# Selected Notes

### MISCELLANEOUS NOTES

These are some notes I have taken through my undergraduate study.

### LECTURE NOTE ON HOMOGENIZATION

(July 14, 2015.)
In summer 2015, while attending a summer course at the University of Science, Vietnam National University, HCMC, Son P.T. Van and I wrote lecture notes for this course.

• Abstract: Consider the linear control system $\displaystyle x'(t) = Ax(t) + Bu(t)$ with the target ${0}$. The function $T(x)$ is the minimum time needed to steer a point $x$ to ${0}$. It is well-known that under the Rank-Kalman condition, $T$ is just continuous in the reachable set. Our main goal is to study the regularity of $T(x)$. We prove that $T$ is a.e. twice differentiable. Moreover, we also obtain an explicit formula for the set of singularities where the optimal control changes its sign.