×

An improvement method for hierarchical model with network structure. (Russian. English summary) Zbl 1345.49023

Summary: Systems of heterogeneous structure are widespread in practice. Currently such systems are the subject of intense study by the representatives of different scientific schools and directions. These systems include systems with variable structure, discrete-continuous, logic-dynamic, hybrid and heterogeneous dynamic systems. In this article, the systems of heterogeneous network structure are considered. For modeling and research, the hierarchical approach is used: a two-level model is created, the lower level of which represents different controlled differential systems of homogeneous structure and the upper level represents a network of operators, providing purposeful interaction of continuous subsystems. This model can be seen as a further development of the discrete-continuous model, proposed and investigated in a number of works of the authors. An optimal control problem is formulated and sufficient conditions of optimality are derived – analogues of the known Krotov sufficient conditions of optimality, which involve resolving functions of Krotov type for each level. On the basis of these conditions and the localization principle, a method of monotone iterative improvements (linear with respect to the state) of the Krotov-type functions is constructed. The involvement of the second derivatives of the control variables in its structure allows to take into account the ravine surface structure of the functional. The method, like the model, has a two-level structure. On the lower level appears a traditional conjugated system of differential equations for the coefficients of the resolving functions, whereas on the upper level, conjugated variables are determined from the linear algebraic system of equations. As an example, we consider the optimization of water protection measures in a river basin for a simplified model with an operator tree. The prototype are the lower flows of the Selenga river. For this problem, a two-level network model is built and the proposed algorithm is applied. The results of calculations are represented.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49M30 Other numerical methods in calculus of variations (MSC2010)
49N90 Applications of optimal control and differential games
93C15 Control/observation systems governed by ordinary differential equations
93A13 Hierarchical systems
90C35 Programming involving graphs or networks
PDFBibTeX XMLCite
Full Text: Link

References:

[1] Anohin, Ju. A.; Gorstko, A. B., Matematicheskie modeli i metody upravlenija krupnomasshtabnym vodnym ob’ektom (in Russian) (1987)
[2] Bortakovskij, A. S., Dostatochnye uslovija optimal’nosti upravlenija determinirovannymi logiko-dinamicheskimi sistemami (in Russian), Informatika Ser. Avtomatizacija proektirovanija, 3, 72-79 (1992)
[3] Vasil’ev, S. N., Teorija i primenenie logiko-upravljaemyh system (in Russian), Trudy 2-oj Mezhdunarodnoj konferencii «Identifikacija sistem i zadachi upravlenija» (SICPRO’03)», 23-52 (2003)
[4] Gurman, V. I., Irkutsk, Izd-vo Irkut. un-ta, 121 (1976)
[5] Gurman, V. I.; Rasina, I. V., Sufficient optimality conditions in hierarchical models of nonuniform systems (in Russian), Avtomat. i telemeh., 12, 15-30 (2013) · Zbl 1284.49021
[6] Gurman, V. I., K teorii optimal’nyh diskretnyh processov (in Russian), Avtomat. i telemeh., 6, 53-58 (1973)
[7] Gurman, V. I.; Rasina, I. V., O prakticheskih prilozhenijah dostatochnyh uslovij sil’nogo otnositel’nogo minimuma (in Russian), Avtomat. i telemeh., 10, 12-18 (1979)
[8] Gurman, V.; Rasina, I., V. Discrete-continuous representations of impulsive processes in the controllable systems (in Russian), Avtomat. i telemeh., 8, 16-29 (2012) · Zbl 1268.93076
[9] Gurman, V. I.; Fesko, O. V.; Rasina, I. V., Modelirovanie vodoohrannyh meroprijatij v bassejne reki (in Russian), Vestnik BGU. Matematika, informatika, 3, 4-15 (2013)
[10] Emel’janov, S. V., Teorija sistem s peremennoj strukturoj (in Russian), Moscow, Nauka (1970)
[11] Krotov, V. F.; Gurman, V. I., Metody i zadachi optimal’nogo upravlenija [Methods and problems of optimal control], 446 (1973) · Zbl 0271.49003
[12] Miller, B. M.; Rubinovich, E. Ja., Optimizacija dinamicheskih sistem s impul’snymi upravlenijami [Optimization of dynamic systems with pulse control], 429 (2005)
[13] Rasina, I. V., Iterative optimization algorithms for discrete-continuous processes (in Russian), Avtomat. i telemeh., 10, 3-17 (2012) · Zbl 1275.49055
[14] Lygeros, J., Lecture Notes on Hyrid Systems (2003)
[15] Van der Shaft, A. J.; Schumacher, H., An Introduction to Hybrid Dynamical Systems (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.