• Previous Article
    Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type
  • EECT Home
  • This Issue
  • Next Article
    Initial boundary value problem for a strongly damped wave equation with a general nonlinearity
June  2022, 11(3): 649-679. doi: 10.3934/eect.2021020

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 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.
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. BiswasS. 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. BiswasS. 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. DoboszczakM. 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. RobinsonJ. 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. SharmaM. 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. BiswasS. 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. BiswasS. 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. DoboszczakM. 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. RobinsonJ. 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. SharmaM. 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.

[1]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110

[2]

Kush Kinra, Manil T. Mohan. Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021061

[3]

Pardeep Kumar, Manil T. Mohan. Well-posedness of an inverse problem for two- and three-dimensional convective Brinkman-Forchheimer equations with the final overdetermination. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022024

[4]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[5]

Varga K. Kalantarov, Sergey Zelik. Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2037-2054. doi: 10.3934/cpaa.2012.11.2037

[6]

Yuncheng You, Caidi Zhao, Shengfan Zhou. The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3787-3800. doi: 10.3934/dcds.2012.32.3787

[7]

Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021059

[8]

Timir Karmakar, Meraj Alam, G. P. Raja Sekhar. Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel. Communications on Pure and Applied Analysis, 2022, 21 (3) : 845-865. doi: 10.3934/cpaa.2022001

[9]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[10]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control and Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003

[11]

Qiangheng Zhang, Yangrong Li. Regular attractors of asymptotically autonomous stochastic 3D Brinkman-Forchheimer equations with delays. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3515-3537. doi: 10.3934/cpaa.2021117

[12]

Shu Wang, Mengmeng Si, Rong Yang. Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1621-1636. doi: 10.3934/cpaa.2022034

[13]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[14]

Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1907-1930. doi: 10.3934/cpaa.2021052

[15]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[16]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control and Related Fields, 2021, 11 (4) : 739-769. doi: 10.3934/mcrf.2020045

[17]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

[18]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445

[19]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323

[20]

Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control and Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (415)
  • HTML views (426)
  • Cited by (0)

Other articles
by authors

[Back to Top]