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1 2001

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An axiomatical approach to the structure of control and optimal control

Buktkovskiy A.G., Babichev A.V.

Systems Science

2001

Vol. 27, nr 1

49-58

The paper deals with an axiomatical approach to the problem of determining and studying the control structures. A number of abstract mathematical structures are considered to define the concept of "space of admissibles" for control systems. It is the basic concept, which covers the convenient notions of spaces of admissible motions, controls, etc. Also the notion of a partition for the admissible and the notion of generalized boundary are defined. In such terms the general optimal control problem is stated. The metric approach to this problem is developed, which yields the necessary and sufficient optimality conditions. On the basis of the concepts of secondary and k-fold metrics the optimality conditions are represented as the structural identities. These results are obtained without using any notion of state space for the system. Such a notion proves to be well defined under the additional condition in terms of the lattice theory. This indicates the dependence of hierarchy of the formal control structures on the axiomatics involved.