This course is an element of the LSU VIGRE project. The project combines graduate and undergraduate education and research. Here is some basic information about the course from the graduate program page:
- MATH 4997-2: Vertically Integrated Research: Lyapunov Functions, Stabilization, and Engineering Applications
- Instructors: Prof. Malisoff (lead), Prof. de Queiroz, and Prof. Wolenski
- Prerequisite: For graduate students: 7320, 7386, or permission of the lead instructor. For undergraduates: 4027, 4340, or permission of the lead instructor.
- Text: Notes and recommended references provided by the instructors
Overview: Mathematical control theory is one of the most central and fast growing areas of applied mathematics. This course will help prepare students for research at the interface of engineering and applied mathematics. The first part provides a self-contained introduction to the mathematics of control systems, focusing on feedback stabilization and Lyapunov functions. The second part will be a series of lectures by control engineering faculty that will discuss open problems in control. The third part will explore ways of solving the problems. The only prerequisite is a graduate or advanced undergraduate course on the theory of differential equations. Students from engineering or mathematics are encouraged to enroll. Here are some related references:
- M.S. de Queiroz, D.M. Dawson, S. Nagarkatti, and F. Zhang, Lyapunov-Based Control of Mechanical Systems. Control Engineering Series, Birkhauser, Cambridge, MA, 2000. ISBN: 0-8176-4086-X
- M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions. Communications and Control Engineering Series, Springer-Verlag London Ltd., London, UK, 2009. ISBN: 978-1-84882-534-5
Here are answers to some questions students had about this course:
Q: I haven't taken an ODE theory course, nor have I taken a PDE course, but I took a PDE-based theoretical grad course at LSU. Can students with such a background still take this VIR course?
A: Yes, that is sufficient background. However, it would be helpful for such students to read Chapters 7 and 17 of [M. Hirsch, S. Smale, and R. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Second Edition, Elsevier, New York, 2004 (ISBN 978-0-123497-03-1)] before the VIR course starts.
Q: I'm a math major and I know I have to take a capstone course to graduate. I want to go to grad school in either math or engineering. Is this VIR course a good course for me to take as a capstone course?
A: Yes, you can take a 4997 VIR course to fulfill your capstone course requirements, and the VIR control course is strongly recommended for students with an interdisciplinary bent.
Q: Could you recommend a brief introduction to control theory so I can get some idea of what it's about?
A: Yes. Chapter 1 of Eduardo Sontag's book Mathematical Control Theory is a nice introduction to the subject at the advanced undergraduate level.
Q: I am not sure if I want to specialize in systems and control, but I plan to work towards a PhD in applied mathematics. Is it still reasonable for me to take this VIR course this fall?
A: Yes, this course would be a reasonable choice for all math PhD students interested in applied mathematics, because control theory is used in many areas; see http://www.ieeecss.org/.
Q: What are the specific course requirements, in terms of exams and grades and so forth?
A: Here is the official syllabus. The classwill meet on Friday August 27th at the usual time and location.
Q: Are there interesting classes of inputs that are neither continuous nor simple step functions?
A: Yes. One could consider so-called chattering controls, which you can read about in the book Theory of Chattering Control by Borisov and Zelikin. There's a copy on reserve at the Middleton Library.
Q: The ISS property you described in class on 8/25 is very interesting. Could you give a specific reference for the paper that introduced ISS?
A: The ISS property seems to have originated in http://dx.doi.org/10.1109/CDC.1988.194352 which should be freely accessible from all LSU computers.
Q: I'm an engineering grad student with a pretty good background in math that includes advanced calculus, linear algebra, and advanced ODEs, but I've never taken any math grad courses. I'm really interested in control theory because I know control systems are used a lot in engineering. Should I still sign up for your course?
A: Yes, you should still sign up because it sounds like you have the proper prerequisites, but double check with your adviser to make sure he or she doesn't want you in a different course this semester.
Q: I didn't find out about the course until 8/26 so I missed the first 2 or 3 days of class. Can I still sign up?
A: Yes, you can. The first few class meetings are just a general overview of the subject that is not essential for following the rest of the course.
Q: Where can I find a proof of the existence and uniqueness theorem that we discussed in class on 8/27?
A: Please check http://www.math.lsu.edu/~malisoff/4997/Fall2010/ExistenceAndUniquenessFromSontagsBook.pdf.
Q: Where can I find a proof that the existence of an iISS Lyapunov function is equivalent to iISS?
A: The paper http://dx.doi.org/10.1109/9.863594 gives this and several other equivalent characterizations of iISS.
Q: Do we have a class meeting on October 1st?
A: There will be a control seminar in Room 117 in the Electrical Engineering building on Friday October 1st at 1:30PM sharp, and you are encouraged to attend. Here is the seminar abstract.
Q: Where can I read about a solution to the adaptive estimation and tracking problem for adaptive systems where there are also unknown parameters multiplying the input?
A: Here is one recent result in that direction.