Numerical approximation for a visibility based pursuit-evasion game

This work addresses a vision-based target tracking problem between a mobile observer and a target in the presence of a circular obstacle. The task of keeping the target in the observer's field-of-view is modeled as a pursuit-evasion game by assuming that the target is adversarial in nature. Due to the presence of obstacles, this is formulated as a game with state constraints. The objective of the observer is to maintain a line-of-sight with the target at all times. The objective of the target is to break the line-of-sight in finite amount of time. First, we establish that the value of the game exists in this setting. Then we reduce the dimension of the problem by formulating the game in relative coordinates, and present a discretization in time and space for the reduced game. Based on this discretization, we use a fully discrete semi-Lagrangian scheme to compute the Kružkov transform of the value function numerically, and show that the scheme converges for our problem. Finally, we compute the optimal control action of the players from the Kružkov transform of the value function, and demonstrate the performance of the numerical scheme by numerous simulations.