Some optimal control applications of real-analytic stratifications and desingularization

Abstract

The name of Stanis law Lojasiewicz has been associated in my mind with mathematical research since early on in my career, because it was through his work that I experienced for the first time in my life the thrill of studying in depth a major piece of new mathematics and following the thoughts of a creative mind. This happened thirty years ago, in 1966, when I reached the last semester of my undergraduate studies in mathematics in Buenos Aires, Argentina. The final requirement was a “trabajo de seminario superior,” for which the student was expected to write a detailed expository monograph based on an important recent research article. My topic was S. Lojasiewicz’s theorem on the division of distributions by real-analytic functions. Subsequently, my path diverged from his until 1977, when I learned from the work of Pavol Brunovsky —[1, 2]— about the possibility of using subanalytic sets in control theory. This was new to me but, thanks to my own previous exposure to semianalytic sets and stratifications through Lojasiewicz’s work, it was something I was ready for. Much later, the name “ Lojasiewicz” entered my life again in a different way, when I collaborated with Prof. Lojasiewicz’s son S. Lojasiewicz Jr. in 1983–84 in [4]. More recently, a brilliant idea of S. Lojasiewicz Jr. has played a decisive role in my own work [11, 12, 13, 14] on the maximum principle of optimal control. So the celebration of Professor Lojasiewicz’s 70th birthday has a special meaning to me, and this paper is dedicated to him with deep admiration and gratitude.