Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

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June 2022, 11(3): 649-679. doi: 10.3934/eect.2021020

Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

Manil T. Mohan ^,

Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA

^*Corresponding author: Manil T. Mohan

Received May 2020 Revised March 2021 Published June 2022 Early access April 2021

The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluids through a rigid, homogeneous, isotropic, porous medium and is given by

$ \partial_t{\boldsymbol{u}}-\mu \Delta{\boldsymbol{u}}+({\boldsymbol{u}}\cdot\nabla){\boldsymbol{u}}+\alpha{\boldsymbol{u}}+\beta|{\boldsymbol{u}}|^{r-1}{\boldsymbol{u}}+\nabla p = {\boldsymbol{f}},\ \nabla\cdot{\boldsymbol{u}} = 0. $

In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent

$ r = 1,2 $

and

$ 3 $

. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case

$ r = 3 $

, we show the second order necessary and sufficient conditions of optimality. We also investigate an another control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is 2D CBF equations, using optimal control techniques.

Keywords: porus medium, convective Brinkman-Forchheimer equations, first order necessary conditions, Pontryagin's maximum principle, optimal control.

Mathematics Subject Classification: Primary 49J20; Secondary 49K20, 35Q35, 76D03.

Citation: Manil T. Mohan. Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, 2022, 11 (3) : 649-679. doi: 10.3934/eect.2021020

References:

[1]	S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Applicable Analysis, 89 (2010), 1805-1825. doi: 10.1080/00036811.2010.495341.
[2]	F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325. doi: 10.1007/BF00271794.
[3]	V. Barbu, Optimal Control of Variational Inequalities, Pitman Res. Notes Math. Ser. 100, Pitman, Boston, 1984.
[4]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Analysis: Theory, Methods & Applications, 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.
[5]	T. Biswas, S. Dharmatti and M. T. Mohan, Pontryagin's maximum principle for optimal control of the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, Analysis (Berlin), 40 (2020), 127-150. doi: 10.1515/anly-2019-0049.
[6]	T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 22 (2020), Article No. 34, 1-42. doi: 10.1007/s00021-020-00493-8.
[7]	T. Biswas, S. Dharmatti and M. T. Mohan, Second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Nonlinear Studies, 28 (2021), 29-43.
[8]	E. Casas and F. Tröltzsch, Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), 406-431. doi: 10.1137/S1052623400367698.
[9]	P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.
[10]	P. Cherier and A. Milani, Linear and Quasi-linear Evolution Equations in Hilbert Spaces, American Mathematical Society Providence, Rhode Island, 2012. doi: 10.1090/gsm/135.
[11]	R. F. Curtain and A. J. Pritchard, Functional Analysis in Modern Applied Mathematics, Mathematics in Science and Engineering, Vol. 132. Academic Press, London-New York, 1977.
[12]	S. Doboszczak, M. T. Mohan and S. S. Sritharan, Existence of optimal controls for compressible viscous flow, Journal of Mathematical Fluid Mechanics, 20 (2018), 199-211. doi: 10.1007/s00021-017-0318-5.
[13]	L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.
[14]	I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.
[15]	H. O. Fattorini and S. S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Roy. Soc. Edinburoh A, 124 (1994), 211-251. doi: 10.1017/S0308210500028444.
[16]	C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, arXiv: 1904.03337.
[17]	B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Lévy noise, Stochastic Analysis and Applications, 31 (2013), 381-426. doi: 10.1080/07362994.2013.759482.
[18]	D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
[19]	A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, , American Mathematical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.
[20]	G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech. Birkhaüser, Basel, 2000, 1-70.
[21]	M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.
[22]	K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, Journal of Differential Equations, 263 (2017), 7141-7161. doi: 10.1016/j.jde.2017.08.001.
[23]	V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054. doi: 10.3934/cpaa.2012.11.2037.
[24]	H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853-1920. doi: 10.1512/iumj.2009.58.3605.
[25]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
[26]	J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.
[27]	J. -L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer Verlag, Berlin, Germany, 1972.
[28]	M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures, Stochastic Analysis and Applications, 38 (2020), 1-61. doi: 10.1080/07362994.2019.1646138.
[29]	M. T. Mohan, First order necessary conditions of optimality for the two dimensional tidal dynamics system, Mathematical Control and Related Fields, 2020. doi: 10.3934/mcrf. 2020045.
[30]	M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.
[31]	M. T. Mohan, The time optimal control of two dimensional convective Brinkman-Forchheimer equations, Applied Mathematics & Optimization, 2021. doi: 10.1007/s00245-021-09748-w.
[32]	J. P. Raymond, Optimal control of partial differential equations. Université Paul Sabatier, Lecture Notes, 2013.
[33]	J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139095143.
[34]	A. K. Sharma, M. K. Khandelwal and P. Bera, Finite amplitude analysis of non-isothermal parallel flow in avertical channel filled with high permeable porous medium, Journal of Fluid Mechanics, 857 (2018), 469-507. doi: 10.1017/jfm.2018.745.
[35]	J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[36]	S. S. Sritharan, Optimal Control of Viscous Flow, SIAM Frontiers in Applied Mathematics, Philadelphia. Society for Industrial and Applied Mathematics, 1998. doi: 10.1137/1.9781611971415.
[37]	T. Tachim Medjo, Second-order optimality conditions for optimal control of the primitive equations of the ocean with periodic inputs, Applied Mathematics and Optimization, 63 (2011), 75-106. doi: 10.1007/s00245-010-9112-y.
[38]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.
[39]	F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 93-119. doi: 10.1051/cocv:2005029.
[40]	L. Wang and P. He, Second order optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations, Acta Mathematica Scientia, 26 (2006), 729-734. doi: 10.1016/S0252-9602(06)60099-4.
[41]	G. Wang, Optimal controls of 3 dimensional Navier-Stokes equations with state constraints, SIAM Journal on Control and Optimization, 41 (2002), 583-606. doi: 10.1137/S0363012901385769.
[42]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.

