Existence results and optimality conditions of time optimal control problem

The paper is concerned with the existence of theorems and optimality conditions for a time-optimal control problem. The first aim deals with the existence theorem of an optimal solution for a time-optimal control problem in the calculus of variations. We prove the existence theorem utilizing a result on the existence of continuous maps from the space of absolutely continuous functions to the set of solutions to a differential inclusion with Lipschitzian right-hand side. The second goal is to formulate the optimality conditions for the time-optimal minimization problem. We examine necessary optimality conditions which play a significant role in finding candidates to be optimal solutions among all admissible solutions via the normal cones.