Harold J. Kushner received the Ph.D. in Electrical Engineering from the University of Wisconsin in 1958. Since then, in ten books and more than two hundred papers, he has established a substantial part of modern stochastic systems theory. These include seminal developments of stochastic stability for both Markovian and non-Markovian systems, optimal nonlinear filtering and effective algorithms for approximating optimal nonlinear filters, stochastic variational methods and the stochastic maximum principle, numerical methods for jump-diffusion type control and game problems (the current methods of choice), efficient numerical methods for Markov chain models, methods for singularly perturbed stochastic systems, an extensive development of controlled stochastic networks such as queueing/communications systems under conditions of heavy traffic, methods for the analysis and approximation of systems driven by wideband noise, large-deviation methods for control problems with small noise effects, stochastic distributed and delay systems, and nearly optimal control and filtering for non-Markovian systems.

His work on stochastic approximations and recursive algorithms has set much of the current framework, and he has contributed heavily to applications of control methods to communications problems.

He is a past Chairman of the Applied Mathematics Department and past Director of the Lefschetz Center for Dynamical Systems, at Brown University, where he is currently a University Professor Emeritus.

**July 1, 2004. Boston, MA**

It is a great honor to receive this award. It is a particular honor that it is in memory of Richard Bellman. I doubt that there are many here who knew Bellman, so I would like to make some comments concerning his role in the field.

Bellman left RAND after the summer of 1965 for the position of Professor of Electrical Engineering, Mathematics, and Medicine at the University of Southern California. This triple title gives you some inkling of how he was viewed at the time. I spent that summer at RAND. My office was right next to Bellman's and we had lots of opportunity to talk.

Bellman was always very supportive of my work. He encouraged me to write my first book, Stochastic Stability and Control, in 1967 for his Academic Press Series. Although naive by modern standards, the book seemed to have a significant impact on subsequent development in that it made many mathematicians realize that there was serious probability to be done in stochastic control, and established the foundations of stochastic stability theory. Numerical methods were among his strong interests. He was well acquainted with my work on numerical methods for continuous time stochastic systems and encouraged me to write my first book on the subject, later updated in two books with Paul Dupuis, and still the methods of choice. Despite his enormous output of published papers, something like 900, he was a strong believer in books since they allowed one to develop a subject with considerable freedom.

There are other connections, albeit indirect, between us. He was a New Yorker, and did his early undergraduate work at CCNY. During those years and, indeed, until the late 50's, CCNY was one of the most intellectual institutions of higher learning in the US. During that time, before the middle class migration out of the city, and the simultaneous opening of opportunities in the elite institutions for the "typical New Yorker," CCNY had the choice of the best of New Yorkers with a serious intellectual bent. Later, he switched to Brooklyn College, which was much closer to his home.

He intended to be a pure mathematician: His primary interest was analytic number theory. When did he become interested in applications? He graduated college at the start of WW2 and the demands of the war exposed him to a great variety of problems. He taught electronics in Princeton and then worked at a sonar lab in San Diego (which kept him out of the Army for a while). He spent the last two years of the war in the army, but assigned to the Manhattan project at Los Alamos. He was a social creature and it was easy for him to meet many of the talented people working on the project. Typically, the physicists considered a mathematician as simply a human calculator, ideally constructed to do numerical computations but not much more. Bellman was asked to numerically solve some PDE's. His mathematical pride refused. To the great surprise of the physicists, he actually managed to integrate some of the equations, obtaining closed form solutions. Holding true to tradition, they checked his solutions, not by verifying the derivation, but by trying some very special cases. Thus his reputation there as a very bright young mathematician was established. This jealously guarded independence and self confidence (and lack of modesty) continued to serve him well. During these years, he absorbed a great variety of scientific experiences. So much was being done due to the needs of the war.

