International Journal of Differential Equations

Stability of Optimal Controls for the Stationary Boussinesq Equations

Gennady Alekseev and Dmitry Tereshko

Full-text: Open access


The stationary Boussinesq equations describing the heat transfer in the viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking-type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.

Article information

Int. J. Differ. Equ., Volume 2011 (2011), Article ID 535736, 28 pages.

Received: 26 May 2011
Accepted: 3 August 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Alekseev, Gennady; Tereshko, Dmitry. Stability of Optimal Controls for the Stationary Boussinesq Equations. Int. J. Differ. Equ. 2011 (2011), Article ID 535736, 28 pages. doi:10.1155/2011/535736.

Export citation


  • M. D. Gunzburger, Perspectives in Flow Control and Optimization, vol. 5 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2003.
  • G. V. Alekseev and D. A. Tereshko, Analysis and Optimization in Hydrodynamics of Viscous Fluid, Dalnauka, Vladivostok, Russia, 2008.
  • T. T. Medjo, R. Temam, and M. Ziane, “Optimal and robust control of fluid flows: some theoretical and computational aspects,” Applied Mechanics Reviews, vol. 61, no. 1-6, pp. 0108021–01080223, 2008.
  • G. V. Alekseev, Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydro-dynamics, Nauchy Mir, Moscow, Russia, 2010.
  • F. Abergel and R. Temam, “On some control problems in fluid mechanics,” Theoretical and Computa-tional Fluid Dynamics, vol. 1, no. 6, pp. 303–325, 1990.
  • M. D. Gunzburger, L. S. Hou, and T. P. Svobodny, “Heating and cooling control of temperature distributions along boundaries of flow domains,” Journal of Mathematical Systems, Estimation, and Control, vol. 3, no. 2, pp. 147–172, 1993.
  • F. Abergel and E. Casas, “Some optimal control problems of multistate equations appearing in fluid mechanics,” Mathematical Modelling and Numerical Analysis, vol. 27, no. 2, pp. 223–247, 1993.
  • K. Ito, “Boundary temperature control for thermally coupled Navier-Stokes equations,” International Series of Numerical Mathematics, vol. 118, pp. 211–230, 1994.
  • M. D. Gunzburger, L. Hou, and T. P. Svobodny, “The Approximation of boundary control problems for fluid flows with an application to control by heating and cooling,” Computers & Fluids, vol. 22, pp. 239–251, 1993.
  • G. V. Alekseev, “Solvability of stationary problems of boundary control for thermal convection equations,” Siberian Mathematical Journal, vol. 39, pp. 844–585, 1998.
  • K. Ito and S. S. Ravindran, “Optimal control of thermally convected fluid flows,” SIAM Journal on Scientific Computing, vol. 19, no. 6, pp. 1847–1869, 1998.
  • A. Căpăţînă and R. Stavre, “A control problem in biconvective flow,” Journal of Mathematics of Kyoto University, vol. 37, no. 4, pp. 585–595, 1997.
  • H.-C. Lee and O. Yu. Imanuvilov, “Analysis of optimal control problems for the 2-D stationary Boussinesq equations,” Journal of Mathematical Analysis and Applications, vol. 242, no. 2, pp. 191–211, 2000.
  • G. V. Alekseev, “Solvability of inverse extremal problems for stationary equations of heat and mass transfer,” Siberian Mathematical Journal, vol. 42, pp. 811–827, 2001.
  • H. C. Lee, “Analysis and computational methods of Dirichlet boundary optimal control problems for 2D Boussinesq equations,” Advances in Computational Mathematics, vol. 19, no. 1–3, pp. 255–275, 2003.
  • G. V. Alekseev, “Coefficient inverse extremal problems for stationary heat and mass transfer equations,” Computational Mathematics and Mathematical Physics, vol. 47, no. 6, pp. 1055–1076, 2007.
  • G. V. Alekseev, “Inverse extremal problems for stationary equations in mass transfer theory,” Computational Mathematics and Mathematical Physics, vol. 42, no. 3, pp. 363–376, 2002.
  • G. V. Alekseev and D. A. Tereshko, “Boundary control problems for stationary equations of heat convection,” in New Directions in Mathematical Fluid Mechanics, A. V. Fursikov, G. P. Galdi, and V. V. Pukhnachev, Eds., pp. 1–21, Birkhäuser, Basel, Switzerland, 2010.
  • G. V. Alekseev and D. A. Tereshko, “Extremal boundary control problems for stationary equations of thermal convection,” Journal of Applied Mechanics and Technical Physics, vol. 51, no. 4, pp. 510–520, 2010.
  • V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Theory and Algorithm, Springer, Berlin, Germany, 1986.
  • A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, vol. 6 of Studies in Mathematics and its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1979.
  • J. Cea, Optimization, Theory and Algorithm, Springer, New York, NY, USA, 1978.