m at h . FA ] 6 M ar 1 99 8 INTERACTIVE GAMES AND REPRESENTATION THEORY

This short note is a conceptual prologue to the series of articles devoted to the analysis of interrelations between the representation theory (especially, its inverse problems) and the control and games theory. This note was appeared as an attempt to formulate rigorously and mathematically the ideas of [1] and simultaneously to treat them in a more (mathematically) general context of control and games theories (see e.g.[2,3]). Definition 1. An interactive system (with n interactive controls) is a control system with n independent controls coupled with unknown or incompletely known feedbacks (the feedbacks as well as their couplings with controls are of a so complicated nature that their can not be described completely). An interactive game is a game with interactive controls of each player. Below we shall consider only deterministic and differential interactive systems. For symplicity we suppose that n = 2. In this case the general interactive system may be written in the form: (1) ˙ ϕ = Φ(ϕ, u 1 , u 2), where ϕ characterizes the state of the system and u i are the interactive controls: u i (t) = u i (u • i (t), [ϕ(τ)]| τ ≤t), i.e. the independent controls u • i (t) coupled with the feedbacks on [ϕ(τ)]| τ ≤t. One may suppose that the feedbacks are integrodifferential on t. Theorem. Each interactive system (1) may be transformed to the form (2) below (which is not, however, unique): (2) ˙ ϕ = ˜ Φ(ϕ, ξ), where the magnitude ξ (with infinite degrees of freedom as a rule) obeys the equation (3) ˙ ξ = Ξ(ξ, ϕ, ˜ u 1 , ˜ u 2),