Transversality condition and optimization of higher order ordinary differential inclusions

In the present paper, a Bolza problem of optimal control theory with a fixed time interval given by convex and nonconvex second-order differential inclusions (P_H) is studied. Our main goal is to derive sufficient optimality conditions for Cauchy problem of sth-order differential inclusions. The sufficient conditions including distinctive transversality condition are proved incorporating the Euler–Lagrange and Hamiltonian type inclusions. The basic concepts involved in obtaining optimality conditions are the locally adjoint mappings. Furthermore, the application of these results is demonstrated by solving the problems with third-order differential inclusions.