Pontryagin's Minimum 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 Semenovich 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 Hamiltonian must be minimized over, the set of all permissible controls. If is the optimal control for the problem, then the principle states that:
where is the optimal state trajectory and is the optimal costate trajectory.
The result was first successfully applied into minimum time problems where the input control is constrained, but it can also be useful in studying state-constrained problems.
Special conditions for the Hamiltonian can also be derived. When the final time is fixed and the Hamiltonian does not depend explicitly on time, then:
and if the final time is free, then:
More general conditions on the optimal control are given below.
When satisfied along a trajectory, Pontryagin's minimum principle is a necessary condition for an optimum. The Hamilton–Jacobi–Bellman equation provides sufficient conditions for an optimum, but this condition must be satisfied over the whole of the state space.
Read more about Pontryagin's Minimum Principle: Maximization and Minimization, Notation, Formal Statement of Necessary Conditions For Minimization Problem
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