On the existence of a Nash equilibrium point forN-person differential games

Abstract

Introduction. Since 1970, it has been known that open-loop Nash equilibrium points exist for N-person differential games in which the controls are unconstrained or have integral constraints (see [3], [6]). Recently, it has been shown that open-loop Nash equilibrium points exist in case the controls take values in compact, convex sets (see [5]). In each of the above papers it was assumed that the dynamics were linear in the state variable as well as in the controls. It was also assumed that the cost functionals were convex in those terms which depended explicitly on the controls. In [3] and [6] it also happens that the analysis leads to restrictions on the duration of the game.In this paper we prove the existence of Nash equilibrium points for differential games with dynamics which are non-linear in the state variable x and have non-convex cost functionals with the controls constrained to take values in compact, convex sets. We also exhibit a connection between finite games played sequentially and differential games. The proof makes use of the weak L 2 compactness of the space of control functions and the weak continuity of the cost functionals, thus these results also hold in case there are integral contraints on the controls. Finally, it should be noted that the notation conforms to that used in [2], pp. 285-288.

Existence of open loop Nash equilibria in certain types of nonlinear differential games

“…The conclusion of the theorem still holds if the two conditions (9) and convexity ofW i (ŵ) are replaced by the conditions that f (i) and g (i) are nondecreasing in x for each (u, t), that f (i) and g (i) are concave in (x, u i ) for each i, and that U i is convex. 4…”

On Nonzero-Sum Game Considered on Solutions of a Hybrid System with Frequent Random Jumps

“…Remark 2. Due to linearity of the functionsf i (z, ω i ) in ω i , i = 1, 2, and due to convexity and compactness of the sets W i , i = 1, 2, a Nash equilibrium of the averaged game exists (see Theorem 1.1 in [14]).…”