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Brief review of tasks and results of programm of creation of unified geometrical theory of control (UGTC)

Butkovskiy A. G., Babichev A. V.

Systems Science

2004

Vol. 30, no 1

37-42

The purpose of this paper is in very much compressed thesis form to depict the results and problems that were obtained during development of "Unified geometrical theory of control (UGTC)", or "Theory of control structures (TCS)". It is emphasized that UGTC deals with control of structures and symmetries, and a number of structures are considered. Because the UGTC treats the basic concepts of control theory, some main philosophical and methodological principles related to the control science and mathematics are discussed. Concrete theoretical results are given in geometrical form, which permits to show the invariance and generality of these statements. Particularly, the geometrical construction of foliation extends to the synthesis in the control theory as well as to the choice axiom in the set theory, demonstrating an important connection between appropriate problems. Also the problems of axiomatic control theory and control in metric and topological spaces are mentioned. For systems with differential structure the various representations in terms of differential forms, partial differential equations and phase portraits of differential inclusions are considered. The relation between optimality principle and physical variable of energy tensor is determined.

Geometrical structures in control theory.

Butkovskiy A.G.

Systems Science

1999

Vol. 25, nr 1

121-126

This work has been done in a frame of the program "Unified Geometrical Theory of Control (UGTC)" or "Theory of Control Structures (TCS)". Topological and nonsymmetric metric spaces are considered as Control Systems where control is associated with structures of orderness and connectedness. There is also considered the optimal control problem (eigen-optimal control) in the sense of minimal length of control in a given metric (distance) function. The concept of secondary and self-secondary metric is introduced in terms of which sufficient conditions of optimality are given.