Solve the problem of optimal control to the origin for the system

where

I have done a bit, but Im not really sure where to go next....

Let

where

Then

where

Then

where

The costate equations are

so maximizes as a function of . As is linear in and then is maximized such that when and when

The eigenvalues are found when

when solving we have two eigenvalues

.

I'm not sure if what I have done is correct, and I don't know where to go from here.