Gatarek, Existence of optimal controls for stochastic evolution systems, in: G. Da Prato et al. , Control of Partial Differential Equations, IFIP WG 7. 2 Conference, Villa Madruzzo, Trento, Italy, January 4-9,1993, New York, Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. , 165 (1994), 81-86.  Google Scholar

[23]

D. Gatarek and J. Sobczyk, On the existence of optimal controls of Hilbert space-valued diffusions, SIAM Control Optim., 32 (1994), 170-175.  doi: 10.1137/S0363012992226260.  Google Scholar

[24]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability, 42 (2014), 80-145.  doi: 10.1214/12-AOP773.  Google Scholar

[25]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.   Google Scholar

[26]

W. Grecksch, Stochastische Evolutionsgleichungen undderen Steuerung, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987.  Google Scholar

[27]

S. B. Kuksin, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2006. doi: 10.4171/021.  Google Scholar

[28]

I. Kukavica, K. Uğurlu and M. Ziane, On the Galerkin approximation and norm estimates of the stochastic Navier-Stokes equations with multiplicative noise, submitted. Google Scholar

[29]

I. Kukavica and V. Vicol, On moments for strong solutions of the 2D stochastic Navier-Stokes equations in a bounded domain, Asymptotic Analysis, 90 (2014), 189-206.   Google Scholar

[30]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity, J. Statist. Phys. , 108 (2002), 1157-1179, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. doi: 10.1023/A:1019799700126.  Google Scholar

[31]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310.  doi: 10.1137/S0036141002409167.  Google Scholar

[32]

R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176.  doi: 10.1214/009117904000000630.  Google Scholar

[33]

J. -L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53.  doi: 10.1007/s00245-002-0734-6.  Google Scholar

[34]

C. Odasso, Spatial smoothness of the stationary solutions of the 3D Navier-Stokes equations, Electron. J. Probab., 11 (2006), 686-699.  doi: 10.1214/EJP.v11-336.  Google Scholar

[35]

G. Da Prato and A. Debussche, Control of the stochastic burgers model of turbulence, SIAM J. Control Optim., 37 (1999), 1123-1149.  doi: 10.1137/S0363012996311307.  Google Scholar

[36]

G. Da Prato and A. Ichikawa, Stability and quadratic control for linear stochastic equations with unbounded coefficients, Boll. Unione Mat. Ital., Ⅵ. Ser. B, 4 (1985), 987-1001.   Google Scholar

[37]

C. Prévôt and M. Röckner, A Concise Course On Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.  Google Scholar

[38]

A. Shirikyan, Analyticity of solutions and Kolmogorov's dissipation scale for 2D NavierStokes equations, Evolution equations (Warsaw, 2001), Banach Center Publ. , vol. 60, Polish Acad. Sci. , Warsaw, 2003, 49-53.  Google Scholar

[39]

S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyper viscosities, Appl. Math. Optim., 41 (2000), 255-308.  doi: 10.1007/s0024599110140.  Google Scholar

[40]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979.  Google Scholar

[41]

C. Tudor, Optimal control for semi-linear evolution equations, Appl. Math. Optim., 20 (1989), 319-331.  doi: 10.1007/BF01447659.  Google Scholar

[42]

C. Tudor, Optimal and optimal control for the stochastic linear-quadratic problem, Math. Nachr., 145 (1990), 135-149.  doi: 10.1002/mana.19901450111.  Google Scholar

[43]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thèse, Université Pierre et Marie Curie, Paris, 1976. Google Scholar

[44]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Ⅲ: Variational Methods and Optimization, Springer, New York, 1985. doi: 10.1007/978-1-4612-5020-3.  Google Scholar

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