Better reply security and existence of equilibria in differential games (Q384051)

A differential game for \(N\) players is considered satisfying the following restrictions: {\parindent=0.6cm\begin{itemize}\item[1)] The strategy set \[ \Big\{v_i\in L^2([0,T],\mathbb{R}^n):v_i(t)\in K_i,t\in [0,T],\mathrm{ a.e.}\Big \} \] of player \(i,\) where \(K_i\subset \mathbb{R}^n\) is a convex and compact set, is a subset of \(L^2([0,T],\mathbb R^n)\); \item[2)] the dynamics of the game is given by a system of the form \[ \dot{s}(t)=B(t)s(t)+C(t)v(t), \quad s(0)=s_0, \] where \(s=(s_1,\dots,s_{N})\), \(s_i\in L^2([0,T],\mathbb{R}^m)\), \(v=(v_1,\dots,v_{N});\) \item[3)] the objective function of player \(i\) has the form \[ \int_{0}^{T}f_i(s(t),v_i(t),v_{-i}(t),t)dt. \] \end{itemize}} Under certain assumptions it is proved that the differential game has a mixed strategy equilibrium or the equilibrium is pure strategy.