# Oral-History:Arthur Krener

## About Arthur Krener

Arthur Krener received a PhD in Mathematics in 1971 from the University of California, Berkeley and joined the faculty of the University of California, Davis, and retired in 2006 as a Distinguished Professor of Mathematics. Krener is a member of the American Mathematical Association, a Fellow of the Society for Industrial and Applied Mathematics and a Life Fellow of the Institute of Electrical and Electronics Engineers, and was recipient of the 2006 IEEE Control System Society Bode Prize Lecture “for fundamental contributions to the foundations of geometric nonlinear control theory”.

In this interview, Krener discusses his work and career at UC Davis, geometric control and software packages.

## About the Interview

ARTHUR KRENER, an oral history conducted in 2015 by Wei Kang, Paris, France at SIAM CT 15.

Interview #757 for the Center for the History of Electrical Engineering, The Institute for Electrical and Electronics Engineers, Inc.

## Copyright Statement

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It is recommended that this oral history be cited as follows:

Arthur Krener, an oral history conducted in 2015 by Wei Kang, Paris, France at SIAM CT 15.

## Interview

INTERVIEWEE: Arthur Krener

INTERVIEWER: Wei Kang

DATE: 8 July 2015

PLACE: Paris, France at SIAM CT 15

### Video

### Introduction

**Kang:**

Hello. My name is Wei Kang. This interview of oral history is a part of the joint effort of IEEE History Center and SIAM CT15. It is my great pleasure to introduce Professor Arthur Krener of U.S. Naval Postgraduate school and the University of California at Davis. Welcome to the interview, Art.

**Krener:**

Thank you, Wei.

**Kang:**

About two weeks ago I saw the news that you just won the IEEE control system award, a very prestigious award and well deserved, of course, congratulations!

**Krener:**

Well, thank you.

### Early Life, Childhood, University

**Kang:**

So, why don’t we start from some kind of background information. Where were you brought up and where did you begin your University studies?

**Krener:**

Well, I was born during the second World War in Brooklyn, New York. My father was off fighting the war. As a baby, I lived with my aunt, and my mother and my grandmother in an extended household. And when my father came back from the war he joined the New York City Police Department. We lived together in a second house as a big extended family for a while. But then when I was eight years old my family moved to Queens, and my brother, my mother and father and myself lived as a nuclear family. It was a working class neighborhood as they didn’t overpay policemen in those days. I don’t think they overpay them today. My mother was a school secretary. So I went to local parochial schools, Catholic schools and then I went to the Jesuit high school in Brooklyn. I had to take the subway an hour each way every day to get to the school. And very often I was late so I had to spend an extra half hour after school in the jug as the priests called it. Then I went on to Holy Cross College in Worcester, Massachusetts, another Jesuit institution. All the time I wanted to be a math major but I had no idea what control theory was, it just wasn’t on my horizon.

**Kang:**

While in college, what was your major? Mathematics?

**Krener:**

I majored in Math. I had been interested in Chemistry and Physics in high school but then I looked at the course schedule. At Holy Cross in those days your four year schedule was completely planned out except for one or two electives. I thought that Math had more electives than Physics and Chemistry. If I were in Physics and Chemistry I would have been in the labs a lot more and I had no desire to do that so I majored in Math. When I graduated in 1964, I applied to several graduate schools. I really wanted to go to Berkeley and I heard acceptances from many schools but never heard from Berkeley. I went to one of the professors in the Math department who had just come back from a sabbatical at Berkeley and told him my situation. He made a phone call or two and the next day I had a TA ship offer. So in the fall of 1964 I moved from Worcester, Massachusetts to Berkeley, California.

**Kang:**

I see. So it seems that you actually started very early to get interested in Mathematics and then-

**Krener:**

Well, yes. Math always came easy for me.

**Kang:**

And under whose direction did you work in Berkeley?

