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 z₁, z₂ in the complex plane, we solve the problem of finding the path C(t) (0⩽t⩽1) such that C(0)=z₁, C(1)=z₂ and ∫₀¹ |C´(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