Pursuit evasion game with costly information (Q1337003)

This paper is devoted to the study of a pursuit-evasion game in finite- dimensional state space with incomplete (for pursuer) information about state. The dynamic of the evader's (E) position in pursuer's (P) coordinate system is described by the simple-motion equation (1): \(\dot x= v- u- w\), \(x(0)= x_0\), \(x, u, v, w\in \mathbb{R}^n\), where the controls \(u(\cdot)\) of P and \(v(\cdot)\) of E are bounded \((|u|\leq a, |v|\leq b, a> b)\) and \(w\) is a fixed drift, \(|w|\leq a\). It is assumed that for each piece of information about the state P loses the constant time \(\delta\) (the cost function). The target set is a ball \(B(0, R_0)\) centered at 0 with a given radius \(R_0\). Using Pontryagin's approach the set \(C_0\) of all initial states \(x_0\) is constructed, such that there exists a control \(u(\cdot)\) of P which transfers the system (1) from \(x_0\) to the target at time \(\tau> \delta\) with only one observation of the state independently on \(v(\cdot)\) of E. In a recursive way, the set \(C\) of initial conditions \(x_0\) is constructed such that P can drive the system with the previous set \(C_{n- 1}\) using no more than \(n\) observations. Next, by the dynamic programming method, the minimized capture time with fixed number of observation is computed.