**Krener:**

Well, that is a complicated story too. I arrived in the fall of 1964 during the Free Speech Movement, which was the first of the disturbances in Berkeley during the 60’s. There were 400 graduate students at Berkeley in Mathematics in those days and about 70 or 80 professors. So to say you got personal attention would be a gross exaggeration. I spent the first two years taking courses. I had to do a lot of adjusting in my study habits because I had always breezed through undergraduate Mathematics courses. All of a sudden the competition was a lot harder and fiercer at Berkeley. But I passed my qualified exams the second year and then looked around for a thesis advisor. I was TAing a course on Numerical Analysis for a Professor named Sherman Lehman. We got to talking and I told him that I was taking a course in linear programming in Industrial Engineering. I expressed my interest in the material and he said that he had been a post doc with Richard Bellman in the 50’s at the Rand Institute. They had worked on time varying version of linear programming. They called it continuous linear programming. Basically the linear operators are linear integral operators. They had found that if you froze one of these continuous linear programs at a fixed time it was a classical linear program. What they had found by a series of examples was that optimal trajectories jumped from one extreme point to another as time marched on and the feasible set rotated. The feasible set of the optimal control problem changes but the optimal solution jumped from one extreme point to another. They had never been able to prove that this was a universal phenomenon. So that was my first thesis problem.

**Kang:**

As you mentioned that it was kind of complicated. It so far sounds like kind of a normal-

**Krener:**

Yes. I was making normal progress and then Sherman Lehman, my advisor, was in a terrible car accident. Somebody ran a red light and smashed into him. And he was very seriously ill and not functioning mathematically for about a year afterwards. So I kind of floated with these 400 graduate students and 80 faculty. Nobody was tracking anybody very well. I kept on coming back to my thesis problem and took courses in functional analysis. I thought the solution to my problem was in functional analysis, a Hahn Banach theorem or something like that. But after about two or three years I started to doubt that the problem was functional analytic . Lehman recovered but he wasn’t really fully functional mathematically. Then he had stroke and the same thing happened. So I was kind of adrift. I had a thesis problem. I thought I had techniques, functional analysis techniques but I couldn’t drive anything home. I had taken a course in differential geometry from SS Chern the famous Chinese American geometer and Chern had taught me what the Lie bracket was in a very fundamental and intuitive way. He was a brilliant teacher and a very kind man. I tried to formulate what I was working on in some kind of geometric terms and went to him and explained to him what I was working on. He said to me that he did not know the answers to the questions I was asking but he said I know who might. He sent me to the Physics Department where Bob Hermann was a visiting professor. For people that don’t know Bob, he was a flamboyant mathematician. Princeton graduate, man of immense scope but very little discipline. He had more ideas than he could ever carry through. Anyhow Bob listened to my question and said oh, you are looking for Chow’s theorem. This theorem is the work of a Chinese mathematician who proved it in Shanghai while the Japanese were bombing. It turned out I was looking for unidirectional version of Chow’s theorem. Chow’s theorem allows trajectories to go backwards and forwards but in control, time marches in only one direction. I was much encouraged. At that time I had a faithful sidekick, my German Shepard dog who would follow me around campus. When I went into Bob’s office for the first time he was chain smoking cigars. The whole room was full of cigar smoke. So between him and his cigars and me and my dog, we were an odd pair.

**Kang:**

There was Berkeley.

