Pontryagin's maximum 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 states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the control Hamiltonian.
[1][2] The maximum principle was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students,[3][4] and its initial application was to the maximization of the terminal speed of a rocket.
[5] The result was derived using ideas from the classical calculus of variations.
[6] After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows.
[7] Widely regarded as a milestone in optimal control theory, the significance of the maximum principle lies in the fact that maximizing the Hamiltonian is much easier than the original infinite-dimensional control problem; rather than maximizing over a function space, the problem is converted to a pointwise optimization.
[8] A similar logic leads to Bellman's principle of optimality, a related approach to optimal control problems which states that the optimal trajectory remains optimal at intermediate points in time.
[9] The resulting Hamilton–Jacobi–Bellman equation provides a necessary and sufficient condition for an optimum, and admits a straightforward extension to stochastic optimal control problems, whereas the maximum principle does not.
[7] However, in contrast to the Hamilton–Jacobi–Bellman equation, which needs to hold over the entire state space to be valid, Pontryagin's Maximum Principle is potentially more computationally efficient in that the conditions which it specifies only need to hold over a particular trajectory.
Consider an n-dimensional dynamical system, with state variable
is the set of admissible controls.
The evolution of the system is determined by the state and the control, according to the differential equation
Let the system's initial state be
and let the system's evolution be controlled over the time-period with values
The latter is determined by the following differential equation: The control trajectory
can be interpreted as the rate of cost for exerting control
can be interpreted as the cost for ending up at state
The constraints on the system dynamics can be adjoined to the Lagrangian
by introducing time-varying Lagrange multiplier vector
, whose elements are called the costates of the system.
This motivates the construction of the Hamiltonian
Here, the trajectory of the Lagrangian multiplier vector
is the solution to the costate equation and its terminal conditions: If