The paper deals with the optimal control problem described by second-order differential inclusions. Based on the infimal convolution concept of convex functions, dual problems for differential inclusions are constructed and the results of duality are proved. In this case, it turns out that Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems. In particular, the linear second-order optimal control problem with the Mayer functional is considered. This problem shows that maximization in the dual problems is realized over the set of solutions of the adjoint equation. Finally, we construct the dual problem to the problem with the second-order polyhedral differential inclusion.