The present paper studies the sufficient conditions of optimality for Cauchy problem of fourth-order differential (PD) inclusions. Mainly our purpose is to derive sufficient optimality conditions for mentioned problems with fourth-order differential inclusions (DFIs) and trans-versality conditions. The basic idea of obtaining optimal conditions is the use of locally adjoint mappings (LAM), defined by Hamiltonian functions. Moreover, in the application of these results the fourth-order linear optimal control problems with linear differential inclusions are considered. We analyze the proposed method for a class of Lagrange problem with integrand of quadratic form involving symmetric non-negative semidefinite matrix. An illustrative example is given. Theoretical analysis and practical results show that our method is simple and easy to implement and is efficient for computing optimal solution of the fourth order differential inclusions. The results reveal that the proposed method is very accurate and efficient.