For those who read my blog a couple of times, most of you would realized that I am currently in the field of control theory. Control theory is one of those fields that requires a lot of mathematics. I have been in the field for roughly 2 years now and I have realized that they are 2 big issues within control theory, namely: (i) identity of a control engineer/theorist, and (ii) stagnation of fundamental theory advancements. In this post, I would like to relate mathematics to the above issues and perhaps generate some discussions.

Mathematics is arguably the most important subject in science and technology. Imagine our current world without calculus and discrete mathematics, this would virtually mean there would be no modern civilization. Calculus is used in building high rise building, cars, computers and more. Discrete mathematics provides means of programming computer and understanding logic. It is not unusual to expect the modern study of science and technology requires huge amount of mathematics.

Of course, we, control engineers (or theorists), use calculus and a bit of discrete mathematics to provide control algorithms for certain systems. Often, this would mean that we will need to use mathematical tools to prove certain things. Sometimes, things will get so theoretical that it is impossible for anyone with undergraduate degree to understand what control theorists are talking about. Yet, there are also theories that are not rigorously proved to work, but simulations show that the scheme is a plausible one. Such a phenomenon causes an identity crisis for us control engineers/theorist. Are we engineers or applied mathematicians? In fact, should we take a more hand-wavy approach (the engineer approach) to control stuff or should it be done in a more rigorous manner (the mathematician approach)? Each of them has their benefits and drawbacks and it unclear that whether we can merge the two approaches together.

Another issue that I have noticed within control theory is the phenomenon of having not enough people working on fundamental theories. For example, we are still using Lyapunov theory, introduced in 1892, to determine the stability of continuous nonlinear dynamical system. Stability is extremely important in control engineering and those, who know Lyapunov theory, would know how troublesome it is to use the theory. Often fundamental theories like Lyapunov theory are very mathematical and it is really sad that there is no big advancement in the stability theory. So, what has gone wrong? Is it because fundamental theories are too hard to study? Or is it because researchers can no longer get funding for research as such? Or is it just because we control theorists/engineers don't interact enough with mathematicians?

It seems that I have generated more questions that I could possibly answer them myself. I would very much appreciate it if readers can help me answer some of it. So, feel free to provide me with your opinion. Thank you.

Stability via the Lyapunov function approach remains an important research direction for some control theorists. There are many extensions, for instance the notion of input/output-stability. Some restrictive conditions of the original Lyapunov stability theory are also relaxed. For instance, exploiting the information of system trajectory, one can prove stability using a positive definite function who derivative w.r.t time needs not to be (semi) negative-definite!

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