About One Unsolved Problem in Matrix Ordinary Differential Equations

Finding the Nash equilibrium situation in linear-quadratic differential game of three persons reduces to construction of explicit form solving of matrix system of Riccati model differential equations. The question is that the existence of such solution, its properties is the unsolved problem. In proposed article this problem is solved only for the game of one player. We tried to apply the Poincare method of small parameter (from the theory of oscillation) but only for special form model of controlled system where two of three players affect in a small way on the rate of change of phase vector. But the question about solving of matrix system of Riccati model equations is open. In cases where Nash equilibrium situation doesn’t exist we recommend to apply other equilibrium concepts (active equilibrium, equilibrium of objections and counterobjections, Berge equilibrium). We note that in the present article we singled out the case (the section 3) when in differential game the Nash equilibrium situation is absent.

Keywords: differential noncooperative games, Nash equilibrium, situation, strategy.