A Group-Theoretic Approach to the Reducibility Problem of Optimal Processes
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NLINEAR SYSTEMS
A Group-Theoretic Approach to the Reducibility Problem of Optimal Processes K. G. Garaev Kazan National Research Technical University (KAI), Kazan, Russia e-mail: [email protected] Received October 16, 2019 Revised January 24, 2020 Accepted January 30, 2020
Abstract—The reducibility problem of optimal processes is posed, and a group-theoretic approach to its solution is proposed. This approach is based on the Lie–Ovsiannikov infinitesimal apparatus [1]. Keywords: optimal control, Bellman equation, theory of Lie groups, reducibility of optimal processes DOI: 10.1134/S0005117920070012
1. INTRODUCTION For the general problem of control, group-theoretical methods were first used in the works of V.G. Pavlov, A.I. Kukhtenko, V.N. Semenov, Yu.N. Pavlovskii, G.N. Yakovenko, K.G. Garaev, G.V. Mozhaev, I.F. Boretskiy, and other researchers. Among foreign authors, the first publications in this field belong to M. Arbib, R. Brockett, S. Barnet, R. Hermann, R. Kalman, and other scientists; for example, see [2, 3]. The group approach allows implementing the ideas of F. Klein’s Erlangen Program in the problem of control. Consider in brief the specific problems that can be studied using the group approach. As is well known, optimal control design leads to a difficult problem called the curse of dimensionality, the colorful term introduced by Bellman. An effective method to solve this problem is to perform decomposition, an operation that splits the original control system into subsystems. As a result, a simpler local problem can be formulated for each of the subsystems [4–9]. A general theory of the decomposition and aggregation of systems with a constant control vector was developed in [10–13], based on the theory of invariants of continuous transformations groups. The problem of aggregation was also studied in [14]. The theory of groups made it possible to classify the types of decomposition as well as to find the number of subsystems and their dimensions, thereby determining the reasonability of decomposition for particular dynamic control systems. In the language of group theory, the influence of perturbations of automatic control systems parameters on their output characteristics, i.e., the sensitivity of systems, was studied in [15–17]. A number of works (e.g., [3, 18–24]) was devoted to the problems of controllability and observability of dynamic systems. In some cases, the concept of an L-system introduced in [25] makes it possible to replace the differential equations of Pontryagin’s maximum principle with equivalent finite relations. A group approach to the design problem of control systems ensuring the invariance of system characteristics with respect to external disturbances was proposed in [19, 20, 22, 26]. The optimal control design problem was the subject of [25, 27, 28]. In particular, the concept of an invariant optimal process was introduced in [27]; moreover, the existence conditions of such 1167
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a process (expressing the invariance of the manifold and the functional with respect to the same group
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