Pontryagin Maximum Principle

A result is Control Theory. Define

$$H(\psi,x,u)\equiv (\psi,f(x,u))\equiv \sum_{a=0}^n \psi_a f^a(x,u).$$

Then in order for a control $u(t)$ and a trajectory $x(t)$ to be optimal, it is Necessary that there exist Nonzero absolutely continuous vector function $\psi(t)=(\psi_0(t), \psi_1(t), \ldots, \psi_n(t))$ corresponding to the functions $u(t)$ and $x(t)$ such that

1. The function $H(\psi(t),x(t),u)$ attains its maximum at the point $u=u(t)$ almost everywhere in the interval $t_0\leq t\leq t_1$,

$$H(\psi(t),x(t),u(t))=\max_{u\in U} H(\psi(t),x(t),u).$$

2. At the terminal time $t_1$, the relations $\psi_0(t_1)\leq 0$ and $H(\psi(t_1),x(t_1),u(t_1))=0$ are satisfied.

References

Iyanaga, S. and Kawada, Y. (Eds.). ``Pontrjagin's [sic] Maximum Principle.'' §88C in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 295-296, 1980.