I am quite interested in any partical answer to the following general (maybe a little bit vague) question: Is there some criterion about the connectedness of the intersection of two connected algebraic subgroups of a linear algebraic subgroup defined over a perfect field?

Indeed, the general question is motivated by the following special case of it: Let $G$ be a semisimple group defined over $k$, which is the algebraic closure of a finite field. Let $U\subset G$ be a connected unipotent subgroup and $H\subset G$ be a parabolic (or any connected) subgroup, then is $U\cap H$ is still connected?

Any comments are very welcome! Thanks in advance!