Hamiltonian of an optimal control problem

Summary

The Hamiltonian of an optimal control problem is a function that combines the dynamics and the running cost with a vector of multipliers, the costate. It has the form $H = p \cdot f - \lambda_0 L$, or $H = L + p \cdot f$ in the sign convention of the minimum principle. Pontryagin's maximum principle says that an optimal control maximizes the Hamiltonian at almost every time. It plays the role that the Hamiltonian of Hamiltonian systems plays in mechanics.