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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

Alexander Arguchintsev and  Vasilisa Poplevko ,

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.

Keywords: Parabolic equation, smooth controls, increment formula for functional, necessary optimality condition, numerical method.
Mathematics Subject Classification: Primary: 49J20, 49M05; Secondary: 49K20.
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. Google Scholar

[2]

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

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

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

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

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

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

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

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

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

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

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