Discrete nonuniform systems and sufficient conditions of optimality

I. V. Rasina

The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 62-74

Résumé

Nonuniform systems are the object of a deep investigation last 15–20 years. They are exemplified by the chemical processes, complicated operations in space, robot dynamics, development of organisms and biological populations. A significant part in the studies of nonuniform systems is related to control optimization problems, when optimal control methods for uniform systems that have already become classical (the Pontryagin's maximum principle, Bellman scheme) cannot be directly applied. On one hand, for this class of optimization problems it is required a mathematical model that takes into account the object's properties, on the other hand, the mathematical apparatus that lets one find solution of the problem. Obviously, there were many researchers who aimed their efforts at modification and refinement of the Pontryagin's maximum principle for this class of problems adding special conditions at the moments of changing description of the system (for example, so called jump conditions). Another approach is related to the Lyapunov vector-function. Some authors use hybrid technique when continuous and discrete components are used for description and control. Besides, some schools actively use in their research the measure theory, generalized functions, and discontinuity time change method. In this work, we propose an alternative approach under traditional assumptions of the optimal control theory. It is based on sufficient optimality conditions of V.F. Krotov for discrete systems set down in terms of arbitrary sets and maps. The proposed specification let us consider sets and maps with variable structure from one step to another, at each stage the control is treated as a combination of some abstract variable and some continuous or discrete process. We consider a class of discrete nonuniform systems which are widespread in practice (economics, ecology). Such systems also arise in process of numerical solution of optimization problems obtained after discretization of continuous controllable systems. For this class a counterpart of Krotov's sufficient conditions is proposed. They are formulated in the Bellman-type form as well. Their specification for linear and liear-quadratic systems w.r.t. state is given.

Détail

Keywords: nonuniform controllable discrete systems, sufficient conditions of optimality, linear and linear-quadratic nonuniform systems.

Bibliographie

[1] Bortakovskij A. S., “Sufficient Conditions of Control Optimality for Determined Logical Dynamic Systems”, Informatika. Ser. Avtomatizacija proektirovanija, 1992, no. 2–3, 72–79

[2] Vasil'ev S. N., “Theory and Application for Logical Controllable Systems”, Trudy. 2-ja Mezhdunarodnaja konferencija «Identifikacija sistem i zadachi upravlenija», SICPRO'03 (Moscow, 2003), 23–52

[3] Rasina I. V., Hierarchical Control Models for Systems with Nonuniform Structure, Fizmatlit, M., 2014, 160 pp.

[4] Fesko O. V., “Optimization of Dynamic Systems with Piecewise Constant Control”, Naukoemkie informacionnye tehnologii, Tr. molodezhnoj nauch.-prakt. konf. (Pereslavl'-Zalesskij, 2009), 206–217

[5] S. V. Emel'yanov, Theory of Systems with Variable Structures, Nauka, M., 1970, 592 pp. | MR

[6] V. I. Gurman, “Theory of Optimum Discrete Processes”, Automation and Remote Control, 34:7-1 (1973), 1082–1087 | Zbl

[7] V. I. Gurman, I. V. Rasina, “Discrete-Continuous Representations of Impulsive Processes in the Controllable Systems”, Automation and Remote Control, 73:8 (2012), 1290–1300 | DOI | MR | Zbl

[8] V. F. Krotov, “Sufficient Optimality Conditions for Discrete Controllable Systems”, Dokl. Akad. Nauk SSSR, 172:1 (1967), 18–21 | MR | Zbl

[9] V. F. Krotov, V. I. Gurman, Methods and Problems of Optimal Control, Nauka, M., 1973, 448 pp. | MR

[10] J. Lygeros, Lecture Notes on Hybrid Systems, University of Cambridge, Cambridge, 2003, 70 pp.

[11] B. M. Miller, E. Ya. Rubinovich, Optimization of the Dynamic Systems with Pulse Controls, Nauka, M., 2005

[12] I. V. Rasina, “Discretization of Continuous Controllable Systems Based on Generalized Solutions”, Automation and Remote Control, 72:6 (2011), 1301–1308 | DOI | MR | Zbl