Tackling coding challenges in constrained optimal control

When I took the graduate course “Numerical Mathematics I” at the University of Tennessee two years ago, my initial understanding was that the phrase “numerical mathematics” (or “numerical analysis”) referred solely to the aspect of writing code to approximate solutions to problems numerically: systems of equations, ordinary or partial differential equations, optimization problems, and so on. The instructor of the course, Abner Salgado (who later became one of my two PhD co-advisors), quickly erased that misconception from my head. As I learned from his courses, numerical analysis is about more than writing code; it also encompasses more theoretical results, namely the theoretical convergence of approximate solutions to a true solution. It is these properties that serve as the starting point for writing code, because they tell us what we want a computer to verify!

All that being said, writing code is still an important part of having a complete picture of the numerical analytical treatment of a problem. This is especially prevalent when there is a specific physical or biological application in mind, and in any case computational results can help attract the attention of a wider audience of researchers, including some whose primary domain is not mathematics.

This is a large part of the reason why Abner wanted me to write some code to supplement our more theoretical optimal control results (an ArXiV preprint link will be placed here once it is made available). In addition to this, he…