Arthur J. Krener received the PhD in Mathematics from the University of California, Berkeley in 1971. From 1971 to 2006 he was at the University of California, Davis. He retired in 2006 as a Distinguished Professor of Mathematics. Currently he is a Distinguished Visiting Professor in the Department of Applied Mathematics at the Naval Postgraduate School. His research interests are in developing methods for the control and estimation of nonlinear dynamical systems and stochastic processes.
Professor Krener is a Life Fellow of the IEEE, a Fellow of IFAC and of SIAM. His 1981 IEEE Transactions on Automatic Control paper with Isidori, Gori-Giorgi and Monaco won a Best Paper Award. The IEEE Control Systems Society chose his 1977 IEEE Transactions on Automatic Control paper with Hermann as one of 25 Seminal Papers in Control in the last century. He was a Fellow of the John Simon Guggenheim Foundation for 2001-2. In 2004 he received the W. T. and Idalia Reid Prize from SIAM for his contributions to control and system theory. He was the Bode Prize Lecturer at 2006 IEEE CDC and in 2010 he received a Certificate of Excellent Achievements from IFAC. His research has been continuously funded since 1975 by NSF, NASA, AFOSR and ONR.
In 1988 he founded the SIAM Activity Group on Control and Systems Theory and was its first Chair. He was again Chair of the SIAG CST in 2005-07. He chaired the first SIAM Conference on Control and its Applications in 1989 and the same conference in 2007 both in San Francisco. He also co-chaired the IFAC Nonlinear Control Design Symposium held at Lake Tahoe in 1996. He has served as an Associate Editor for the SIAM Journal on Control and Optimization and for the SIAM book series on Advances in Design and Control.
It is a honor to receive the 2012 Richard E. Bellman Control Heritage Award. I am deeply humbled to join the very distinguished group of prior winners. At this conference there are so many people whose work I have admired for years. To be singled out among this group is a great honor.
I did not know Richard Bellman personally but we are all his intellectual descendants. Years ago my first thesis problem came from Bellman and currently I am working on numerical solutions to Hamilton-Jacobi-Bellman partial differential equations.
I began graduate school in mathematics at Berkeley in 1964, the year of the Free Speech Movement. After passing my oral exams in 1966, I started my thesis work with R. Sherman Lehman who had been a postdoc with Bellman at the Rand Corporation in the 1950s. Bellman and Lehman had worked on continuous linear programs also called bottleneck problems in Bellman’s book on Dynamic Programming. These problems are dynamic versions of linear programs, with linear integral transformations replacing finite dimensional linear transformations. At each frozen time they reduce to a standard linear program. Bellman and Lehman had worked out several examples and found that often the optimal solution was basic, at each time an extreme point of the set of feasible solutions to the time frozen linear program. These extreme points moved with time and the optimal solution would stay on one moving extreme point for awhile and then jump to another. It would jump from one bottleneck to another.
Lehman asked me to study this problem and find conditions for this to happen. We thought that it was a problem in functional analysis and so I started taking advanced courses in this area. Unfortunately about a year later Lehman had a very serious auto accident and lost the ability to think mathematically for some time. I drifted, one of hundreds of graduate students in Mathematics at that time. Moreover, Berkeley in the late 1960s was full of distractions and I was distractable. After a year or so Lehman recovered and we started to meet regularly. But then he had a serious stroke, perhaps as a consequence of the accident, and I was on my own again.
I was starting to doubt that my thesis problem was rooted in functional analysis. Fortunately I had taken a course in differential geometry from S. S. Chern, one of the pre-eminent geometers of his generation. Among other things, Chern had taught me about the Lie bracket. And one of my graduate student colleagues told me that I was trying to prove a bang-bang theorem in Control Theory, a field that I had never heard of before. I then realized that my problem was local in nature and intimately connected with flows of vector fields so the Lie bracket was an essential tool. I went to Chern and asked him some questions about the range of flows of multiple vector fields. He referred me to Bob Hermann who was visiting the Berkeley Physics Department at that time.
I went to see Hermann in his cigar smoked-filled office accompanied by my faithful companion, a German Shepherd named Hogan. If this sounds strange, remember this was Berkeley in the 1960s. Bob was welcoming and gracious, he gave me galley proofs of his forthcoming book which contained Chow’s theorem. It was almost the theorem that I had been groping for. Heartened by this encounter I continued to compute Lie brackets in the hope of proving a bang-bang theorem.
Time drifted by and I needed to get out of graduate school so I approached the only math faculty member who knew anything about control, Stephen Diliberto. He agreed to take me on as a thesis student. He said that we should meet for an hour each week and I should tell him what I had done. After a couple of months, I asked him what more I needed to do to get a PhD. His answer was ”write it up”. My ”proofs” fell apart several times trying to accomplish this. But finally I came up with a lemma that might be called Chow’s theorem with drift that allowed me to finish my thesis.
I am deeply indebted to Diliberto for getting me out of graduate school. He also did another wonderful thing for me, he wrote over a hundred letters to help me find a job. The job market in 1971 was not as terrible as it is today but it was bad. One of these letters landed on the desk of a young full professor at Harvard, Roger Brockett. He had also realized that the Lie bracket had a lot to contribute to control. Over the ensuing years, Roger has been a great supporter of my work and I am deeply indebted to him.
Another Diliberto letter got me a position at Davis where I prospered as an Assistant Professor. Tenure came easily as I had learned to do independent research in graduate school. I brought my dog, Hogan, to class every day, he worked the crowds of students and boosted my teaching evaluations by at least a point. After 35 wonderful years at Davis, I retired and joined the Naval Postgraduate School where I continue to teach and do research. I am indebted to these institutions and also to the NSF and the AFOSR for supporting my career.
I feel very fortunate to have discovered control theory both for the intellectual beauty of the subject and the numerous wonderful people that I have met in this field. I mentioned a few names, let me also acknowledge my intellectual debt to and friendship with Hector Sussman, Petar Kokotovic, Alberto Isidori, Chris Byrnes, Steve Morse, Anders Lindquist, Wei Kang and numerous others.
In my old age I have come back to the legacy of Bellman. Two National Research Council Postdocs, Cesar Aguilar and Thomas Hunt, have been working with me on developing patchy methods for solving the Hamilton-Jacobi-Bellman equations of optimal control. We haven’t whipped the ”curse of dimensionality” yet but we are making it nervous.
The first figure shows the patchy solution of the HJB equation to invert a pendulum. There are about 1800 patches on 34 levels and calculation took about 13 seconds on a laptop. The algorithm is adaptive, it adds patches or rings of patches when the residual of the HJB equation is too large. The optimal cost is periodic in the angle. The second figure shows this. Notice that there is a negatively slanted line of focal points. At these points there is an optimal clockwise and an optimal counterclockwise torque. If the angular velocity is large enough then the optimal trajectory will pass through the up position several times before coming to rest there.
What are the secrets to success? Almost everybody at this conference has deep mathematical skills. In the parlance of the NBA playoffs which has just ended, what separates researchers is “shot selection” and ”follow through”. Choosing the right problem at the right time and perseverance, nailing the problem, are needed along with good luck and, to paraphrase the Beatles, ”a little help from your friends”.