**Krener:**

There was Berkeley. There was definitely Berkeley. There were a series of disturbances in Berkeley, People’s Park, Free Speech Movement, Third World, College, Anti Vietnam War, etc. I got interested in student politics. I was the shop steward for the math department TAs and got arrested one day on a picket line. I was released two hours later. The police were just harassing us. It was an exciting time. Then I met my wife, my wife to be and I realized that I had wasted a lot of time enjoying the distractions of Berkeley. So I knuckled down and told myself that I had to find a second thesis advisor. The only professor in the math department that knew anything about optimal control or control in general was a man named Steven Diliberto, he was a student of Solomon Lefschitz and had written a paper of two on the Pontryagin Maximum Principle. I went to see him and I explained to him what I was working on. By that time I knew what I was trying to prove a bang bang theorem in optimal control. If there is an optimal solution then there is a bang bang optimal solution. This information was conveyed to me by one of my fellow graduate students and that really marks my transition to becoming a control theorist. That is the first time I realized that I was doing control. I explained to Diliberto what I was working on and asked if he would take me on as a student. We started meeting every week. Unfortunately Diliberto was going through a divorce. So he didn’t show up every week. He showed up every other week. I was getting desperate. Finally I said to him how much more do I have to do to get out of graduate school. And he said to me, write it up. Of course in the process of writing it up I realized a lot of things that I thought were obvious and true were not obvious and not true and after several false starts I finally managed to produce a PhD. I graduated in 1971. One of the many nice things Diliberto did for me was to write a 100 recommendation letters. This was impressive in those days because this was before word processors. He had a secretary type 100 letters of recommendation and send them out. One of them landed on the desk of Roger Brockett at Harvard. Roger had also been interacting with Bob Hermann and had come to the same conclusion that Bob and I had come to which was that Lie bracket had a lot to contribute to nonlinear control. Roger called me up and we chatted a little . I had a wedding, a family wedding in New York at the time so I agreed to go up and meet with him and we had a very nice chat. He said keep in touch and we will see what happens. I was able to grab one of the last tenured track jobs of the early 70’s. The job market was very generous in the 60’s, the late 60’s but by 1971, ’72. it was really drying up.

### UC Davis

**Kang:**

Sorry, I don’t mean to interrupt. So basically when you were at Berkeley as a graduate student you already made up your mind about being a professor in university, rather than going to industry. So you had a clear career plan.

**Krener:**

Yes. I always wanted to be a mathematician. An academic mathematician. So fortunately UC Davis, 70 miles away from Berkeley, made me a job offer. I snapped it up. In the fall of ’71 I started at Davis. I was at Davis for 35 years until I retired in 2006. But along the way Roger invited me to come back to Harvard as a postdoctoral fellow. I was already on a tenure track position and I wasn’t about to resign a tenure track position so I just told my university that I was taking a leave of absence to work at Harvard. Roger represented me as a postdoctoral fellow to the Harvard administration and I got time on the tenure clock for my year in the Harvard ’74-’75. Roger had a profound influence on me. He is a remarkable person, very generous, very enthusiastic. A deep thinker and so he really supplied a lot of the education, the training in control that I didn’t get as a graduate student.

**Kang:**

You spent 35 years in Davis and I know that you visited many other places and universities for your sabbatical leaves and projects. Where else have you worked and visited?

### Geometric control theory

**Krener:**

Well, in 1973 there was a famous meeting in geometric control theory, nonlinear geometric control theory organized by Roger Brockett and David Mayne in London. I met Alberto Isidori there who became a very close friend and collaborator over the years. A couple of years later I visited Italy for a meeting and Alberto asked if I would care to come for a longer period. I was just going through a divorce at that time so I was very flexible. I said sure. So my first big trip away from home besides Harvard was three months I spent with Alberto in Rome in 1979.

**Kang:**

I see. Yes. Go ahead.

**Krener:**

We had a very successful collaboration. We were able to produce a paper that won the best paper award in the IEEE Journal on Automatic Control.

**Kang:**

Yes, I certainly heard a lot about this London workshop. From Harvard to the London workshop and Rome visit, during this period of time you were mostly focusing on the geometric control.

**Krener:**

Almost completely. And frankly by 1978, early 1979 some of the fundamental questions of reachability had been dealt with by geometric methods. So the field was getting a little stale and new impetuses were needed. One was my collaboration with Alberto, it really opened up a broader horizon. Alberto had a very clear goal of bringing to nonlinear systems many of the geometric concepts of linear systems. He particularly particularly interested in generalizing the work of Wonham and Morse. That is what we had worked on when I was in Rome. L later he and Chris Byrnes generalized some work of Wonham’s student, Bruce Francis, in their famous paper on nonlinear regulation.

**Kang:**

I remember that in the ‘80s you spent a few years, kind of away from the geometric control, working on reciprocal processes. Is there-

**Krener:**

Oh, yes.

**Kang:**

- anything you want to talk about?

**Krener:**

Well, frankly I think it is the best work I have ever did.

**Kang:**

Wow.

