Games with pay-off function and pulse influence (Q2784868)

The author considers a differential game governed by the following impulsive dynamical system: \(\dot z=f(u,v,z)\), \(z\neq \Gamma_\tau \); \(\Delta z|_{(t,z)\in \Gamma_\tau}=A_\tau z-z.\) Here \(z\in \mathbb R^n\), \(\Gamma_\tau\) is a hyper-surface in \(R^n\), \(f:\mathbb R^n\times U\times V \mapsto \mathbb R^n\), \(A_\tau :\mathbb R^n\times U_\tau \times V_\tau \mapsto \mathbb R^n\), \(\tau \geq 0\), the sets \(U\), \(V ,U_\tau\), \(V_\tau \) being compact subsets of Euclidean spaces, and the mapping \(A_\tau \) being invertible. The controls \(u\in U,v\in V\) are at the disposal of the pursuer (player \(P\)) and the evader (player \(E\)), respectively. Let \(\theta \geq\tau \) be the time of the game termination. The goal of the player \(R\) is to minimize a given functional \(\varphi(z(\theta))\). The author proves the existence of optimal \(\varepsilon \)-strategies of the players. NEWLINENEWLINENEWLINESee also the paper of \textit{N. A. Perestyuk} and \textit{E. V. Ostapenko} [Ukr. Math. J. 52, 1274-1281 (2000; Zbl 0973.91010)].