Interception Strategies for the Protection of a Target Against Penetration-Pursuers Moving Along Two Parallel Lines

摘要

Consider a random motion of three points Me'Mp1 and Mp2 in an open and bounded domain Do in the plane. The velocity (u1,u2) of Me is essentially directed towards a disc Dc at the center of Do but is continually perturbed by an r squared-valued Gaussian white noise. In order to prevent the point Me from reaching the set Dc, the points Mpi, i=1,2, move on parallel lines Li, i=1,2, respectively, in Do. The velocity vi of mpi is also perturbed by a Gaussian white noise. Denote by epsilon the following event: epsilon=(Me is intercepted by Mp1 or Mp2 before Me reaches the set Dc and before either of the points Me, Mp1 and Mp2 has left Do). The problem dealt with here is to choose for Mpi, i=1,2, velocity strategies (v1,v2) such that Prob. epsilon will be maximized on a given class of admissible velocity strategies (v1,V2). Sufficient conditions on optimal strategies of a dynamic programming type, are derived. These conditions require the existence of a smooth solution to a nonlinear partial differential equation, which is solved here by applying a finite-difference scheme. An example is dealt with numerically.