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This paper deals with optimal control problems for dynamical systems governed by general functional differential inclusions of neutral type with endpoint constraints. First, a sequence of well-posed discrete optimization problems is constructed by developing discrete approximations to neutral functional-differential inclusions. Second, necessary optimality conditions for discrete optimization problems of neutral type are established using advanced generalized differentiation tools of variational analysis. Finally, necessary optimality conditions of both Euler-Lagrange and Hamiltonian types for the original problem are derived from passing to the limit in necessary optimality conditions of discrete-time systems.