ABSTRACT
A Hamiltonian function is involved in dynamic programming of an objective function on the state variables in optimal control theory. Problems in optimal control theory involve continuous time, a finite time horizon, and fixed endpoints. A Hamiltonian is similar to a Lagrangian in concave programming and requires first-order conditions. The Hamiltonian is the operator corresponding to the total energy of the system in most cases. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation with the time-evolution of a system, it is very important in most formulations of quantum theory. The Hamiltonian in optimal control theory is distinct from its quantum mechanical definition. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian.
