On certain problems in optimal control theory

V. I. Gurman

The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 26-43

Résumé

Traditionally, in mathematical optimal control theory a significant part in the studies is related to investigation methods of ready mathematical problem settings with various complexity. Since all attention is centered on problem solution then the question of adequateness of used mathematical model to the real process usually is not under consideration. The mathematical problem settings itself in terms of their practical content and application of obtained solutions in general is out of the question; this aggravates mutual understanding between theorists and empirics which leads to depreciation of their results. Standard assumptions for variables in the mathematical problem setting obtained as the result of a certain idealization of the object’s real properties do not correspond to behavior and the features of the object for which the mathematical model was constructed. Mathematical idealization of the object’s real properties actually is simplification of the real problem for its effective investigation using mathematical methods. In this light the solution of a formal problem is an approximate solution of a real problem. The borders between real and formal problems are conditional as the borders between more detailed and less detailed models of the real object. The question now arises of whether it possible to simplify ready formal problem on the principle of elimination of passive differential constraint used in the theory of degenerate problems? One of the answers is the turnpike approach to the search for an approximate globally optimal solutions. However, consideration of such methodological problems dealing with a foundation of the optimal control theory usually not paid enough attention in literature. The purpose of this work is to attract attention of researchers to these problems and offer a solution to one of them. In this work, we show that the optimal control problem for differential system in a general form doesn't have a solution in the classical sense in respect to the object's real model taking into account the continuity of proceeding processes in it but traditional solution may be considered as an approximate solution of the turnpike type with less order for the real problem. The existence problem for solutions in variational calculus and optimal control theory is encompassed. We give both methodical and meaningful examples.

Détail

Keywords: mathematical model, formal problem, real problem, approximate turnpike solution.

Bibliographie

[1] Gurman V. I., Rasina I. V., Guseva I. S., “Preobrazovanija differencial'nyh upravljaemyh sistem dlja poiska priblizhenno-optimal'nogo upravlenija”, Program. Sist.: Teor. i Prilozheniya, 5:4(22) (2014), 123–157

[2] Gurman V. I., Trushkova E. A., Practical methods of optimization, University of Pereslavl, Pereslavl-Zalessky, 2009, 160 pp.

[3] V. I. Gurman, Degenerate Problems of Optimal Control, Nauka, M., 1977 | MR | Zbl

[4] V. I. Gurman, “Turnpike Solutions in the Procedures Seeking Optimal Controls”, Automation and Remote Control, 64:3 (2003), 399–408 | DOI | MR | Zbl

[5] V. I. Gurman, Models of Natural Resource Control, Nauka, M., 1981

[6] V. I. Gurman, I. V. Rasina, O. V. Fesko, I. S. Guseva, “On Certain Approaches to Optimization of Control Processes. I”, Automation and Remote Control, 77:8 (2016), 1370–1385 | DOI | MR | Zbl

[7] V. I. Gurman, I. V. Rasina, O. V. Fesko, I. S. Guseva, “On Certain Approaches to Optimization of Control Processes. II”, Automation and Remote Control, 77:9 (2016), 1544–1556 | DOI | Zbl

[8] V. I. Gurman, M. Y. Ukhin, “Turnpike Solutions in Optimization of Regional Development Strategies”, Automation and Remote Control, 65:4 (2004), 603–611 | DOI | Zbl

[9] M. I. Zelikin, V. F. Borisov, “Optimal Chattering Feedback Control”, Journal of Mathematical Sciences, 114:3 (2003), 1227–1344 | DOI | MR

[10] V. F. Krotov, V. I. Gurman, Methods and Problems of Optimal Control, Nauka, M., 1973 | MR

[11] V. F. Krotov, V. Z. Bukreev, V. I. Gurman, New Methods of Variational Calculus in the Flight Dynamics, Mashinostroenie, M., 1969

[12] A. M. Letov, “Analytic Controller Design. II”, Automation and Remote Control, 21:5 (1960), 389–393 | Zbl

[13] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Nauka, M., 1983, 392 pp. | MR | Zbl

[14] L. Stoleryu, Equilibrium and Economical Growth, Statistics, M., 1974, 474 pp.

[15] V. I. Gurman, The Extension Principle in the Problems of Sustainable Development, Fizmatlit, M., 2005, 128 pp. | MR

[16] V. I. Gurman, L. N. Nikiforova, “Principle of Expansion in Modeling and Optimal Control: Case Study for Helicopter Maneuvering”, Proceeding of 10-th Workshop IFAC (Haifa, Izrael, 19–21 December, 1995), Elsevier, London, 1996

[17] L. McKenzie, “Turnpike Theory”, Econometrica, 44, Nov. (1976), 841–866 | DOI | MR

[18] P. A. Samuelson, “A Catenary Turnpike Theorem Involving Consumption and the Golden Rule”, American Economic Review, 55 (1965), 486–496

[19] P. A. Samuelson, “Maximum Principles in Analytical Economics”, American Economic Review, 62:3, June (1972), 249–262

[20] E. Trelat, “Optimal control and Applications to Aerospace: Some Results and Challenges”, Journal of Optimization Theory and Applications, 154:3 (2012), 713–758 | DOI | MR | Zbl

[21] E. Trelat, E. Zuazua, “The Turnpike Property in Finite-Dimensional Nonlinear Optimal Control”, J. Differential Equations, 258 ( 2015), 81–114 | DOI | MR | Zbl

[22] A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2005, 407 pp. | MR