$[$Stopping a pendulum in minimum time$]$

Consider the system dynamics $$$($t$)=$sin$($$($t$))+$u$($t$)$ where $$ is a small constant. It models

the angle from vertical of a pendulum with the addition of a control. Consider the problem

of selecting a control u such that $|$u$($t$)|1$ for all t $0$ in order to minimize the time needed

to reach the resting state $($t$1)=$ $($t$1)=0.$

$($a$)$ Letting x$1=$ and x$2=$ $,$ write out the equations involving the state and co$-$state

variables implied by the minimum principle. Identify the optimal control as a function

of the co$-$state variables. $($You don$$t need not solve the equations. Even with zero

control the state trajectories of the pendulum can$$t be expressed in terms of elementary

functions.$)$

$($b$)$ The total $($kinetic plus potential$)$ energy of the system, up to an additive constant, is

E$($$,$ $)=1$

$2$

$2$ cos$($$).$ Calculate d

dtE$($t$)$ and then, based on your calculation, suggest a

heuristic control.

$2$