Optimal controls that maximize the probability of hitting a set of targets: A numerical study

Abstract

A two-dimensional random motion of a point is dealt with. The point velocity (v cos ϑ,v sin ϑ) is subjected to two different kinds of perturbations, the first represented by a vector of independent standard Wiener processes and the second by a generalized type of Poisson process. The control function is ϑ, whilev is kept fixed. We assume given a configuration ofn+1 target sets,A ₀,...,A _n , in the plane, all of these sets being surrounded by an open and bounded setD. We denote by ℙ _x (ℙ(ϑ)∈A _i the probability thatX _t , the location of the point, whereX ₀=x∈D, will reach the setA _i beforeX _i reaches any other setA _j ,i≠j, and before it leavesD. The problem dealt with here is to find an optimal control law ϑ*, ϑ*=ϑ*(x),x∈D, such that the function

$$V\left( {x;\theta } \right) = \sum\limits_{i = 0}^n {\lambda _i \mathbb{P}_x \left( {X_\tau \left( \theta \right) \in A_i } \right),}$$

where λ₁,i=0,...,n, are given nonnegative numbers, will be maximized on a given class of admissible control laws. Sufficient conditions on optimal controls, of a dynamic programming type, are derived. These conditions require the existence of a smooth solution to a nonlinear partial integrodifferential equation, which is solved here by applying a finite-difference scheme. Two examples are dealt with numerically.