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December  2021, 10(4): 937-963. doi: 10.3934/eect.2020097

Optimal distributed control of the three dimensional primitive equations of large-scale ocean and atmosphere dynamics

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  July 2019 Revised  September 2020 Published  December 2021 Early access  October 2020

Fund Project: This work was supported by the National Science Foundation of China Grant (11401459, 11871389), the Natural Science Foundation of Shaanxi Province (2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

The main objective of this paper is to study the optimal distributed control of the three dimensional non-autonomous primitive equations of large-scale ocean and atmosphere dynamics. We apply the well-posedness and regularity results of solutions for this system as well as some abstract results from the nonlinear functional analysis to establish the existence of optimal controls as well as the first-order necessary optimality condition for an associated optimal control problem in which a distributed control is applied to the temperature.

Citation: Bo You. Optimal distributed control of the three dimensional primitive equations of large-scale ocean and atmosphere dynamics. Evolution Equations and Control Theory, 2021, 10 (4) : 937-963. doi: 10.3934/eect.2020097
References:
[1]

A. Belmiloudi, Mathematical analysis and optimal control problems for the perturbation of the primitive equations of the ocean with vertical viscosity, J. Appl. Anal., 8 (2002), 153-200.  doi: 10.1515/JAA.2002.153.

[2]

C. S. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267.  doi: 10.4007/annals.2007.166.245.

[3]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[4]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250.  doi: 10.1137/140994800.

[5]

A. V. FursikovM. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case, SIAM J. Control Optim., 43 (2005), 2191-2232.  doi: 10.1137/S0363012904400805.

[6]

H. J. Gao and C. F. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[7]

B. L. Guo and D. W. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207.

[8]

B. L. Guo and D. W. Huang, On the existence of atmospheric attractors, Sci. China, Ser. D, 51 (2008), 469-480. 

[9]

B. L. Guo and D. W. Huang, 3D stochastic primitive equations of the large-scale ocean: global well-posedness and attractors, Comm. Math. Phys., 286 (2009), 697-723.  doi: 10.1007/s00220-008-0654-7.

[10]

B. L. Guo and D. W. Huang, On the 3D viscous primitive equations of the large-scale atmosphere, Acta Math. Sci. Ser. B, 29 (2009), 846-866.  doi: 10.1016/S0252-9602(09)60074-6.

[11]

B. L. Guo and D. W. Huang, Existence of the universal attractor for the 3D viscous primitive equations of large-scale moist atmosphere, J. Differential Equations, 251 (2011), 457-491.  doi: 10.1016/j.jde.2011.05.010.

[12]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.

[13]

C. B. HuR. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chinese Ann. Math. Ser. B, 23 (2002), 277-292.  doi: 10.1142/S0252959902000262.

[14]

N. Ju, The global attractor for the solutions to the three dimensional viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179.  doi: 10.3934/dcds.2007.17.159.

[15]

N. Ju, The finite dimensional global attractor for the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 36 (2016), 7001-7020.  doi: 10.3934/dcds.2016104.

[16]

N. Ju, On $H^2$ solutions and $z$-weak solutions of the 3D primitive equations, Indiana Univ. Math. J., 66 (2017), 973-996.  doi: 10.1512/iumj.2017.66.6065.

[17]

N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity, J. Nonlinear Sci., 25 (2015), 131-155.  doi: 10.1007/s00332-014-9223-8.

[18]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.  doi: 10.1088/0951-7715/20/12/001.

[19]

J. L. LionsO. P. ManleyR. Temam and S. Wang, Physical interpretation of the attractor dimension for the primitive equations of atmospheric circulation, J. Atmospheric Sci., 54 (1997), 1137-1143.  doi: 10.1175/1520-0469(1997)054<1137:PIOTAD>2.0.CO;2.

[20]

J. L. LionsR. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.  doi: 10.1088/0951-7715/5/5/002.

[21]

T. T. Medjo, Maximum principle of optimal control of the primitive equations of the ocean with two point boundary state constraint, Appl. Math. Optim., 62 (2010), 1-26.  doi: 10.1007/s00245-009-9092-y.

