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 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 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 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.
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 to be the state of the dynamical system with input , such that
where is the set of admissible controls and is the terminal (i.e., final) time of the system. The control must be chosen for all to minimize the objective functional which is defined by the application and can be abstracted as
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 defined for all by:
where is the transpose of .
Pontryagin's minimum principle states that the optimal state trajectory , optimal control , and corresponding Lagrange multiplier vector must minimize the Hamiltonian so that
for all time and for all permissible control inputs . It must also be the case that
Additionally, the costate equations
must be satisfied. If the final state is not fixed (i.e., its differential variation is not zero), it must also be that the terminal costates are such that
These four conditions in (1)-(4) are the necessary conditions for an optimal control. Note that (4) only applies when is free. If it is fixed, then this condition is not necessary for an optimum.
- Lagrange multipliers on Banach spaces, Lagrangian method in calculus of variations
- I. M. Ross A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009.
- See ref. below for first published work.
- See p.13 of the 1962 book of Pontryagin et al. referenced below.
- V.G. Boltyanskii, R.V. Gamkrelidze. L.S. Pontryagin: Towards a theory of optimal processes, (Russian), Reports Acad. Sci. USSR, vol.110(1), 1956
- Pontryagin L.S, Boltyanskii V.G, Gamkrelidze R. V, Mishchenko E. F, The Mathematical Theory of Optimal Processes (Russian), English translation: Interscience 1962. ISBN 2-88124-077-1 and ISBN 978-2-88124-077-5
- Fuller A.T. Bibliography of Pontryagin's maximum principle, J. Electronics & Control vol.15 no.5 Nov. 1963 pp. 513–517
- Kirk, D.E. Optimal Control Theory, An Introduction, Prentice Hall, 1970. ISBN 0-486-43484-2
- Sethi, S. P. and Thompson, G. L. Optimal Control Theory: Applications to Management Science and Economics, 2nd edition, Springer, 2000. ISBN 0-387-28092-8 and ISBN 0-7923-8608-6. Slides are available at http://www.utdallas.edu/~sethi/OPRE7320presentation.html
- Geering, H.P. Optimal Control with Engineering Applications, Springer, 2007. ISBN 978-3-540-69437-3
- Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.
- Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
- Hazewinkel, Michiel, ed. (2001), "Pontryagin maximum principle", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4