"May an undergraduate student assign his/her intellectual property rights to LSU?"
Yes, to the extent that an undergraduate student owns rights in intellectual property independently of LSU, he/she may assign such rights to LSU by mutual agreement. It is important to note LSU evaluates such inventions on a case-by-case basis and does not always agree to accept a proposed assignment by a student.
"When three undergraduate students in a senior design project in mechanical engineering
and a mechanical engineering faculty member are joint inventors of an invention, who
owns the invention?"
LSU owns the faculty member's interest in the invention. The students will usually own their interests in the invention, but may find it advantageous to assign their rights to LSU in exchange for a share of royalties, if doing so is mutually agreeable to the students and to LSU.