Order Reduction of Optimal Control Systems
  The paper presents necessary and sufficient conditions for the order reduction of optimal control systems. Exploring the corresponding Hamiltonian system allows to solve the order reduction problem in terms of dynamical systems, observability and invariant differential forms. The approach is applicable to non-degenerate optimal control systems with smooth integral cost function. The cost function is defined on the trajectories of a smooth dynamical control system with unconstrained controls and fixed boundary conditions. Such systems form a category of Lagrangian systems with morphisms defined as mappings preserving extremality of the trajectories. Order reduction is defined as a factorization in the category of Lagrangian systems. 
Igor Borovikov