Games with controllable impulse effect at fixed instants (Q2768855)

The author considers a differential game governed by the following impulsive dynamical system: \(\dot z=f(z,u,v), t\neq \tau; \Delta z|_{t=\tau}=\Gamma_\tau(z,u_\tau,v_\tau)-z.\) Here \(z\in \mathbb R^n, f:\mathbb R^n\times U\times V \mapsto \mathbb R^n, \Gamma :\mathbb R^n\times U_\tau \times V_\tau \mapsto \mathbb R^n\), the sets \(U, V ,U_\tau, V_\tau \) being compact subsets of Euclidean spaces. The controls \(u\in U,v\in V\) are at the disposal of the pursuer (player \(P\)) and the evader (player \(E\)), respectively. The differential game is also characterized by a closed set of phase restrictions \(N\subset\mathbb R^n\) and by a closed terminal set \(M\subset N\). The game with fixed termination time \(\theta \) is studied. The goal of the player \(P\) is to reach the inclusions: \(z(\theta)\in M, z(t)\in N \forall t\in [0,\theta]\). The authors construct a mapping \(\Phi_{N,\theta}\) acting from the set of bounded subsets of \(\mathbb R^n\) into itself with the following property. If \(z_0\in Z_0:=\Phi_{N,\theta}M\) then there exists such an \(\epsilon \)-strategy of the pursuer, that the solution \(z(t), z(0)=z_0\), has the property: \(z(t)\in N\) for all \(t\in [0,\theta)\) and \(z(\theta)\in M\). If \(z_0\not \in Z_0\), then there exists a favorable strategy of the evader: for corresponding solution \(z(t)\), either \(z(\theta)\notin M\) or there exists such \(t\in [0,\theta)\) that \(z(t)\notin N\). In addition, the case of linear game is examined in more detail.