Multiplicative Optimal Control Problem
DOI:
https://doi.org/10.5564/jasea.v4i1.2684Keywords:
Pontriyagin’s principle, quasiconcave, quasiconvexAbstract
In this paper, we consider a multiplicative optimal control problem subject to a system of linear differential equation.It has been shown that product of two concave functions defined positively over a feasible set is quasiconcave. It allows us to consider the original problem from a view point of quasiconvex maximization theory and algorithm. Global optimality conditions use level set of the objective function and convex programming as subproblem. The objective function is product of two concave functions. We consider minimization of the objective functional. The problem is nonconvex optimal control and application of Pontriyagin’s principle does not always guarantee finding a global optimal control. Based on global optimality conditions, we develop an algorithm for solving the minimization problem globally.
Downloads
146
References
F. L. Chernous’ko, Otsenivaniye fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov [Estimation of the phase state of dynamic systems. Ellipsoid method], M.: Nauka[ Nauka, Moscow], 1988. [In Russian]
T. Bayartugs and R. Enkhbat, On the maximization of product of two concave functions, Journal of Mongolian Mathematical Society, 18 (2014), pp. 23-26.
T. Bayartugs and R. Enkhbat, On the minimization of product of two concave functions, Journal of Mongolian Mathematical Society, 17 (2013), pp. 50-55.
R. Enkhbat, Quasiconvex Programming and Its Applications, Lambert Publisher, 2009.
R. Enkhbat and T. Ibaraki, On the maximization and minimization of a quasiconvex function, Journal of Nonlinear and Convex Analysis, (2003), pp. 43-76.
R. Enkhbat Rentsen, J. Zhou and K. L. Teo, A global optimization approach to fractional optimal control, Journal of Industrial and Management Optimization, 12-1 (2016), pp. 73-82. https://doi.org/10.3934/jimo.2016.12.73
Yu.G. Evtushenko and M.A. Posypkin, Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy, Computational Mathematics and Mathematical Physics, 53 (2013), pp. 144–157. https://doi.org/10.1134/S0965542513020061
R. Gabasov, N. M. Dmitruk and F. M. Kirillova, The indirect optimal control of dynamical systems, Comput. Math. Math. Phys., 44-3 (2004), pp. 418–439.
A. Y. Gornov, Vychislitel’nyye tekhnologii resheniya zadach optimal’nogo upravleniya [Computing technologies for solving optimal control problems], Rossiyskaya akad. nauk, Sibirskoye otd-niye, In-t dinamiki sistem i teorii upravleniya, Novosibirsk : Nauka [Russian acad. Sciences, Siberian Branch, Institute of System Dynamics and Control Theory. -Novosibirsk: Science],2009. [In Russian]
A. Y. Gornov, T. S. Zarodnyuk, E. A. Finkelstein et al., The method of uniform monotonous approximation of the reachable set border for a controllable system, J Glob Optim 66 (2016), pp. 53–64. https://doi.org/10.1007/s10898-015-0346-8
R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, Berlin, 1993.
P. P. Khargonekar and M. A. Rotea, Multiple objective optimal control of linear systems: the quadratic norm case, IEEE Transactions on Automatic Control, 36-1 (1991), pp. 14-24. https://doi.org/10.1109/9.62264
I. Bykadorov, A. Ellero, S. Funari et al., Dinkelbach approach to solving a class of fractional optimal control problems, J Optim Theory Appl, 142 (2009), pp. 55–66. https://doi.org/10.1007/s10957-009-9540-5
Jung-Fa Tsai, Global optimization of nonlinear fractional programming problems in engineering design, Engineering Optimization , 37-4 (2005), pp. 399-409. https://doi.org/10.1080/03052150500066737
B. Kheirfam, Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming, Journal of Industrial and Management Optimization, 6-2 (2010), pp. 347 - 361. https://doi.org/10.3934/jimo.2010.6.347
O. L. Mangosarian, Non-linear programming, McGaw-Hill, New York, 1969.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical theory of optimal processes, Nauka, Moscow, 1976.
S. Schaible, Fractional programming: Applications and algorithms, European Journal of Operational Research, 7 (1981), pp. 111-120. https://doi.org/10.1016/0377-2217(81)90272-1
K.L. Teo, C.J. Goh and K.H. Wong, A Unified Computational Approach to Optimal Control Problems, 1st Edition, Longman Scientific and Technical, New York, 1991.
F. P. Vasiliev, Metody resheniya ekstremal’nykh zadach [Methods for solving extremal problems], Nauka, Moskva [Nauka, Moscow] 1981. [In Russian]
W. E. Zahgwill, Non-linear Programming: A unified approach, Prentice-Hall, Englewood Gliffs, New York, 1960.
A. Zhang and S. Hayashi, Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints, Journal of Industrial and Management Optimization, 1-1 (2011), pp. 83 - 98. https://doi.org/10.3934/naco.2011.1.83
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Enkhbat Rentsen, Bayartugs Tamjav, Ulziibayar Vandandoo,
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright on any research article in the Journal of Applied Science and Engineering A is retained by the author(s).
The authors grant the Journal of Applied Science and Engineering A a license to publish the article and identify itself as the original publisher.
Articles in the Journal of Applied Science and Engineering A are Open Access articles published under a Creative Commons Attribution 4.0 International License CC BY.
This license permits use, distribution and reproduction in any medium, provided the original work is properly cited.