### Mathematical Problems in Image Processing

**Krener:**

But it is completely ignored. It hasn’t raised a blip. In about 1980 or so I attended a workshop at the Naval Postgraduate school on Mathematical Problems in Image Processing. The dominant theme of that meeting was to generalize Kalman filtering from one dimension to two dimensions. When I say dimensions I mean independent variables , one dimensional time, two dimensional time. People were doing Kalman filtering for partial difference models that were first order hyperbolic or parabolic. So they had some sort of causality where the information structure in the process would advance from upper left to lower right in hyperbolic models and line-by-line in parabolic models. I thought a fundamental question had not been addressed in this field. The most significant thing about first or second order of partial difference equations is this characteristic structure of the equations. Whether they are characteristics and how many of them, are they real and so on. The question I asked many people at that conference was how does the interplay between the characteristic and the stochastic structure work itself out. How can you tell from the covariance of the process what is the characteristic structure of the underlying difference equation. How could you deduce what kind of differential equation or difference equation it came from? Surprisingly nobody had thought to ask this question. So I said well, there is a fun question. I will just knock it off in a weekend. Well, I started doing it.

**Kang:**

It was a long weekend.

**Krener:**

It was a very long weekend. So I said well, maybe what I should do is back up. I always tell my graduate students, Wei was my graduate student, I always tell my graduate students to first attack the simplest problem that you don’t understand. Don’t try to do the general problem right away. Just get your feet on the lowest rung on the ladder and work your way up. So I took a standard one dimensional process, A, B, C, D model and instead of putting initial conditions on the process, I put boundary conditions. So between the state at times 0 and the state at time T, there are 2n degrees of freedom. Let’s constrain the value of n linear functionals of those 2n degrees of freedom to be Gaussian random vector . You have to worry about whether the process is well posed but that is a fairly straightforward question. So I got these boundary value models which I was very happy with. I did a lot of work in classifying the stationary Gaussian processes that they generated . But I could never get my hand on the stochastic characterization of these models. If initial conditions were imposed on the process then it was a Markov process but generally boundary conditions destroyed the Markovian property. What was the stochastic replacement for that? I stumbled on a paper of Gene Wong that mentioned reciprocal processes. He referenced a paper by Benton Jamison. I found his paper and he claimed to have classified all one-dimensional stationary reciprocal processes. Stationary in the sense that the covariance was a function of t minus s rather than t and s. So I compared his list with the list that I had worked out for the boundary value processes and there was considerable overlap. But each list had one process that the other one didn’t have. So I was very frustrated. I didn’t know what to make of this. But about a year later I woke up one morning. You know how serendipity sometimes hits. I woke up one morning with a proof in my head that all these boundary value processes were actually reciprocal processes. So I went back to the literature and did a forward search from Jamison’s article and found that there had been two corrections of Jamison’s list. When you took the full list, my set of boundary value processes was a subset of Jameson’s set of reciprocal processes. Every boundary value process was a reciprocal process. But what about the one additional process from Jamison’s list that wasn’t a boundary value process? This question led me into reciprocal processes and stochastic differential equations of second order. The analog for reciprocal processes of the Fokker-Planck equation for Markov processes is a series of conservation laws very similar to the conservation laws of quantum mechanics and it is a beautiful theory. I discovered later that the whole idea of reciprocal processes dates back to the Russian mathematician, Serge Bernstein, who was trying to reformulate some ideas of Schroedinger. Schroedinger had been trying to interpret quantum mechanics as a stochastic process of some sort. He studied some pinned Markov processes. Bernstein gave the rigorous definition of what Schroedinger was doing. A process parameterized by one dimensional parameter is reciprocal if what happens in between two times is independent of what happens outside of the two times, conditioned on the value of the process at the two times. It turns out every Markov process is reciprocal but not vice versa. So this got me off on reciprocal processes and we thought at one point we had achieved Schroedinger’s goal of stochasticizing quantum mechanics. But it only worked for the ground state of the quantum mechanical oscillator. Once we went to first excited state it all fell apart but it was fun.

### Collaboration

**Kang:**

Wow. So far you mentioned many collaborators, you know, from Alberto Isidori to Roger Brockett. I know that you have a very impressive list of collaborators and some collaborations led to award winning work. So, tell us about some of your close collaborators?

