On the evadable sets of differential evasion games (Q1106753)

The author considers an autonomous differential game described by the equation \(\dot z=f(z,u,v)\), \(z(0)=z_ 0\), \(u\in U\in comp R^ p\), \(v\in V\in compR^ q\). Here \(z\in R^ n\), u is the pursuit control, v is the evasion control. The terminal set of the game is a linear subspace M of \(R^ n\) with dim \(M\leq n-2\). Player P chooses his control \(u(\cdot)=(u(t)\), \(0\leq t<\infty)\) in the class \({\mathcal U}_ h[0,\infty)\) defined as follows: \({\mathcal U}_ h[0,\infty):=\{u:\) [0,\(\infty)\to U\), u is measurable; \(\forall \delta \in [0,\infty)\) \(\| u(t)-u(t- \delta)\| \leq h(\delta)\) a.e. \(t\in [\delta,\infty)\}\) where h: [0,\(\infty)\to [0,\infty)\) is a nondecreasing function, \(h(0)=0\). Player E forms his strategy \(S_{\delta}\) having complete information about \(z(\cdot)=(z(s)\), \(0\leq s\leq t-\delta)\) and \(u(\cdot)=(u(s)\), \(0\leq s\leq t-\delta)\) where the parameter \(\delta >0\) describes an information delay. The purpose of the paper is to discuss the evasion game with the mentioned information delay and to get some sufficient conditions for evadability and strict evadability. The results of the paper extend the results obtained earlier by L. S. Pontryagin, N. Satimov, B. N. Pshenichnyj and other authors.