Equilibria for noncooperative differential games (Q2704014)

Continuing his own research in [\textit{È. R. Smol'yakov}, Differ. Uravn. 31, No. 11, 1880-1885 (1995; Zbl 0868.90149); 33, No. 11, 1523-1527 (1997; Zbl 0949.91006)], the author proves several results concerning different types of equilibria for \(N\)-person differential games which consist in the maximization of each of the functionals: NEWLINE\[NEWLINE J_i(q):=\int_T dt\int_Uf^i(u,x,t) dq, \quad i=1,2,\dots, N, \;q=q_1\dots, q_N,NEWLINE\]NEWLINE over corresponding sets \(Q_i\) of mixed strategies \(q_i\), \(i=1,2,\dots, N\), for which the solution \(x(.)\) of the differential equation: NEWLINE\[NEWLINE x'(t)=\int_Uf(u,x,t) dq, \quad t\in T=[t_0,t_1], NEWLINE\]NEWLINE satisfies boundary conditions of the form: NEWLINE\[NEWLINE x_j(t_0)=x_j^0, \quad j=1,2,\dots,N, \qquad x_k(t_1)=x_k^1, \quad k\in M\subseteq \{1,2,\dots, N\}. NEWLINE\]NEWLINE After recalling the definitions of complete and absolute equilibria, the author introduces the apparently new concept of ``\(A'\)-absolute equilibrium'' and proves its existence in the case the data are of the form: \(f(u,x,t)= A(t)x+h(u,t), \;f^k(u,x,t)=g^k(x)+h^k(u,t)\).