**Krener:**

Well, I was very fortunate. A characteristic that I found in the controls community is that people are very open, very accepting and very generous and uh, not competitive, you know not excessively competitive. Everybody wants to do well. But I was able collaborate several times with people in papers, for example, Isidori. I worked with some graduate students on the reciprocal processes. Bernard Levy in the Electrical Engineering department at Davis also collaborated on reciprocal processes. He made some fundamental insights. And then later on I got into power series methods and worked with Mont Hubbard in Mechanical Engineering at Davis. Chris Byrnes was a big influence on me. His work with Isidori opened eyes, opened my eyes. So it is not just people that you collaborate with, it is the people who work near you whose work really impresses you. There were a lot of them.

**Kang:**

Talking about people near you, there are influences and collaborations. So, who do you think has had the most profound effect on your career, and why?

**Krener:**

Well, I mean there are two or three choices here. I think Roger Brockett would be the first really seminal figure in my career and then Alberto Isidori would be the second seminal figure in my career. You contributed a lot.

**Kang:**

It is my honor. Thank you.

**Krener:**

Those are the names that come off the top of my head.

**Kang:**

For your 45 years or more, depending on counting the graduate school or not for your career, there are, of course, exciting moments, not so exciting moments and so on. So what do you consider to be the highlights of your career?

**Krener:**

Well, getting out of graduate school after being on the edge of failure for seven years, getting out and getting a job was a very important thing. Working with Roger. Bob Herman had a major influence in both a positive and a negative sense. Bob would start projects but would never finish them. He would have great ideas and if you would just listen to him you could mine his ideas. The paper Bob and I wrote on nonlinear controllability and observability - - it was initially his idea but I put all the work into it. Getting tenure. I mean I got tenure and the ‘70s were not kind to academics. There were not a whole lot of jobs. The baby boomers went to college in the 60’s, and all the universities, the new universities had filled up their ranks of their faculty. So that boom in faculty hiring was a brief thing. Also things stagnate for a while and then you need to get get a new idea, you get a new direction and you take off with that.

**Kang:**

Yes, I heard that during the early 70’s and late 60’s the job market was very hard but you were lucky to have a professor who wrote 100 letters.

**Krener:**

Oh, yes.

**Kang:**

Maybe this is related. You had good time, bad time. You talked about the highlights. How about challenges? Any challenges you feel-

**Krener:**

Well, I mean I have pounded my head against theorems for many days sometimes. You know, I am very stubborn and maybe sometimes too stubborn and I don’t see what I am trying to prove couldn’t possibly be true until several weeks and months have gone by. But I did have a rough spot in my in my early 50’s. I went through a bout of depression. I had developed an allergic reaction to alcohol and fruit juice and I was getting migraine headaches almost every day.

**Kang:**

Wow.

**Krener:**

And so this led to depression, divorce. So you know, I suffered but - - .

### Software packages

**Kang:**

Let’s maybe come back to the topics. I noticed that you mentioned the geometric control, of course that was the topic attracted me to get into this field the first place, and later the reciprocal process. I know that you also had some work on software packages. That is kind of very different from your theoretical work in terms of style, right?

**Krener:**

Yes. I have become a born again Christian, if you excuse the phrase, for computation. I think too much work even at a meeting like this, too much of the papers that are presented have no hope of ever being computable. And I think our discipline needs to be more informed about choices of problems that will have computable outcomes. Not necessarily all, I am not Neanderthal about computation. But eventually everything that we do should be computable and should become a tool for the broader community. As mathematicians we are tool creators in control and we make very specialized tools that may not be useful unless we can do calculations based on those tools. But these tools are not easily transferred into mass production. And the counter example to what I am saying here is what Matlab has done to linear systems theory. It is has completely revolutionized LQR Kalman filtering and so on by making relatively easy, stable, widely available packages to do fundamental control related computations. A larger portion of the nonlinear community needs to devote more effort into making mass produced tools. Ones that you can ship out the door and your neighbor down the block can use it with minimal help.

