Stability of optimal controls for the stationary Boussinesq equations. (English) Zbl 1236.49064
Summary: 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.
MSC:
| 49K40 | Sensitivity, stability, well-posedness |
| 35Q93 | PDEs in connection with control and optimization |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
local stability of optimal controls; stationary Boussinesq equations; inhomogeneous Dirichlet boundary conditions; tracking-type functionalsReferences:
| [1] | 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. · Zbl 1088.93001 · doi:10.1137/1.9780898718720 |
| [2] | G. V. Alekseev and D. A. Tereshko, Analysis and Optimization in Hydrodynamics of Viscous Fluid, Dalnauka, Vladivostok, Russia, 2008. · Zbl 1272.76075 |
| [3] | 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. · Zbl 1145.76341 · doi:10.1115/1.2830523 |
| [4] | G. V. Alekseev, Optimization in Stationary Problems of Heat and Mass Transfer and Magnetic Hydrodynamics, Nauchy Mir, Moscow, Russia, 2010. |
| [5] | F. Abergel and R. Temam, “On some control problems in fluid mechanics,” Theoretical and Computational Fluid Dynamics, vol. 1, no. 6, pp. 303-325, 1990. · Zbl 0708.76106 · doi:10.1007/BF00271794 |
| [6] | 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. |
| [7] | 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. · Zbl 0769.49002 |
| [8] | K. Ito, “Boundary temperature control for thermally coupled Navier-Stokes equations,” International Series of Numerical Mathematics, vol. 118, pp. 211-230, 1994. · Zbl 0813.76018 |
| [9] | 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. · Zbl 0778.76081 · doi:10.1016/0045-7930(93)90056-F |
| [10] | G. V. Alekseev, “Solvability of stationary problems of boundary control for thermal convection equations,” Siberian Mathematical Journal, vol. 39, pp. 844-585, 1998. · Zbl 0919.49004 · doi:10.1007/BF02672906 |
| [11] | 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. · Zbl 0918.49004 · doi:10.1137/S1064827596299731 |
| [12] | A. C\uap\ua\ua and R. Stavre, “A control problem in biconvective flow,” Journal of Mathematics of Kyoto University, vol. 37, no. 4, pp. 585-595, 1997. · Zbl 0937.76098 |
| [13] | 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. · Zbl 0951.49006 · doi:10.1006/jmaa.1999.6651 |
| [14] | 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. · Zbl 0998.35042 · doi:10.1023/A:1011940606843 |
| [15] | 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. · Zbl 1031.35118 · doi:10.1023/A:1022872602498 |
| [16] | 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. · Zbl 1292.76031 · doi:10.1134/S0965542507060115 |
| [17] | 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. · Zbl 1055.80001 |
| [18] | 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. · Zbl 1195.35249 |
| [19] | 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. · Zbl 1272.76231 · doi:10.1007/s10808-010-0067-1 |
| [20] | V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Theory and Algorithm, Springer, Berlin, Germany, 1986. · Zbl 0585.65077 |
| [21] | 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. · Zbl 0407.90051 |
| [22] | J. Cea, Optimization, Theory and Algorithm, Springer, New York, NY, USA, 1978. |
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