The present paper studies a new class of problems of optimal control theory with special differential inclusions described by higher-order linear differential operators (HLDOs). There arises a rather complicated problem with simultaneous determination of the HLDOs and a Mayer functional depending of high-order derivatives of searched functions. The sufficient conditions, containing both the Euler-Lagrange and Hamiltonian type inclusions and transversality conditions at the endpoints t = -1, 0 and t =1 are derived. One of the key features in the proof of sufficient conditions is the notion of locally adjoint mappings. Then, we demonstrate how these conditions can be transformed into Pontryagin's maximum principle in some particular cases.