On a class of pursuit and evasion problems

Abstract

where each “pursuit control” z, from a given class Y yields a pursuit trajectory q( *, w). A function h : E,, x E,, -+ El is preassigned, with h(& , Q) > 0. If the evader chooses a control u E 6 and follows the corresponding trajectory f(*, U) then he “evades capture” before b if b E [0, tl] and h(f(t, u), T(t, v)) > 0 for all t E [0, b] and v E V. (For example, if h(wl , ws) = 1 w1 w2 I2 a2 then [(a, u) evades capture so long as he remains at a distance of at least 6 from every pursuing r](*, v).) We may consider several related problems based on the above model. In Problem I, a closed set A, C En is given and the evader wishes to choose IL E 9 and b E [0, tI] so as to maximize b while insuring that [(b, u) E A, and he evades capture before b. In another problem, for a given function ho : En x En -+ El , the evader may seek to maximize infhO(f(1, , u), T(t, , v))(w E Y) while evading capture before tl . In fact, our methods lend