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An approximate method for solving optimal control problems for discrete systems based on local approximation of an attainability set
The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 75-88 Cet article a éte moissonné depuis la source Math-Net.Ru

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An optimal control problem for discrete systems is considered. A method of successive improvements along with its modernization based on the expansion of the main structures of the core algorithm about the parameter is suggested. The idea of the method is based on local approximation of attainability set, which is described by the zeros of the Bellman function in the special problem of optimal control. The essence of the problem is as follows: from the end point of the phase is required to find a path that minimizes functional deviations of the norm from the initial state. If the initial point belongs to the attainability set of the original controlled system, the value of the Bellman function equal to zero, otherwise the value of the Bellman function is greater than zero. For this special task Bellman equation is considered. The support approximation and Bellman equation are selected. The Bellman function is approximated by quadratic terms. Along the allowable trajectory, this approximation gives nothing, because Bellman function and its expansion coefficients are zero. We used a special trick: an additional variable is introduced, which characterizes the degree of deviation of the system from the initial state, thus it is obtained expanded original chain. For the new variable initial nonzero conditions is selected, thus obtained trajectory is lying outside attainability set and relevant Bellman function is greater than zero, which allows it to hold a non-trivial approximation. As a result of these procedures algorithms of successive improvements is designed. Conditions for relaxation algorithms and conditions for the necessary conditions of optimality are also obtained.
Keywords: discrete systems, optimal control, attainability set, improvement method. 
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V. A. Baturin. An approximate method for solving optimal control problems for discrete systems based on local approximation of an attainability set. The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 75-88. http://geodesic.mathdoc.fr/item/IIGUM_2017_19_a5/

[1] Baturin V. A., Goncharova E. V., “An improvement method based on an approximate representation of the attainable set. A relaxation theorem”, Avtomat. i Telemekh., 1999, no. 11, 19–29 (in Russian)

[2] Goncharova E. V., Baturin V. A., “An iterative method for solving discrete problems of optimal control”, Computational Technologies, 8 (2003), 269–275 (in Russian)

[3] Gurman V. I., Konstantinov G. N., Attainability sets of control systems. The relation with the Bellman equation, No 4038-81, All-Russian Institute for Scientific and Technical Information, 1981 (in Russian)

[4] Gurman V. I., Baturin V. A., Improved algorithm based on estimates for attainability set, No 651-85, All-Russian Institute for Scientific and Technical Information, 1985 (in Russian)

[5] Gurman V. I., The Extension Principle in Control Problems, 2nd edition, Fizmatlit, M., 1997 (in Russian)

[6] Gurman V. I., Rasina I. V., “Sufficient optimality conditions in hierarchical models of nonuniform systems”, Automation and Remote Control, 74:12 (2013), 1935–1947 | DOI | MR | Zbl

[7] Gurman V. I., Rasina I. V., Fesko O. V., Usenko O. V., “Methods for improving the control of hierarchical models of systems with network structure”, Izv. Irkutsk. Gos. Univ. Ser. Mat., 8 (2014), 71–85 (in Russian)

[8] Konstantinov G. N., Rationing effects on dynamic systems, Irkutsk State Univ., Irkutsk, 1983 (in Russian)

[9] Krotov V. F., Bulatov A. V., Baturina O. V., “Optimization of linear systems with controllable coefficients”, Automation and Remote Control, 72:6 (2011), 1199–1212 | DOI | MR | Zbl

[10] Lotov A. V., “On the concept of generalized attainability sets and their construction for a linear control system”, Doklady Academy of Sciences, 1980, no. 5, 1081–1083 (in Russian) | Zbl

[11] V. Baturin, E. Goncharova, “An Optimal Control Algorithm Based on Reachability Set Approximation and Linearization”, Autom. Remote Control., 63:7 (2002), 1043–1050 | DOI | MR | Zbl

[12] T. Pescvardi, K. S. Arenda, “Reachable sets for linear dynamic systems”, Inform. and Control., 19:4 (1971), 319–344 | DOI | MR

[13] R. Vinter, “A characterization of the reachable set for nonlinear control systems”, Siam J. Contr. and Optim., 18:6 (1980), 599–610 | DOI | MR | Zbl