An optimal control problem for automatic air collision avoidance

In this paper, we discuss an automatic air collision avoidance problem from an optimal control theory standpoint. The objective is to find the combination of escape maneuvers between two aircraft that attains the largest minimum separation distance among all the possible choices of escape maneuvers. To achieve this objective, a terminal optimal control problem with a quadratic performance criteria is formulated where the terminal time is set to be the time at the minimum separation. The commands for an escape maneuver consist of a normal acceleration command and a roll rate command. To simplify the problem, the maximum allowable normal acceleration is commanded in the pitch axis, hence the command inputs appeared in the cost functional is formulated only with the roll rate commands. The optimal roll rate commands are determined by solving the Euler-Lagrange equations.