Controllable pulse influence in games with fixed termination time (Q2740353)

The author considers a conflict-controlled process governed by the system with pulse action \(\dot z=f(z,u,v), t\neq \tau; \Delta z|_{t=\tau }=\Gamma_\tau(z,u_\tau,v_\tau)-z.\) Here \(z\in \mathbb R^n, f:\mathbb R^n\times U\times V \mapsto \mathbb R^n, \Gamma :\mathbb R^n\times U_\tau \times V_\tau \mapsto \mathbb R^n\), and \(U, V ,U_\tau, V_\tau \) are compact sets. 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 set of phase restrictions \(N\subset\mathbb R^n\) and by a terminal set \(M\subset N\), \(M\) and \(N\) being closed. The game with fixed termination time \(\theta \) is studied. The goal of the player \(P\) is to reach inclusions \(z(\theta)\in M, z(t)\in N \forall t\in [0,\theta ]\). NEWLINENEWLINENEWLINEThe authors construct such an initial set \(Z_0\) that there exist an optimal \(\epsilon \)-strategy of the pursuer, provided that the ~initial point \(z_0\) of the trajectory belongs to \(Z_0\). If \(z_0\not \in Z_0\), then there exists a favorable strategy of the evader.NEWLINENEWLINENEWLINEIn addition the case of linear game is examined in more detail.