Optimal paths for avoiding a radiating source

We consider the problem of navigating between points in the plane so as to minimize the exposure to a radiating source. Specifically, given two points z(1), z(2) in the complex plane, we solve the problem of finding the path C(t) (0 less than or equal to t less than or equal to 1) such that C(0) = z(1), C(1) = z(2) and integral(0)(1) [C-1(t)]/[C(t)](k) dt is minimized. The parameter k specializes to a number of interesting cases: in particular k = 2 pertains to the passive sensor avoidance problem and k = 4 entails the active radar avoidance problem. The avoidance paths which minimize exposure may have infinite arc-length. To overcome this problem we introduce a weighted exposure and path length optimization problem whose solution requires a variational approach. The optimal trajectory results we obtain are surprisingly intuitive in the cases of interest.