By Andrei I. Subbotin (auth.), Martino Bardi, T. E. S. Raghavan, T. Parthasarathy (eds.)
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Extra resources for Stochastic and Differential Games: Theory and Numerical Methods
Sample text
Let an initial point Xo E G and a number e E (0, w) be given. 16). By definition of minimax solution, there exists a lower solution v of this problem such that -In{1- v(xo)) E (e,w]. 21). 29) and ya(x) is a point at which the maximum defining ua(x) is attained. 1. 3. 16). 30) is valid, provided diam (~) :::; 80 . 28). 3 imply the following assertion. 1. The value Val of the time-optimal differential game exists. 16) is related to the value by the equality Val (xo) = -In(l - u(xo)). 28) are sub-optimal for the evader provided that the function v is chosen so that the difference u(xo) - v(xo) is sufficiently small.
Let an initial position (to, xo) E [0,0] x Rn be given. Let (U,~) be a step-by-step procedure chosen by the player P (here U : [0,0] x Rn 1-+ P is a feedback strategy, and ~ := {to < tl < ... < tm+1 = O} is a partition of the time interval [to, 0]). , Subbotin where p(t) = U(ti' x(ti)), ti ::; t < ti+l, i E 0, m. O where the symbol lim sup stands for the upper topological limit. This means that the set X(to, Xo, U) consists of all limits of the form limk_ooxk(·) where XkO E X(to,xO,U'~k)' diam~k 10, and where the convergence is uniform on [to, 0].
Since ~ + /*8 = X(T + 8), we obtain Now we will estimate the quantity h. The function WE is differentiable; therefore, where s* = DXWE(T,~,T/) = -DyWE(T,~,T/). Recall that p* = UE(T,~) =Po(T,~,S'). 3), we have (S',/(T,~,p',q)) - g(T,~,p',q) ~ H(T,~,S') Consequently (s', f*) - g* ~ H(T,~, SO) for all q E Q. + (5(8). Let us note also that q* = qO(T, T/, SO). ) - g. 2: H(T, T/, SO) - (6(8). 4), we obtain for allp E P. 1. 23) and come to the inequality U,,(T,O ;::: U,,(T + 8, X(T + 8)) - (g* - (5(8) - (6(8))8.