Deriving costate equation and control law in optimal control

[mad mathematician]

Homework Statement

below

Relevant Equations

see below.

and its proposed solution:

Now the first equation seems to be $\dot{\lambda}=-H_{x}$, but they seem to forget to divide by two $(3x^2+u^2)$, and the second equation, I am not sure how did they derive the second equation of $u=-\lambda$? anyone here knows how?
thanks!

 

[Gavran]

and the second equation, I am not sure how did they derive the second equation of $u=-\lambda$? anyone here knows how?

thanks!

There are two optimality conditions and one of them is a necessary condition for the optimal control input $$ \frac{\partial H}{\partial u}=0 $$.

 

[mad mathematician]

There are two optimality conditions and one of them is a necessary condition for the optimal control input $$ \frac{\partial H}{\partial u}=0 $$.

Yeah, I know that. Forgot to reply that I found my answer... I hope to get a good grade in this course.
I tell you the exam was an open book; and I got so absent minded as to how to answer a few questions.
It's a pitty because the exam was really easy. (I had three exams this week in a row... phou ).