show all references

References:

[1]	S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Applicable Analysis, 89 (2010), 1805-1825. doi: 10.1080/00036811.2010.495341.
[2]	F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325. doi: 10.1007/BF00271794.
[3]	V. Barbu, Optimal Control of Variational Inequalities, Pitman Res. Notes Math. Ser. 100, Pitman, Boston, 1984.
[4]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Analysis: Theory, Methods & Applications, 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.
[5]	T. Biswas, S. Dharmatti and M. T. Mohan, Pontryagin's maximum principle for optimal control of the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, Analysis (Berlin), 40 (2020), 127-150. doi: 10.1515/anly-2019-0049.
[6]	T. Biswas, S. Dharmatti and M. T. Mohan, Maximum principle and data assimilation problem for the optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 22 (2020), Article No. 34, 1-42. doi: 10.1007/s00021-020-00493-8.
[7]	T. Biswas, S. Dharmatti and M. T. Mohan, Second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn-Hilliard-Navier-Stokes equations, Nonlinear Studies, 28 (2021), 29-43.
[8]	E. Casas and F. Tröltzsch, Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), 406-431. doi: 10.1137/S1052623400367698.
[9]	P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.
[10]	P. Cherier and A. Milani, Linear and Quasi-linear Evolution Equations in Hilbert Spaces, American Mathematical Society Providence, Rhode Island, 2012. doi: 10.1090/gsm/135.
[11]	R. F. Curtain and A. J. Pritchard, Functional Analysis in Modern Applied Mathematics, Mathematics in Science and Engineering, Vol. 132. Academic Press, London-New York, 1977.
[12]	S. Doboszczak, M. T. Mohan and S. S. Sritharan, Existence of optimal controls for compressible viscous flow, Journal of Mathematical Fluid Mechanics, 20 (2018), 199-211. doi: 10.1007/s00021-017-0318-5.
[13]	L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.
[14]	I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.
[15]	H. O. Fattorini and S. S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. Roy. Soc. Edinburoh A, 124 (1994), 211-251. doi: 10.1017/S0308210500028444.
[16]	C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, arXiv: 1904.03337.
[17]	B. P. W. Fernando and S. S. Sritharan, Nonlinear filtering of stochastic Navier-Stokes equation with Lévy noise, Stochastic Analysis and Applications, 31 (2013), 381-426. doi: 10.1080/07362994.2013.759482.
[18]	D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
[19]	A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, , American Mathematical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.
[20]	G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech. Birkhaüser, Basel, 2000, 1-70.
[21]	M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.
[22]	K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, Journal of Differential Equations, 263 (2017), 7141-7161. doi: 10.1016/j.jde.2017.08.001.
[23]	V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054. doi: 10.3934/cpaa.2012.11.2037.
[24]	H. Kozono and T. Yanagisawa, $L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains, Indiana Univ. Math. J., 58 (2009), 1853-1920. doi: 10.1512/iumj.2009.58.3605.
[25]	O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
[26]	J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.
[27]	J. -L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer Verlag, Berlin, Germany, 1972.
[28]	M. T. Mohan, Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures, Stochastic Analysis and Applications, 38 (2020), 1-61. doi: 10.1080/07362994.2019.1646138.
[29]	M. T. Mohan, First order necessary conditions of optimality for the two dimensional tidal dynamics system, Mathematical Control and Related Fields, 2020. doi: 10.3934/mcrf. 2020045.
[30]	M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.
[31]	M. T. Mohan, The time optimal control of two dimensional convective Brinkman-Forchheimer equations, Applied Mathematics & Optimization, 2021. doi: 10.1007/s00245-021-09748-w.
[32]	J. P. Raymond, Optimal control of partial differential equations. Université Paul Sabatier, Lecture Notes, 2013.
[33]	J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139095143.
[34]	A. K. Sharma, M. K. Khandelwal and P. Bera, Finite amplitude analysis of non-isothermal parallel flow in avertical channel filled with high permeable porous medium, Journal of Fluid Mechanics, 857 (2018), 469-507. doi: 10.1017/jfm.2018.745.
[35]	J. Simon, Compact sets in the space $\mathrm{L}^p(0, T;\mathrm{B})$, Annali di Matematica Pura ed Applicata, 146 (1986), 65-96. doi: 10.1007/BF01762360.
[36]	S. S. Sritharan, Optimal Control of Viscous Flow, SIAM Frontiers in Applied Mathematics, Philadelphia. Society for Industrial and Applied Mathematics, 1998. doi: 10.1137/1.9781611971415.
[37]	T. Tachim Medjo, Second-order optimality conditions for optimal control of the primitive equations of the ocean with periodic inputs, Applied Mathematics and Optimization, 63 (2011), 75-106. doi: 10.1007/s00245-010-9112-y.
[38]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.
[39]	F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 93-119. doi: 10.1051/cocv:2005029.
[40]	L. Wang and P. He, Second order optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations, Acta Mathematica Scientia, 26 (2006), 729-734. doi: 10.1016/S0252-9602(06)60099-4.
[41]	G. Wang, Optimal controls of 3 dimensional Navier-Stokes equations with state constraints, SIAM Journal on Control and Optimization, 41 (2002), 583-606. doi: 10.1137/S0363012901385769.
[42]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Analysis: Theory, Methods & Applications, 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.

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