Pontryagin's maximum principle

Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students. It has as a special case the Euler–Lagrange equation of the calculus of variations.

The principle states, informally, that the control Hamiltonian must take an extreme value over controls in the set of all permissible controls. Whether the extreme value is maximum or minimum depends both on the problem and on the sign convention used for defining the Hamiltonian. The normal convention, which is the one used in Hamiltonian, leads to a maximum hence maximum principle but the sign convention used in this article makes the extreme value a minimum.

If ${\displaystyle {\mathcal {U}}}$ is the set of values of permissible controls then the principle states that the optimal control ${\displaystyle u^{*}}$ must satisfy:

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