Towards a Framework for Tracking Multiple Targets: Hybrid Systems meets Computational Geometry

We investigate a variation of the art gallery problem in which a team of mobile guards tries to track an unpredictable intruder in a simply-connected polygonal environment. In this work, we use the deployment strategy for diagonal guards originally proposed in [1]. The guards are confined to move along the diagonals of a polygon and the intruder can move freely within the environment. We define critical regions to generate event-triggered strategies for the guards. We design a hybrid automaton based on the critical regions to model the tracking problem. Based on reachability analysis, we provide necessary and sufficient conditions for tracking in terms of the maximal controlled invariant set of the hybrid system. We express these conditions in terms of the critical curves to find sufficient conditions for n/4 guards to track the mobile intruder using the reachability analysis.