On the connection between costate and the annihilator of the Hamiltonian vector field in optimal control problems

Within the framework of finite-horizon optimal control problems involving nonlinear, input-affine dynamics, a connection between the costate variable and generating functions of the annihilating codistribution of the underlying Hamiltonian vector field is established. It is shown that the inverse mapping of any collection of n such generating functions coincides, for any time and for a certain constant vector, with the costate of the optimal process. In particular, the corresponding constant vector is determined by solving a parameterized boundary value problem in the state space of the original plant alone, rather than in the extended state/costate space of the Hamiltonian dynamics.