Pontryagin's minimum principle

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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 $\mathcal{U}$ , the set of all permissible controls. If $u^*\in \mathcal{U}$ is the optimal control for the problem, then the principle states that:

$H(x^*(t),u^*(t),\lambda^*(t),t) \leq H(x^*(t),u(t),\lambda^*(t),t), \quad \forall u \in \mathcal{U}, \quad t \in [t_0, t_f]$

where $x^*\in C^1[t_0,t_f]$ is the optimal state trajectory and $\lambda^* \in BV[t_0,t_f]$ 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 $t f$ is fixed and the Hamiltonian does not depend explicitly on time ( $\frac{\partial H}{\partial t} \equiv 0$ ), then:

$H(x^*(t),u^*(t),\lambda^*(t)) \equiv \mathrm{constant}\,$

and if the final time is free, then:

$H(x^*(t),u^*(t),\lambda^*(t)) \equiv 0.\,$

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.

[edit] 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.

[edit] 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

$\dot{x}=f(x,u), \quad x(0)=x_0, \quad u(t) \in \mathcal{U}, \quad t \in [0,T]$

where $\mathcal{U}$ is the set of admissible controls and $T$ is the terminal (i.e., final) time of the system. The control $u \in \mathcal{U}$ must be chosen for all $t \in [0,T]$ to maximize the functional $J$ , which is defined by

$J=\Psi(x(T))+\int^T_0 L(x(t),u(t)) dt$

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 $t \in [0,T]$ by:

$H(\lambda(t),x(t),u(t),t)=\lambda'(t)f(x(t),u(t))+L(x(t),u(t)) \,$

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

$(1) \qquad H(x^*(t),u^*(t),\lambda^*(t),t)<H(x^*(t),u(t),\lambda^*(t),t) \,$

for all time $t \in [0,T]$ where $u$ for all suitable controls $u$ . It must also be the case that

$(2) \qquad \Psi_T(x(T))+H(T)=0 \,$

Additionally, the costate equations

$(3) \qquad -\dot{\lambda}'(t)=H_x(x^*(t),u^*(t),\lambda^*(t),t)=\lambda'(t)f_x(x^*(t),u^*(t))+L_x(x^*(t),u^*(t))$

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

$(4) \qquad \lambda'(T)=\Psi_x(x(T)) \,$

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.

$\Psi_T(x(T))= \frac{\partial \Psi(x)}{\partial T}|_{x=x(T)} \,$

$\Psi_x(x(T))=\begin{bmatrix} \frac{\partial \Psi(x)}{\partial x_1}|_{x=x(T)} & \cdots & \frac{\partial \Psi(x)}{\partial x_n} |_{x=x(T)} \end{bmatrix}$

$H_x(x^*,u^*,\lambda^*,t)=\begin{bmatrix} \frac{\partial H}{\partial x_1}|_{x=x^*,u=u^*,\lambda=\lambda^*} & \cdots & \frac{\partial H}{\partial x_n}|_{x=x^*,u=u^*,\lambda=\lambda^*} \end{bmatrix}$

$L_x(x^*,u^*)=\begin{bmatrix} \frac{\partial L}{\partial x_1}|_{x=x^*,u=u^*} & \cdots & \frac{\partial L}{\partial x_n}|_{x=x^*,u=u^*} \end{bmatrix}$

$f_x(x^*,u^*)=\begin{bmatrix} \frac{\partial f_1}{\partial x_1}|_{x=x^*,u=u^*} & \cdots & \frac{\partial f_1}{\partial x_n}|_{x=x^*,u=u^*} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_n}{\partial x_1}|_{x=x^*,u=u^*} & \ldots & \frac{\partial f_n}{\partial x_n}|_{x=x^*,u=u^*} \end{bmatrix}$