The Hamiltonian of Control Compared To The Hamiltonian of Mechanics
William Rowan Hamilton defined the Hamiltonian as a function of three variables:
where is defined implicitly by
Hamilton then formulated his equations as
In contrast the Hamiltonian of control theory (as defined by Pontryagin) is a function of 4 variables
and the associated conditions for a maximum are
This difference is somewhat confusing, nevertheless a specific problem, such as the Brachystochrone problem, can be solved by either method. For details, see the article by Sussmann and Willems.
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