[22]

T. T. Medjo, Optimal control of the primitive equations of the ocean with state constraints, Nonlinear Anal., 73 (2010), 634-649.  doi: 10.1016/j.na.2010.03.043.

[23]

T. T. Medjo, Second-order optimality conditions for optimal control of the primitive equations of the ocean with periodic inputs, Appl. Math. Optim., 63 (2011), 75-106.  doi: 10.1007/s00245-010-9112-y.

[24]

T. T. Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 265-291.  doi: 10.3934/dcds.2012.32.265.

[25]

T. T. Medjo, Averaging of a 3D primitive equations with oscillating external forces, Appl. Anal., 92 (2013), 869-900.  doi: 10.1080/00036811.2011.640628.

[26]

M. Nodet, Optimal control of the primitive equations of the ocean with Lagrangian observations, ESAIM Control Optim. Calc. Var., 16 (2010), 400-419.  doi: 10.1051/cocv/2009003.

[27]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[28]

S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998. doi: 10.1137/1.9781611971415.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, American Mathematical Society Providence, Rhode Island, 2010.

[30]

B. You and F. Li, Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics, Z. Angew. Math. Phys., 69 (2018), Page No. 114, 13pp. doi: 10.1007/s00033-018-1007-9.

[31]

G. L. Zhou, Random attractor for the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637.  doi: 10.1016/j.jde.2018.12.009.

show all references

References:
[1]

A. Belmiloudi, Mathematical analysis and optimal control problems for the perturbation of the primitive equations of the ocean with vertical viscosity, J. Appl. Anal., 8 (2002), 153-200.  doi: 10.1515/JAA.2002.153.

[2]

C. S. Cao and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large-scale ocean and atmosphere dynamics, Ann. of Math., 166 (2007), 245-267.  doi: 10.4007/annals.2007.166.245.

[3]

A. DebusscheN. Glatt-HoltzR. Temam and M. Ziane, Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity, 25 (2012), 2093-2118.  doi: 10.1088/0951-7715/25/7/2093.

[4]

S. FrigeriE. Rocca and J. Sprekels, Optimal distributed control of a nonlocal Cahn-Hilliard/Navier-Stokes system in two dimensions, SIAM J. Control Optim., 54 (2016), 221-250.  doi: 10.1137/140994800.

[5]

A. V. FursikovM. D. Gunzburger and L. S. Hou, Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case, SIAM J. Control Optim., 43 (2005), 2191-2232.  doi: 10.1137/S0363012904400805.

[6]

H. J. Gao and C. F. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3053-3073.  doi: 10.3934/dcdsb.2016087.

[7]

B. L. Guo and D. W. Huang, Existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics, J. Math. Phys., 47 (2006), 083508, 23pp. doi: 10.1063/1.2245207.

[8]

B. L. Guo and D. W. Huang, On the existence of atmospheric attractors, Sci. China, Ser. D, 51 (2008), 469-480. 

[9]

B. L. Guo and D. W. Huang, 3D stochastic primitive equations of the large-scale ocean: global well-posedness and attractors, Comm. Math. Phys., 286 (2009), 697-723.  doi: 10.1007/s00220-008-0654-7.

[10]

B. L. Guo and D. W. Huang, On the 3D viscous primitive equations of the large-scale atmosphere, Acta Math. Sci. Ser. B, 29 (2009), 846-866.  doi: 10.1016/S0252-9602(09)60074-6.

[11]

B. L. Guo and D. W. Huang, Existence of the universal attractor for the 3D viscous primitive equations of large-scale moist atmosphere, J. Differential Equations, 251 (2011), 457-491.  doi: 10.1016/j.jde.2011.05.010.

[12]

N. Glatt-Holtz, I. Kukavica, V. Vicol and M. Ziane, Existence and regularity of invariant measures for the three dimensional stochastic primitive equations, J. Math. Phys., 55 (2014), 051504, 34pp. doi: 10.1063/1.4875104.

