Pontryagin's Minimum Principle - Formal Statement of Necessary Conditions For Minimization Problem

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

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

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

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 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:

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

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

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

for all time and for all permissible control inputs . 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 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 is free. If it is fixed, then this condition is not necessary for an optimum.

Read more about this topic:  Pontryagin's Minimum Principle

Famous quotes containing the words formal, statement, conditions and/or problem:

    “True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
    —Marcel Proust (1871–1922)

    “He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
    —Margaret Halsey (b. 1910)

    “We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
    —Urie Bronfenbrenner (b. 1917)

    “We have heard all of our lives how, after the Civil War was over, the South went back to straighten itself out and make a living again. It was for many years a voiceless part of the government. The balance of power moved away from it—to the north and the east. The problems of the north and the east became the big problem of the country and nobody paid much attention to the economic unbalance the South had left as its only choice.”
    —Lyndon Baines Johnson (1908–1973)