Optimal control problems via a direct method

In this paper we deal with the minimization problem of a cost functional associated to a nonlinear boundary value control problem of a general form, defined in the fixed time interval [0, 1]. Specifically, we first give conditions which ensure that the nonlinear boundary value control problem is solvable and we study the structure of the relative solution set. Then, based on the properties of this set, we establish conditions ensuring both the existence of quasisolutions and that of solutions of the minimization problem under consideration. Such conditions will depend also on the choice of the control space Lr([0, 1],Rm) where 1⩽ r ⩽ + ∞