American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Vladimir Srochko, Vladimir Antonik, Elena Aksenyushkina. Sufficient optimality conditions for extremal controls based on functional increment formulas. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013
[2]

Franck Boyer, Víctor Hernández-Santamaría, Luz De Teresa. Insensitizing controls for a semilinear parabolic equation: A numerical approach. Mathematical Control & Related Fields, 2019, 9 (1) : 117-158. doi: 10.3934/mcrf.2019007
[3]

Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2020, 16 (1) : 55-70. doi: 10.3934/jimo.2018140
[4]

Sylvain Ervedoza, Enrique Zuazua. A systematic method for building smooth controls for smooth data. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1375-1401. doi: 10.3934/dcdsb.2010.14.1375
[5]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497
[6]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355
[7]

Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095
[8]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279
[9]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[10]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279
[11]

U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021032
[12]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[13]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[14]

Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
[15]

Sebastian Engel, Karl Kunisch. Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach. Mathematical Control & Related Fields, 2020, 10 (3) : 591-622. doi: 10.3934/mcrf.2020012
[16]

Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure & Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009
[17]

Zhenghuan Gao, Peihe Wang. Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete & Continuous Dynamical Systems, 2022, 42 (3) : 1201-1223. doi: 10.3934/dcds.2021152
[18]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[19]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[20]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

 Impact Factor: 

Tools

Metrics

  • PDF downloads (149)
  • HTML views (375)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy