Two-dimensional pursuit-evasion game with penalty on turning rates

A two-dimensional pursuit-evasion game, characterized by a difficulty level intermediate between that of the «Simple Motions Game» (with freely and instantaneously orientable velocities) and that of the «Game of Two Cars» (with lower bounds on curvature radii), is formulated and then solved by an approach both analytical and numerical. Each player's velocity has a constant modulus. The meneuvers (turns) are penalized by introducing, in the performance index, an integral term of the squared velocity vector angular rate. The performance index to be mini-maximized for this zero-sum game than takes the form: J = tf-t0 + 1/M1 ∫t 0 t fω2 2 dt with t0, tf : initial and final times ω : velocity vector angular rate 1/M ⩾ 0 : penalty coefficient ()1 : related to pursuer P (minimizing player) ()2 : related to evader E (maximizing player). The problem formulated in this way is more realistic than the «Simple Motions Game» i.e. «Simple Pursuit Game», since it takes into account the maneuverability limitation; on the other hand, it is less complex than the «Game of Two Cars», because the trajectories curvatures change more continuously than in this latter game, in which appears a great number of switches, i.e. sudden, maximum amplitude, direction changes.