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$1)=0.$

$1.$ 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.$)$

$2.$ The total $($kinetic plus potential$)$ energy of the system, up to an additive constant, is E$($$,$$)=$

$1$

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

dt E$($t$)$ and then, based on your calculation, suggest a heuristic control.