Pontryagin's minimum principle is used in
optimal control theory to find the best possible control for taking a
dynamic system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated by the Russian mathematician
Lev Semenovich Pontryagin and his students. It has as a general 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:
![H(x^*(t),u^*(t),\lambda^*(t),t) \leq H(x^*(t),u,\lambda^*(t),t), \quad \forall u \in \mathcal{U}, \quad t \in [t_0, t_f]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v5NLnrKFMn-uxoEUZksfckYZ7JU14u9n5svhZ4Cf_I5tdHJrlxl5nFOKD0DyXzjBAK2w3ysGZRMUSVUSfSWlHG9e1dUtF4QtP2fwjXBAd3kI7cC0kPZmmQWM6T8TI7Eqpe3-QHyBDSFbxkxcl1=s0-d)
where
![x^*\in C^1[t_0,t_f]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uNAa6S9mRWrMN7iyESnYifxdg_zG3LIbzkeS7JtKGeJ4u5MblyLWco62Bn4GRX8JIkQZc54fuM_fxz3dMKZaV1-Z0L5nVtU0NZGwMFzBaGAQe1kvYDHprckp7h314uASzBVxpwFZ5WLNRebbqJnQ=s0-d)
is the optimal state trajectory and
![\lambda^* \in BV[t_0,t_f]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vLkEbVxG_IO96rDMK6efSNqhyrnxmfRmcRlpGsQ7HaangAIiXeai4ZxJP5k2KYwze8VhGhKhKDyZzz5bo6T3wfH2rPz0GJ-K3xgik3UXzyipP-y5HolgNT_S7IjkjkAsAA58Xbjftp3fGoV5hPVA=s0-d)
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
tf 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.
Pontryagin's minimum principle is a
necessary condition for an optimum. The
Hamilton–Jacobi–Bellman equation provides
sufficient conditions for an optimum.
Maximization and minimization
The results here are sometimes known as Pontryagin's maximum principle. This is because Pontryagin's original work focused on maximizing a benefit functional rather than minimizing a cost functional, the proof of the minimum principle is historically based on maximizing the Hamiltonian rather than minimizing the Hamiltonian. In this framework, to minimize the cost functional instead of maximizing a benefit functional, the functional should be multiplied by
− 1. Modern applications of this work focus on the minimization problem.
Formal statement of necessary conditions for minimization problem
Here the necessary conditions are shown for minimization of a functional. Take
x to be the state of the
dynamical system with input
u, such that
![\dot{x}=f(x,u), \quad x(0)=x_0, \quad u(t) \in \mathcal{U}, \quad t \in
[0,T]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tSFbX121iJ7-M5nAfKCuAW5IPLVlwL0mULG6NXytpRVVyQ6Skg77CBcQYIyhDNlAXqO_q6EbcE7rhi4kCreEKr1Cbj6BlQsgIPG6RNUP-Nh3k1eJFnilHgtKTLzu_wKCuxj4FGwQEXGXQOabLx=s0-d)
where

is the set of admissible controls and
T is the terminal (i.e., final) time of the system. The control

must be chosen for all
![t \in [0,T]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_urr4nVFH3eUy4ntAknEb6_b2MML3fT-HvzoH00jCUJNOTVt-TeaQQ1dNIwlUQyNdTQGeuVg4RwtYIyL6q3iSsXpQyJ9HdhVBdTUf5h-IUv7AUN-H47lKEK9d5XtkqOkfNheELf7zqjgrzE-HJl=s0-d)
to maximize the objective functional
J which is defined by the application and can be abstracted as

The constraints on the system dynamics can be adjoined to the
Lagrangian L by introducing time-varying
Lagrange multiplier vector
λ, whose elements are called the costates of the system. This motivates the construction of the
Hamiltonian H defined for all
![t \in [0,T]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_urr4nVFH3eUy4ntAknEb6_b2MML3fT-HvzoH00jCUJNOTVt-TeaQQ1dNIwlUQyNdTQGeuVg4RwtYIyL6q3iSsXpQyJ9HdhVBdTUf5h-IUv7AUN-H47lKEK9d5XtkqOkfNheELf7zqjgrzE-HJl=s0-d)
by:

where
λ' is the transpose of
λ.
Pontryagin's minimum principle states that the optimal state trajectory
x * , optimal control
u * , and corresponding Lagrange multiplier vector
λ * must minimize the Hamiltonian
H so that

for all time
![t \in [0,T]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_urr4nVFH3eUy4ntAknEb6_b2MML3fT-HvzoH00jCUJNOTVt-TeaQQ1dNIwlUQyNdTQGeuVg4RwtYIyL6q3iSsXpQyJ9HdhVBdTUf5h-IUv7AUN-H47lKEK9d5XtkqOkfNheELf7zqjgrzE-HJl=s0-d)
and for all permissible control inputs

. It must also be the case that

Additionally, the
costate equations

must be satisfied. If the final state
x(T) 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
x(T) is free. If it is fixed, then this condition is not necessary for an optimum.
The notation used above is defined below.





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