June 2018, 8(2): 193-202. doi: 10.3934/naco.2018011

An optimal control problem by parabolic equation with boundary smooth control and an integral constraint

Institute of Mathematics, Economics and Computer Science, Irkutsk State University, Irkutsk, Russia

* Corresponding author: Vasilisa Poplevko

This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26, 2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey

Received  November 2016 Revised  May 2017 Published  May 2018

Fund Project: This research is partially supported by the Russian Foundation for Basic Research, grant No. 14-0100564

In the paper, we consider an optimal control problem by differential boundary condition of parabolic equation. We study this problem in the class of smooth controls satisfying certain integral constraints. For the problem under consideration we obtain a necessary optimality condition and propose a method for improving admissible controls. For illustration, we solve one numerical example to show the effectiveness of the proposed method.

Citation: Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
References:
[1]

A. Arguchintsev, Optimal Control by Hyperbolic Systems, Fizmatlit, Moscow, 2007.

[2]

A. Arguchintsev, Optimal Control by Initial Boundary Condition of Hyperbolic Systems, Irkutsk State University, Irkutsk, 2003.

[3]

A. Butkovsky, Theory of Optimal Control by Distributed Systems, Nauka, Moscow, 1965.

[4]

A. Butkovsky, Methods of Control by Systems with Distributed Parameters, Nauka, Moscow, 1975.

[5]

N. Demidenko, Modelling and Optimization by Systems with Distributed Parametres, Nauka, Novosibirsk, 2006.

[6]

N. Demidenko, Control Distributed Systems, Nauka, Novosibirsk, 1999.

[7]

A. Egorov, Optimal Control by Thermal Processes and Diffusion, Nauka, Moscow, 1978.

[8]

J.-L. Lions, Optimal Control by Systems Described by Partial Differential ations, Mir, Moscow, 1972.

[9]

J.-L. Lions, Control by Singular Distributed Systems, Nauka, Moscow, 1987.

[10]

P. Neittaanmaki, Optimal Control of Nonlinear Parabolic Systems, Marcel Dekker Inc, New York - Basel - Hong Kong, 1994.

[11]

A. Tikhonov, Equations of Mathematical Physics, Nauka, Moscow, 1977.

[12]

F. Vasiliev, Methods of Optimization, Factorial Press, Moscow, 2002.

show all references

References:
[1]

A. Arguchintsev, Optimal Control by Hyperbolic Systems, Fizmatlit, Moscow, 2007.

[2]

A. Arguchintsev, Optimal Control by Initial Boundary Condition of Hyperbolic Systems, Irkutsk State University, Irkutsk, 2003.

[3]

A. Butkovsky, Theory of Optimal Control by Distributed Systems, Nauka, Moscow, 1965.

[4]

A. Butkovsky, Methods of Control by Systems with Distributed Parameters, Nauka, Moscow, 1975.

[5]

N. Demidenko, Modelling and Optimization by Systems with Distributed Parametres, Nauka, Novosibirsk, 2006.

[6]

N. Demidenko, Control Distributed Systems, Nauka, Novosibirsk, 1999.

[7]

A. Egorov, Optimal Control by Thermal Processes and Diffusion, Nauka, Moscow, 1978.

[8]

J.-L. Lions, Optimal Control by Systems Described by Partial Differential ations, Mir, Moscow, 1972.

[9]

J.-L. Lions, Control by Singular Distributed Systems, Nauka, Moscow, 1987.

[10]

P. Neittaanmaki, Optimal Control of Nonlinear Parabolic Systems, Marcel Dekker Inc, New York - Basel - Hong Kong, 1994.

[11]

A. Tikhonov, Equations of Mathematical Physics, Nauka, Moscow, 1977.

[12]

F. Vasiliev, Methods of Optimization, Factorial Press, Moscow, 2002.

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