1. Extend Lemma 2.8 in Krstic et al. to systems of the form
where is continuously differentiable. Assume that
and are available such that , where
is a positive definite function.
Hint: Note that, by the mean-value theorem, the continuous differentiablility of
implies that there exists a continuous function such that
where .
2. Apply the result from Problem 1 to stabilize the system
3. Apply the result from Problem 1 and 2 to stabilize the system
4. Compute the Sontag formula for the system from Problem 2 using the backstepping clf.