Dynamic programming approach to a minimum distance optimal control problem

An optimal control problem with minimum-type (non-additive) functional is considered. Such problem has several applications, including air collision avoidance problem for two aircraft. It is known that the Bellman optimality principle is not fulfilled globally for this problem, so that the dynamic programming technique works only in a part of the problem´s phase space. The boundary of this part is unknown and has to be found as an element of the solution of a dynamic programming problem with unknown boundary. In some problems this boundary contains optimal (singular) trajectories. The equations for such paths are derived by applying the method of singular characteristics. Some other necessary conditions of optimality are discussed in terms of Bellman equation and Hamiltonian. Examples are given for which the unknown boundary includes and does not include optimal paths. An aircraft collision avoidance problem is discussed.