Abstract
Minimizing sequences for degenerate optimal control problems are constructed from turnpike solutions. Two variational approximation schemes for the turnpike solution are described: the first is a direct improvement of the simple approximation of piecewise-continuous turnpikes by the solutions of the initial differential system and the second consists of constructing an approximate optimal control in the neighborhood of a turnpike by a parametric curve in the state space. Since a sequence of state-linear feedback controls with variable coefficients is generated in both variants, turnpikes can be easily realized in practice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Optimal Control Problems), Moscow: Nauka, 1977.
Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.
Gurman, V.I., Printsip rasshireniya v zadachakh optimal’nogo upravleniya (The Extension Principle in Optimal Control Problems), Moscow: Nauka, 1997.
Gurman, V.I., Turnpike Solutions in the Procedures Seeking Optimal Controls, Avtom. Telemekh., 2003, no. 3, pp. 61–71.
Gurman, V.I. and Ukhin, M.Yu., The Extension Principle in Control Problems. Constructive Methods and Applied Problems, Moscow: Fizmatlit, 2005.
Tikhonov, A.N. and Arsenin, V.Ya., Metody resheniya nekorrektnykh zadach (Solution Methods for Ill-defined Problems), Moscow: Nauka, 1986.
Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie metody v teorii singulyarnykh vozmushchenii (Asymptotic Methods in Singular Perturbation Theory), Moscow: Vysshaya Shkola, 1990.
Baturin, V.A. and Urbanovich, D.E., Priblizhennye metody optimal’nogo upravleniya, osnovannye na printsipe rasshireniya (Approximate Optimal Control Methods based on the Extension Principle), Novosibirsk: Nauka, 1997.
Gaishun, I.V., Vpolne razreshimye mnogomernye differentsial’nye uravneniya (Completely Solvable Multidimensional Differential Equations), Minsk: Nauka i Tekhnika, 1983.
Dmitriev, M.G. and Ni Min Quang, Contrast Structures in a Simple Vector Variational Problem and Their Asymptotec Behavior, Avtom. Telemekh., 1998, no. 5, pp. 41–52.
Miller, B.M., Optimization of Dynamic Systems with Generalized Control, Avtom. Telemekh., 1989, no. 6, pp. 23–35.
Miller, B.M., Generalized Solutions of Nonlinear Optimization Problems with Impulse Controls, Avtom. Telemekh., 1995, no. 4, pp. 81–95.
Author information
Authors and Affiliations
Additional information
Original Russian Text © Ni Min Quang, M. Yu. Ukhin, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 54–60.
This work was supported by the Russian Foundation for Basic Research, project no. 05-01-00260 and the Shanghai Municipal Committee for Education, project no. E03004.
This paper was recommended for publication by B.M. Miller, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Quang, N.M., Ukhin, M.Y. Turnpike solutions of optimal control problems. Autom Remote Control 67, 880–886 (2006). https://doi.org/10.1134/S000511790606004X
Received:
Issue Date:
DOI: https://doi.org/10.1134/S000511790606004X