Hamiltonian (control theory)

The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. It was inspired by, but is distinct from, the Hamiltonian of classical mechanics. 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. For details see Pontryagin's maximum principle.

A control ${\displaystyle u(t)}$ is to be chosen so as to minimize the objective function

where ${\displaystyle x(t)}$ is the system state, which evolves according to the state equations

and the control must satisfy the constraints

where ${\displaystyle \lambda (t)}$ is a vector of costate variables of the same dimension as the state variables ${\displaystyle x(t)}$.

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