Abstract
The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Halsted, New York, 1977; Nauka, Moscow, 1986).
G. I. Marchuk, Mathematical Models in Environmental Problems (Nauka, Moscow, 1982; Elsevier, New York, 1986).
O. M. Alifanov, E. A. Artyukhin, and S. V. Rumyantsev, Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems (Nauka, Moscow, 1988; Begell House, New York, 1995).
K. Ito and K. Kunish, “Estimation of the convection coefficient in elliptic equations,” Inverse Probl. 14, 995–1013 (1997).
V. I. Agoshkov, F. P. Minuk, A. S. Rusakov, and V. B. Zalesny, “Study and solution of identification problems for nonstationary 2D- and 3D-convection–diffusion–reaction,” Russ. J. Numer. Anal. Math. Model. 20, 19–43 (2005).
G. V. Alekseev and E. A. Adomavichus, “Theoretical analysis of inverse extremal problems of admixture diffusion in viscous fluid,” J. Inverse Ill-Posed Probl. 9, 435–468 (2001).
P. A. Nguyen and J.-P. Raymond, “Control problems for convection-diffusion equations with control localized on manifolds,” ESAIM Control Optim. Calc. Var. 6, 467–488 (2001).
G. V. Alekseev, “Inverse extremal problems for stationary equations in mass transfer theory,” Comput. Math. Math. Phys. 42 (3), 363–376 (2002).
G. V. Alekseev, “Coefficient inverse extremum problems for stationary heat and mass transfer equations,” Comput. Math. Math. Phys. 47 (6), 1007–1028 (2007).
G. V. Alekseev, O. V. Soboleva, and D. A. Tereshko, “Identification problems for a steady-state model of mass transfer,” J. Appl. Mech. Tech. Phys. 49 (4), 537–547 (2008).
G. V. Alekseev and D. A. Tereshko, Analysis and Optimization in Viscous Fluid Dynamics (Dal’nauka, Vladivostok, 2008) [in Russian].
G. V. Alekseev and D. A. Tereshko, “Two-parameter extremum problems of boundary control for stationary thermal convection equations,” Comput. Math. Math. Phys. 51 (9), 1539–1557 (2011).
G. V. Alekseev, I. S. Vakhitov, and O. V. Soboleva, “Stability estimates in identification problems for the convection–diffusion–reaction equation,” Comput. Math. Math. Phys. 52 (12), 1635–1649 (2012).
A. E. Kovtanyuk, A. Y. Chebotarev, N. D. Botkin, and K.-H. Hoffmann, “Theoretical analysis of an optimal control problem of conductive–convective–radiative heat transfer,” J. Math. Anal. Appl. 412 (1), 520–528 (2014).
A. Yu. Chebotarev, “Extremal boundary value problems of the dynamics of a viscous incompressible fluid,” Sib. Math. J. 34 (5), 972–983 (1993).
G. V. Alekseev, “Control problems for stationary equations of magnetic hydrodynamics,” Dokl. Math. 69, 310–313 (2004).
V. V. Penenko, “Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment,” Numer. Anal. Appl. 2, 341–351 (2009).
E. V. Dement’eva, E. D. Karepova, and V. V. Shaidurov, “Recovery of a boundary function from observation data for the surface wave propagation problem in an open basin,” Sib. Zh. Ind. Mat. 16 (1), 10–20 (2013).
A. I. Korotkii and D. A. Kovtunov, “Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid,” Proc. Steklov Inst. Math. 255, suppl. 2, 81–92 (2006).
R. V. Brizitskii and Zh. Yu. Saritskaya, “Boundary value and extremal problems for the nonlinear convection–diffusion–reaction equation,” Sib. Elektron. Mat. Izv. 12, 447–456 (2015).
G. V. Alekseev, R. V. Brizitskii, and Zh. Yu. Saritskaya, “Stability estimates of solutions to extremal problems for nonlinear convection–diffusion–reaction equation,” J. Appl. Ind. Math. 10 (3), 155–167 (2016).
A. E. Kovtanyuk, A. Yu. Chebotarev, N. D. Botkin, and K.-H. Hoffmann, “Unique solvability of a steady-state complex heat transfer model,” Commun. Nonlinear Sci. Numer. Simul. 20, 776–784 (2015).
A. E. Kovtanyuk and A. Yu. Chebotarev, “Steady-state problem of complex heat transfer,” Comput. Math. Math. Phys. 54 (4), 719–726 (2014).
A. E. Kovtanyuk and A. Yu. Chebotarev, “Stationary free convection problem with radiative heat exchange,” Differ. Equations 50 (12), 1592–1599 (2014).
A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications (Nauchnaya Kniga, Novosibirsk, 1973; Am. Math. Soc., Providence, R.I., 2000).
J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineires (Dunod, Paris, 1969; Mir, Moscow, 1972).
A. Kufner and S. Fucik, Nonlinear Differential Equations (Elsevier, Amsterdam 1980; Nauka, Moscow, 1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © R.V. Brizitskii, Zh.Yu. Saritskaya, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 12, pp. 2042–2053.
Rights and permissions
About this article
Cite this article
Brizitskii, R.V., Saritskaya, Z.Y. Stability of solutions to extremum problems for the nonlinear convection–diffusion–reaction equation with the Dirichlet condition. Comput. Math. and Math. Phys. 56, 2011–2022 (2016). https://doi.org/10.1134/S096554251612006X
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S096554251612006X