  We propose a method to optimally position a sensor system, which consists of
multiple sensors, each has limited range and viewing angle, and they may fail
with a certain failure rate. The goal is to find the optimal locations as well
as the viewing directions of all the sensors and achieve the maximal
surveillance of the known environment. We setup the problem using the level set
framework. Both the environment and the viewing range of the sensors are
represented by level set functions. Then we solve a system of ordinary
differential equations (ODEs) to find the optimal viewing directions and
locations of all sensors together. Furthermore, we use the intermittent
diffusion, which converts the ODEs into stochastic differential equations
(SDEs), to find the global maximum of the total surveillance area. The
numerical examples include various failure rates of sensors, different rate of
importance of surveillance region, and 3-D setups. They show the effectiveness
of the proposed method.

Optimization and Control