Linear differential games (Q5899800)

This is a brief survey of four papers by the author. Differential games as mathematical idealizations of practical collisions are discussed. Pursuit and evasion game problems are stated within the classes of closed-loop strategies using information about the opponent's control on an interval adjoining to a current time instant from the left. The case of the linear motion equation \(\dot z=Cz-u+v\), \(t\geq 0\), and a terminal subspace M is considered especially; here u and v are a pursuer's and an evader's controls varying within convex compacta P and Q resp. To solve the problems one introduces (i) the sets \(P_{\tau}\) \(=\pi e^{\tau C}P\) and \(Q_{\tau}=\pi e^{\tau C}Q\) where \(\pi\) is the projector on the subspace \({\mathcal L}\) orthogonal to M, (ii) the integral of a set- valued function, (iii) the geometrical difference \(A-^{*}B=\{x:\) \(x+B_ t\subset A\}\), (iv) T(z) being the minimum of \(t\geq 0\) such that \(\pi e^{tC}z\in \int^{t}_{0}(P_{\tau}-^{*}Q_{\tau})d\tau\), (v) the sets \(\hat P_{\tau}={\hat \pi}e^{\tau C}P\) and \(\hat Q_{\tau}={\hat \pi}e^{\tau C}Q\) where \({\hat \pi}\) is the projector on a two-dimensional subspace \(\hat {\mathcal L}\subset {\mathcal L}\), (vi) the relation \(B\subset^{*}A:\) \(x+B\subset A\) for some \(x\in {\mathcal L}\). It is shown that (1) the game of pursuit from an initial state \(z_ 0\) can be completed up to time \(T(z_ 0)\), and (2) if \(\mu\) \(\hat P_{\tau}\subset \hat Q_{\tau}\) for all small \(\tau >0\) and the condition \(\hat Q_{\tau}\subset^{*}{\mathcal L}^*\) (for all small \(\tau >0)\) is violated for any straight line \({\mathcal L}^*\subset {\mathcal L}\), then for every initial state \(z_ 0\not\in M\) there exists a strategy of evasion.