# zbMATH — the first resource for mathematics

On the evadable sets of differential evasion games. (English) Zbl 0651.90108
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.
Reviewer: A.Kleimenov

##### MSC:
 91A24 Positional games (pursuit and evasion, etc.) 91A23 Differential games (aspects of game theory) 91A99 Game theory
Full Text:
##### References:
  Kaśkosz, B., On a nonlinear evasion problem, SIAM J. control optim., 15, 661, (1977) · Zbl 0373.90098  Kaśkosz, B., A differential game of evasion with delays, J. optim. theory appl., 44, 231-268, (1984) · Zbl 0534.90109  Ostapenko, V.V.; Ostapenko, V.V., A nonlinear escape problem, Kibernetika (kiev), Cybernetics, 14, 594-601, (1978), English transl. · Zbl 0396.90118  Pontryagin, L.S.; Pontryagin, L.S., A linear differential escape game, (), 27-60, English transl. · Zbl 0261.90081  Pontryagin, L.S.; Mishchenko, E.F.; Pontryagin, L.S.; Mishchenko, E.F., Deviation from coincidence in linear differential games, Differentsial’nye uravneniya, Differential equations, 7, 335-352, (1971), English transl. · Zbl 0269.90059  Pshenichnyi, B.N.; Pshenichnyi, B.N., The flight problem, Kibernetika (kiev), Cybernetics, 11, 642-651, (1975), English transl.  Satimov, N.; Satimov, N., On the theory of differential games of escape, Mat. sb., Math. USSR-sb., 32, 145, 371-383, (1977), English transl. · Zbl 0396.90117  Yong, J., On differential evasion games, SIAM J. control optim., 26, 1-22, (1987) · Zbl 0645.90109  \scJ. Yong, Evasion with weak superiority, to appear. · Zbl 0664.90107  Yong, J., On differential games of evasion and pursuit, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.