Differential antagonistic games with lexicographic vector-payoffs

In this paper the existence problem of the equilibrium situation in differential antagonistic games with perfect information and lexicographic payoffs or in a -dimensional vector-payoffs' game where criteria are strictly ranged with preference relation is studied. The players' dinamic is defined by vector differential x=f( t,x ,u ),y=g( t,y ,v ) equations, respectively control functions u( .), v(. ) and ∈〔0.T〕time interval. This is a game ΓL(x0, y0)=(Γ1,...Γm ) where x0, y0 are starting positions in t=0 moment respectively the first and second players'. x(t)and y(t) are trajectories, the players final aim is finding their optimal variants. A lexicographic ε -equilibrium situation is defined in the game and the conditions of its existence are investiga-terd. These conditions are mainly about f and g functi-ons. The main definitions are introduced and some results are formulated from theory of differential games with scalar payoff functions and independent move-ments, they are the main for getting results for analogic differential games in the case of lexicographic payoffs. Some auxiliary statements correctness are also establi-shed, on its basic it is proved that in ΓL(x0, y0) game for any ε>0 there exists a lexicographic ε-equilibrium situation in pure strategies.