[13]

C. B. HuR. Temam and M. Ziane, Regularity results for linear elliptic problems related to the primitive equations, Chinese Ann. Math. Ser. B, 23 (2002), 277-292.  doi: 10.1142/S0252959902000262.

[14]

N. Ju, The global attractor for the solutions to the three dimensional viscous primitive equations, Discrete Contin. Dyn. Syst., 17 (2007), 159-179.  doi: 10.3934/dcds.2007.17.159.

[15]

N. Ju, The finite dimensional global attractor for the 3D viscous primitive equations, Discrete Contin. Dyn. Syst., 36 (2016), 7001-7020.  doi: 10.3934/dcds.2016104.

[16]

N. Ju, On $H^2$ solutions and $z$-weak solutions of the 3D primitive equations, Indiana Univ. Math. J., 66 (2017), 973-996.  doi: 10.1512/iumj.2017.66.6065.

[17]

N. Ju and R. Temam, Finite dimensions of the global attractor for 3D primitive equations with viscosity, J. Nonlinear Sci., 25 (2015), 131-155.  doi: 10.1007/s00332-014-9223-8.

[18]

I. Kukavica and M. Ziane, On the regularity of the primitive equations of the ocean, Nonlinearity, 20 (2007), 2739-2753.  doi: 10.1088/0951-7715/20/12/001.

[19]

J. L. LionsO. P. ManleyR. Temam and S. Wang, Physical interpretation of the attractor dimension for the primitive equations of atmospheric circulation, J. Atmospheric Sci., 54 (1997), 1137-1143.  doi: 10.1175/1520-0469(1997)054<1137:PIOTAD>2.0.CO;2.

[20]

J. L. LionsR. Temam and S. Wang, On the equations of the large-scale ocean, Nonlinearity, 5 (1992), 1007-1053.  doi: 10.1088/0951-7715/5/5/002.

[21]

T. T. Medjo, Maximum principle of optimal control of the primitive equations of the ocean with two point boundary state constraint, Appl. Math. Optim., 62 (2010), 1-26.  doi: 10.1007/s00245-009-9092-y.

[22]

T. T. Medjo, Optimal control of the primitive equations of the ocean with state constraints, Nonlinear Anal., 73 (2010), 634-649.  doi: 10.1016/j.na.2010.03.043.

[23]

T. T. Medjo, Second-order optimality conditions for optimal control of the primitive equations of the ocean with periodic inputs, Appl. Math. Optim., 63 (2011), 75-106.  doi: 10.1007/s00245-010-9112-y.

[24]

T. T. Medjo, Non-autonomous 3D primitive equations with oscillating external force and its global attractor, Discrete Contin. Dyn. Syst., 32 (2012), 265-291.  doi: 10.3934/dcds.2012.32.265.

[25]

T. T. Medjo, Averaging of a 3D primitive equations with oscillating external forces, Appl. Anal., 92 (2013), 869-900.  doi: 10.1080/00036811.2011.640628.

[26]

M. Nodet, Optimal control of the primitive equations of the ocean with Lagrangian observations, ESAIM Control Optim. Calc. Var., 16 (2010), 400-419.  doi: 10.1051/cocv/2009003.

[27]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

[28]

S. S. Sritharan, Optimal Control of Viscous Flow, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1998. doi: 10.1137/1.9781611971415.

[29]

F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, American Mathematical Society Providence, Rhode Island, 2010.

[30]

B. You and F. Li, Global attractor of the three-dimensional primitive equations of large-scale ocean and atmosphere dynamics, Z. Angew. Math. Phys., 69 (2018), Page No. 114, 13pp. doi: 10.1007/s00033-018-1007-9.

[31]

G. L. Zhou, Random attractor for the 3D viscous primitive equations driven by fractional noises, J. Differential Equations, 266 (2019), 7569-7637.  doi: 10.1016/j.jde.2018.12.009.

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