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Initial boundary value problem for a strongly damped wave equation with a general nonlinearity
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 |
$ \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. $ |
$ r = 1,2 $ |
$ 3 $ |
$ r = 3 $ |
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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