There is one more indirect connection between us. Bellman was a student of Solomon Lefschetz at Princeton, head of the Math. Dept. at the time, a very tough minded mathematician and one of the powerhouses of American mathematics, and impressed with Bellman's ability. While at Los Alamos in WW2 Bellman worked out various results on stability of ODE's. Although he initially intended to do a thesis with someone else on a number theoretic problem, Lefschetz convinced him that those stability results were the quickest way to a thesis, which was in fact true. It took only several months and was the basis of his book on stability of ODE's. I was the director of the Lefschetz Center for Dynamical Systems at Brown University for many years, with Lefschetz our patron saint. Some of you might recall the book (not the movie) "A Beautiful Mind" about John Nash, a Nobel Laureate in Game Theory, which describes Lefschetz's key role in mathematics during Nash's time at Princeton.

Bellman spent the summer of 1948 at RAND, where an amazing array of talent was gathered, including David Blackwell, George Dantzig, Ted Harris, Sam Karlin, Lloyd Shapley, and many others, who provided the foundations of much of decision and game theory. The original intention was to do mathematics with some of the RAND talent on problems of prior interest. But Bellman turned out to be fascinated and partially seduced by the excitement in OR, and the developing role of mathematics in the social and biological sciences. His mathematical abilities were widely recognized. He was a tenured Associate Professor at Stanford at 28, after being an Associate Professor at Princeton, where all indications were that he would have had an assured future had he remained there. He began to have doubts about the payoff for himself in number theory and returned to the atmosphere at RAND often, where he eventually settled and became fully involved in multistage decision processes, having been completely seduced, and much to our great benefit.

Here is a non mathematical item that should be of interest. To work at RAND one needed a security clearance, even though much of the work did not involve "security." Due to an anonymous tip, Bellman lost his clearance for a while: His brother-in-law, whom Bellman had not seen since he (his brother-in-law) was about 13, was rumored to be a communist? This was an example of a serious national problem that was fed, exploited, and made into a national paranoia by unscrupulous politicians.

Bellman was a remarkable person, thoroughly a man of his time and renaissance in his interests, with a fantastic memory. Some epochs are represented by individuals that are towering because of their powerful personalities and abilities. People who could not be ignored. Bellman was one of those. He was one of the driving forces behind the great intellectual excitement of the times.

The word programming was used by the military to mean scheduling. Dantzig's linear programming was an abbreviation of "programming with linear models." Bellman has described the origin of the name "dynamic programming" as follows. An Assistant Secretary of the Air Force, who was believed to be strongly anti-mathematics was to visit RAND. So Bellman was concerned that his work on the mathematics of multi-stage decision process would be unappreciated. But "programming" was still OK, and the Air Force was concerned with rescheduling continuously due to uncertainties. Thus "dynamic programming" was chosen a politically wise descriptor. On the other hand, when I asked him the same question, he replied that he was trying to upstage Dantzig's linear programming by adding dynamic. Perhaps both motivations were true.

If one looks closely at scientific discoveries, ancient seeds often appear. Bellman did not quite invent dynamic programming, and many others contributed to its early development. It was used earlier in inventory control. Peter Dorato once showed me a (somwhat obscure) economics paper from the late thirties where something close to the principle of optimality was used. The calculus of variations had related ideas (e.g., the work of Caratheodory, the Hamilton-Jacobi equation). This led to conflicts with the calculus of variations community. But no one grasped its essence, isolated its essential features, and showed and promoted its full potential in control and operations research as well as in applications to the biological and social sciences, as did Bellman.

Bellman published many seminal works. It is sometimes claimed that many of his vast number of papers are repetitive and did not develop the ideas as far as they could have been. Despite this criticism, his works were poured over word for word, with every comment and detail mined for ideas, technique, and openings into new areas. His work was a mother lode. It was clearly the work of someone with a superb background in analysis as well as a facile mind and sharp eye for aplications. There are lots of examples, with broad coverage, accessible, and usually simple assumptions. His writing is articulate. It flows very smoothly through the problem formulation and mathematical analysis, and he is in full command of it.

We still owe a great debt to him.