An Optimal Search Framework for the Maximum-Probability Target-Detection Problem in Discrete Time and Space

This study presents an optimal control framework designed to maximize the probability of detecting a target using a sensor-equipped agent. The proposed approach incorporates recursive Bayesian updates of a spatial probability distribution regarding the target's location, facilitating a dynamic interaction between trajectory optimization and the evolving belief state. By applying the discrete-time Pontryagin's Minimum Principle, we reformulate the optimal search problem as a two-point boundary value problem, which enables us to solve the optimal control problem indirectly. Numerical simulations are conducted to evaluate the effectiveness of the proposed framework, comparing its performance with direct optimization methods. The resulting trajectories efficiently concentrate search efforts in areas with higher target likelihood, thereby maximizing the expected terminal reward while satisfying the optimality conditions. The resulting trajectories focus search efforts on areas with higher target likelihood, thereby maximizing the expected terminal reward while meeting the optimality conditions. Overall, this framework enhances our understanding of optimal search strategies, positioning itself as a valuable asset for search and surveillance operations.