On Chattering Solutions for the Maximum Principle Boundary-Value Problem in the Optimal Control Problem in Microeconomics

被引：1
作者：
Grigor'eva E.V. [1 ]
Khailov E.N. [2 ]
机构：
[1] Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX
[2] Lomonosov Moscow State University, Moscow
来源：
Computational Mathematics and Modeling | 2014年 / 25卷 / 2期
基金：
俄罗斯基础研究基金会;
关键词：
Maximum Principle; Optimal Control Problem; Switching Cost; Optimal Trajectory; Switch Point;
D O I：
10.1007/s10598-014-9216-3
中图分类号：
学科分类号：
摘要：
We consider the problem of optimal control of the maximization of income for the nonlinear mathematical model of microeconomical system used for the description of production, storage, and selling of consumer goods. For this model, we formulate the boundary-value problem of the maximum principle giving the necessary conditions of optimality. We consider conditions for the parameters characterizing the original system under which the solution of the boundary-value problem of maximum principle possesses a section of singular mode. It is shown that this section of singular mode neighbors with nonsingular sections containing containing infinitely many switchings on finite periods of time. For the investigation of this phenomenon, we use the theory developed by M. Zelikin and V. Borisov. It is shown that the solution of the boundary-value problem of maximum principle is locally optimal in the analyzed problem of optimal control. Hence, the corresponding optimal trajectory has three sections. The first part is a nonsingular section in which the original system with infinitely many switchings passes to the singular section for a finite period of time. The second part is the section of singular mode and the last (third) part is a section of the trajectory in which the analyzed system leaves the singular mode with more and more frequent switchings in the inverse time. © 2014 Springer Science+Business Media New York.
引用
收藏
页码：158 / 168
页数：10
相关论文
共 11 条
  • [1] Gorskii A.A., Kolpakova I.G., Lokshin B.Y., A dynamical model of the process of production, storage, and sale, Izv. RAN, Tekhn. Kibern, 1, pp. 190-193, (1992)
  • [2] Gorskii A.A., Kolpakova I.G., Lokshin B.Y., A dynamical model of the process of production, storage, and sale of a consumer good, Izv. RAN, Teor. Sist. Uprav, 1, pp. 144-148, (1998)
  • [3] Khailov E.N., Grigor'eva E.V., Reachability set of a nonlinear system on a plane, Vestnik MGU, Vychil. Matem. I Kibern, 4, pp. 27-32, (2001)
  • [4] Fleming W., Rishel R., Deterministic and Stochastic Optimal Control [Russian Translation], (1978)
  • [5] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F., Mathematical Theory of Optimal Processes [in Russian], (1983)
  • [6] Kelley H.J., Kopp R.E., Moyer H.S., Singular extremals, Topics in Optimization, pp. 63-103, (1967)
  • [7] Zelikin M.I., Borisov V.F., Chattering modes in optimal control problems, Trudy Mat. Inst. AN SSSR, 197, pp. 85-167, (1991)
  • [8] Zelikin M.I., Borisov V.F., Synthesis of optimal controls with switch-point accumulation, Itogi Nauki I Tekhniki, Ser. Sov. Mat. Pril., 90, pp. 5-189, (2002)
  • [9] Zelikin M.I., Borisov V.F., Singular optimal modes in problems of mathematical economics, Sov. Mat. Pril, 11, pp. 3-161, (2003)
  • [10] Stewart D.E., A numerical algorithm for optimal problems with switching costs, J. Australian Math. Soc., Ser. B, 34, pp. 212-228, (1992)
12