The present paper deals with the optimal control theory given by second-order differential inclusions (P (C) ) with a non-fixed time interval and endpoint constraints. Our aim is to establish well-verifiable sufficient conditions of optimality for second-order differential inclusions. Thus, the sufficient conditions, including distinctive t (1)-attainability condition ones, are formulated by using the Euler-Lagrange and Hamiltonian type of inclusions. Here, the basic apparatus of locally adjoint mappings (L A M s) is suggested. Application of these results is illustrated by solving some linear control problem with second-order differential inclusions.