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

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

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 $\left(\tfrac{\partial H}{\partial t} \equiv 0\right)$, 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.

When satisfied along a trajectory, Pontryagin's minimum principle is a necessary condition for an optimum. The Hamilton–Jacobi–Bellman equation provides sufficient conditions 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 velocity 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.

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

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

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

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

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

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 minimize the objective functional $J$ which is defined by the application and can be abstracted as

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

The constraints on the system dynamics can be adjoined to the Lagrangian $L$ by introducing time-varying Lagrange multiplier vector $\lambda$, 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 $\lambda'$ is the transpose of $\lambda$.

Pontryagin's minimum principle states that the optimal state trajectory $x^*$, optimal control $u^*$, and corresponding Lagrange multiplier vector $\lambda^*$ must minimize the Hamiltonian $H$ so that

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

for all time $t \in [0,T]$ and for all permissible control inputs $u \in \mathcal{U}$. 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.

See also

• Lagrange multipliers on Banach spaces, Lagrangian method in calculus of variations

References

• 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 2881240771 and ISBN 978-2881240775
• 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 0486434842
• Sethi, S. P. and Thompson, G. L. Optimal Control Theory: Applications to Management Science and Economics, 2nd edition, Springer, 2000. ISBN 0387280928 and ISBN 0792386086. 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. (http://www.ElissarGlobal.com free chapter on Pontryagin's Principle).