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doi: 10.3934/eect.2021020
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## Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations

 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 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.
Citation: Manil T. Mohan. Optimal control problems governed by two dimensional convective Brinkman-Forchheimer equations. Evolution Equations & Control Theory, 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.  Google Scholar [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.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Pitman Res. Notes Math. Ser. 100, Pitman, Boston, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [9] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar [14] I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.   Google Scholar [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.  Google Scholar [16] C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, arXiv: 1904.03337. Google Scholar [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.  Google Scholar [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.   Google Scholar [19] A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, , American Mathematical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.  Google Scholar [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.  Google Scholar [21] M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [26] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.  Google Scholar [27] J. -L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer Verlag, Berlin, Germany, 1972.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted. Google Scholar [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.  Google Scholar [32] J. P. Raymond, Optimal control of partial differential equations. Université Paul Sabatier, Lecture Notes, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] V. Barbu, Optimal Control of Variational Inequalities, Pitman Res. Notes Math. Ser. 100, Pitman, Boston, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [9] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM Philadelphia, 2013.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [13] L. C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar [14] I. Ekeland and T. Turnbull, Infinite-dimensional Optimization and Convexity, The University of Chicago press, Chicago and London, 1983.   Google Scholar [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.  Google Scholar [16] C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, arXiv: 1904.03337. Google Scholar [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.  Google Scholar [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.   Google Scholar [19] A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications, , American Mathematical Society, Rhode Island, 2000. doi: 10.1090/mmono/187.  Google Scholar [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.  Google Scholar [21] M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM's Advances in Design and Control series, Philadelphia, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.  Google Scholar [26] J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.  Google Scholar [27] J. -L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications, Springer Verlag, Berlin, Germany, 1972.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted. Google Scholar [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.  Google Scholar [32] J. P. Raymond, Optimal control of partial differential equations. Université Paul Sabatier, Lecture Notes, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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