Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 13
175
Views
0
CrossRef citations to date
0
Altmetric
Research Article

First-order necessary conditions of optimality for the optimal control of two-dimensional convective Brinkman–Forchheimer equations with state constraints

ORCID Icon
Pages 3861-3907 | Received 03 Jul 2020, Accepted 16 Apr 2021, Published online: 13 May 2021

References

  • Sharma AK, Khandelwal MK, Bera P. Finite amplitude analysis of non-isothermal parallel flow in avertical channel filled with high permeable porous medium. J Fluid Mech. 2018;857:469–507.
  • Hajduk KW, Robinson JC. Energy equality for the 3D critical convective Brinkman–Forchheimer equations. J Differ Equ. 2017;263:7141–7161.
  • Antontsev SN, de Oliveira HB. The Navier–Stokes problem modified by an absorption term. Appl Anal. 2010;89(12):1805–1825.
  • Kalantarov VK, Zelik S. Smooth attractors for the Brinkman–Forchheimer equations with fast growing nonlinearities. Commun Pure Appl Anal. 2012;11:2037–2054.
  • Fefferman CL, Hajduk KW, Robinson JC. Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces. Available from: https://arxiv.org/abs/1904.03337.
  • Mohan MT. On the convective Brinkman–Forchheimer equations. Submitted.
  • Abergel F, Temam R. On some control problems in fluid mechanics. Theor Comput Fluid Dyn. 1990;1:303–325.
  • Barbu V. Optimal control of variational inequalities. Boston: Pitman; 1984. (Pitman research notes in mathematics series; vol. 100).
  • Doboszczak S, Mohan MT, Sritharan SS. Pontryagin maximum principle for the optimal control of linearized compressible Navier–Stokes equations with state constraints. Published online in Evol Equ Control Theory. 2020. Available from: https://doi.org/10.3934/eect.2020110.
  • Fursikov AV. Optimal control of distributed systems: theory and applications. Rhode Island: American Mathematical Society; 2000.
  • Gunzburger MD. Perspectives in flow control and optimization, Advances in Design and Control, 5. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM); 2003.
  • Lions J-L. Optimal control of systems governed by partial differential equations. New York-Berlin: Springer-Verlag; 1971.
  • Raymond JP. Optimal control of partial differential equations. Université Paul Sabatier; 2013. (Lecture notes). Available from: https://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf
  • Sritharan SS. Optimal control of viscous flow. Philadelphia: Society for Industrial and Applied Mathematics; 1998. (SIAM frontiers in applied mathematics).
  • Barbu V. Optimal control of Navier–Stokes equations with periodic inputs. Nonlinear Anal Theory Methods Appl. 1998;31(1–2):15–31.
  • Casas E. An optimal control problem governed by the evolution Navier–Stokes equations. In: Sritharan SS, editor. Optimal control of viscous flow. Philadelphia: SIAM; 1998. p. 79–95.
  • Casas E, Chrysafinos K. Analysis of the velocity tracking control problem for the 3D evolutionary Navier–Stokes equations. SIAM J Control Optim. 2016;54(1):99–128.
  • Fattorini HO, Sritharan SS. Necessary and sufficient conditions for optimal controls in viscous flow problems. Proc R Soc Edinb A. 1994;124:211–251.
  • Tröltzsch F, Wachsmuth D. Second-order sufficient optimality conditions for the optimal control of Navier–Stokes equations. ESAIM Control Optim Calc Var. 2006;12(1):93–119.
  • Wang G. Optimal controls of 3 dimensional Navier–Stokes equations with state constraints. SIAM J Control Optim. 2002;41(2):583–606.
  • Wang G, Wang L. Maximum principle of state-constrained optimal control governed by fluid dynamic systems. Nonlinear Anal Theory Methods Appl. 2003;52(8):1911–1931.
  • Mohan MT. Optimal control problems governed by the two dimensional convective Brinkman–Forchheimer equations. Published online in Evol Equ Control Theory. 2020. Available from: https://doi.org/10.3934/eect.2021020.
  • Mohan MT. The time optimal control of two dimensional convective Brinkman–Forchheimer equations. Published online in Appl Math Optim. 2020. Available from: https://doi.org/10.1007/s00245-021-09748-w.
  • Tachim-Medjo T. Optimal control of a two-phase flow model with state constraints. Math Control Relat Fields. 2016;6(2):335–362.
  • Temam R. Navier–Stokes equations and nonlinear functional analysis. Philadelphia: SIAM; 1983.
  • Temam R. Navier–Stokes equations, theory and numerical analysis. Amsterdam: North-Holland; 1984.
  • Fujiwara D, Morimoto H. An Lr-theorem of the Helmholtz decomposition of vector fields. J Fac Sci Univ Tokyo Sect IA Math. 1977;24:685–700.
  • Ladyzhenskaya OA. The mathematical theory of viscous incompressible flow. New York: Gordon and Breach; 1969.
  • Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa. 1959;3(13):115–162.
  • Agmon S. Lectures on elliptic boundary value problems. Providence (RI): AMS Chelsea Publishing; 1965.
  • Ciarlet PG. Linear and nonlinear functional analysis with applications. Philadelphia: SIAM; 2013.
  • Barbu V. Analysis and control of nonlinear infinite dimensional systems. Boston (MA): Academic Press Inc.; 1993. (Mathematics in science and engineering, vol. 190).
  • Mohan MT. Deterministic and stochastic equations of motion arising in Oldroyd fluids of order one: existence, uniqueness, exponential stability and invariant measures. Stoch Anal Appl. 2020;38(1):1–61.
  • Mohan MT. Stochastic convective Brinkman–Forchheimer equations. Submitted. Available from: https://arxiv.org/abs/2007.09376.
  • Fernando BPW, Sritharan SS. Nonlinear filtering of stochastic Navier–Stokes equation with Lévy noise. Stoch Anal Appl. 2013;31:1–46.
  • Li X, Yong J. Optimal control theory for infinite dimensional systems. Boston: Birkhäuser; 1995.
  • Barbu V, Pavel N. Flow-invariance closed set with respect to nonlinear semigroup flows. Nonlinear Differ Equ Appl. 2003;10:57–72.
  • Simon J. Compact sets in the space Lp(0,T;B). Ann Mat Pura ed Appl. 1986;146:65–96.
  • Mitrinović DS, Pecarić JE, Fink AM. Inequalities involving functions and their integrals and derivatives. Dordrecht: Kluwer Academic Publishers Group; 1991. (Mathematics and its applications, vol. 53).
  • Vishik MI, Fursikov AV. Mathematical problems of statistical hydromechanics. Dordrecht: Kluwer Academic Publishers Group; 1988.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.