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.
Contents
- Maximization and minimization
- Notation
- Formal statement of necessary conditions for minimization problem
- References
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
where
The result was first successfully applied to 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
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 a necessary and sufficient condition for an optimum, but this condition must be satisfied over the whole of the state space.
Maximization and minimization
The principle was first known as Pontryagin's maximum principle and its proof is historically based on maximizing the Hamiltonian. The initial application of this principle was to the maximization of the terminal speed of a rocket. However, as it was subsequently mostly used for minimization of a performance index it has here been referred to as the minimum principle. Pontryagin's book solved the problem of minimizing a performance index.
Notation
In what follows we will be making use of the following notation.
Formal statement of necessary conditions for minimization problem
Here the necessary conditions are shown for minimization of a functional. Take
where
The constraints on the system dynamics can be adjoined to the Lagrangian
where
Pontryagin's minimum principle states that the optimal state trajectory
for all time
Additionally, the costate equations
must be satisfied. If the final state
These four conditions in (1)-(4) are the necessary conditions for an optimal control. Note that (4) only applies when