Consistent approximation of an optimal search problem

This paper focuses on the problem of optimizing the trajectories of multiple searchers attempting to detect a non-evading moving target whose motion is conditionally deterministic. This problem is a parameter-distributed optimal control problem, as it involves an integration over a space of stochastic parameters as well as an integration over the time domain. In this paper, we consider a wide range of discretization schemes to approximate the integral in the parameter space by a finite summation, which results in a standard control-constrained optimal control problem that can be solved using existing techniques in optimal control theory. We prove that when the sequence of solutions to the discretized problem has an accumulation point, it is guaranteed to be an optimal solution of the original search problem. We also provide a necessary condition that accumulation points of this sequence must satisfy.