**Kang:**

The control theory is somewhere in between mathematics and engineering. You can go to applied side or pure theoretical side. I know that your work was funded by different agencies. Anything you can tell us about your funding situation or sponsors?

**Krener:**

Well, I think I was very lucky with funding. I got my first NSF grant two or three years after I arrived at Davis and basically have been with funding ever since. Not a lot of funding. I have never been comfortable with running a large research enterprise. This is the difference between the style of mathematics professors and the style of many engineering professors. I not putting their style down but I am just saying it is not my style. My style is to work relatively alone or with one or two graduate students with a low budget and relatively stable funding so that you don’t have the headache of waking up the next morning and saying I just lost my grant. I got five post docs whose mouths I have to feed. I am risk adverse that way. I have great respect for the people in our field that have really generated steady streams of outstanding graduate students and I will mention a few of them. Roger Brockett in the ‘70s and ‘80s particularly but throughout his career generated a large number of very successful graduate students. Almost everybody of my age plus or minus five years who is an nonlinear control theorist was touched in some way by Roger. After Roger, Shankar Sastry produced many great students. Shankar invited me to give a graduate course in nonlinear systems in Berkeley in the mid ‘80s and with his enthusiasm and his mixture of engineering and mathematics, he had assembled and mobilized a whole group of very talented people, Andy Teel, Richard Murray, Dawn Tilbury, etc. The list goes on and on and they are just fantastic. Petar Kokotovic is another guy who generated a tremendous stream of really outstanding graduate students over the years. So that I think that this style of doing business is in some ways the life blood of the discipline. Jacques-Louis Lions falls into that category. But it wasn’t my style.

**Kang:**

Okay, a lot of serious topics. Let’s go to something like, for example, do you have any anecdotes or stories you want to share, you know, whatever -

**Krener:**

You should have brought a bottle of wine. I don’t think I want to say-

**Kang:**

Okay. So we talked a lot about past, your 47 years of career. Now let’s talk a little bit about now and the future. Currently what control areas do you find exciting?

**Krener:**

Well, I think the field has really changed. And if I were giving advice to a young person coming into the field I think the role of theoreticians like myself has been greatly diminished. We have attacked and solved a lot of the fundamental problems in linear and nonlinear systems. So just, we don’t need to produce a lot of theorists in those areas. What we need to do is have people carry the message from our discipline to other disciplines. And so I am very encouraged by seeing and meeting people going into mathematical biology. Systems biology is basically set up for our talents, also big data and big systems. Control of the large complicated structures like electricity distribution systems. These are all topics that are made for our skill set and I am very encouraged to see people going into that. And again I come back to computation. I think those people that want to stay as theorists have a responsibility to produce usable tools for other people. If you are just going to produce a theory that doesn’t go anywhere then there really isn’t a whole lot of need for that. What we need is a software package like the Controls Systems Tool Box for nonlinear systems.

**Kang:**

Yes. You kind of already touched the topic. But anyway, for any field including control area, attracting young talented people is always very important. So, specifically what advice would you give to young people thinking about a career in the control field?

**Krener:**

Well, think fundamental. The more fundamental that you can think, the better off that you are. Don’t be afraid of hard work. I spent I don’t know how many years at Berkeley trying to prove that Bang Bang theorem and it wasn’t true. I went through reams of paper. I would go to the computer center and get the printouts that had been discarded. In those days they had batch processing of computer programs. There would be these stacks of paper and I would grab a stack and take it home and fill it all up on the back with computations. So hard work, there is no easy road to success. What else can I say? Well, choose your problem and the people that you associate with very carefully. I think that most of the people in this conference have a fundamental set of skills and we all have it pretty much to the same degree. So what separates people out is their shot selection. Choosing an interesting fundamental problem to work on and then sticking with it. I think that would be the best advice I could give anybody, particularly, anybody that is mathematically oriented. If you are engineeringly oriented then there is other advice but I don’t want to give that advice because I am not an engineer.

**Kang:**

Okay. I think it’s been an honor and great pleasure talking to you. You have provided a lot of very valuable insights and view about very important issues about control area. Thank you very much.

**Krener:**

Thank you, Wei.