Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems

Considers the relation between the optimal control problem and the classical calculus of variations problem with constraints. Some variational problems concerning control systems modeled on a state space of dimension n may be attacked by either formulation. The Pontryagin maximum principle gives rise to necessary conditions formulated in terms of a system of 2n Hamiltonian equations, while the Euler-Lagrange equations describing the necessary conditions in the classical calculus of variations formulation give rise to more equations. Clearly, in this case, the classical Legendre condition does not link the two formulations. The authors describe a technique to reduce the Euler-Lagrange equations to a system of 2n equations and describe the transformation which links the resulting system with the corresponding Hamiltonian equations. The authors describe some examples in detail and specifically address the situation where the equations describing the necessary conditions may be reduced due to the presence of symmetries