# Richard Ernest Bellman

### Quick Info

Brooklyn, New York City, USA

Los Angeles, California, USA

**Richard Bellman**was an American applied mathematician who made important advances in dynamic programming.

### Biography

**Richard Bellman**'s father was John James Bellman and his mother was Pearl Saffian. Both sides of the family came from Jewish descent, with both John Bellman's father having emigrated from Russia and Pearl Saffian's father having emigrated from Poland. Despite the Jewish descent, the family that Richard was born into were agnostics.

The Great Depression began in 1929, when Richard was nine years old, and by 1932 one quarter of the workers in the United States were unemployed. The Depression of the 1930s saw low wages and there was much anti-Semitism. John Bellman was ruined by the Depression but, despite great hardship, managed to see Richard obtain a good education. Richard first met mathematics at the age of eleven when he studied Schultze's

*Elementary and advanced algebra*. He was not only thrilled with this early encounter with mathematics but as a boy he enjoyed other pursuits such as reading avidly, going round the museums of New York, and spending happy hours in the 42nd Street Public Library.

Richard attended Abraham Lincoln High School in Brooklyn where he represented his school on the mathematics team and in his final year was rewarded with achieving the top rank among all New York school pupils. After High School Bellman entered the City College of New York in January 1937. At this stage he had made up his mind to become a theoretical physicist and he took courses at the College with this in mind. In 1938 he moved from City College to Brooklyn College where he now decided to make mathematics his main area of study. He represented Brooklyn College in the three man team in the Lowell Putman mathematics competition in his final two years at Brooklyn College. He graduated with a B.A. in mathematics in 1941 and in September of that year he entered Johns Hopkins University in Baltimore to undertake postgraduate studies.

The United States entered World War II after the Japanese attacked the American fleet at Pearl Harbour on 7 December 1941. By 11 December the United States was at war with Germany. Bellman left Johns Hopkins University early in 1942 to take up a position as Instructor in Military Electronics at the University of Wisconsin. While teaching electronics as part of the war effort, he undertook his own studies in mathematics and was awarded a Master's Degree from Wisconsin in 1943. Continuing to undertake war work with his teaching, Bellman next went to Princeton University where he taught in the Army Specialized Training Program. He was able to continue the undertake graduate work in mathematics but in December 1944 he was drafted into the army and assigned o the Manhattan Project in Los Alamos. There he worked on problems in theoretical physics until his discharge in 1946.

Bellman returned immediately to Princeton where he completed his doctoral studies under Lefschetz's supervision. His doctoral dissertation on the stability of differential equations was concerned with the behaviour of the solutions of real differential equations as the independent variable t tends to infinity. It was submitted to Princeton later that year and he was awarded his Ph.D. Results from his dissertation appeared in the book

*Stability theory of differential equations*which he published in 1953. A reviewer praised its:-

... lucid and attractive manner of presentation.He remained at Princeton as Assistant Professor of Mathematics after the award of his doctorate but in 1948 he left to take up the position of Associate Professor of Mathematics at Stanford University. During the following summer he first worked at the RAND Corporation. He wrote [1]:-

I was very eager to go to RAND in the summer of 1949 ... I became friendly with Ed Paxson and asked him what RAND was interested in. He suggested that I work on multistage decision processes. I started following that suggestion.After a second period of time at RAND, Bellman spent a year on leave from Stanford, working at Princeton on H-bomb research. While there he began to ponder whether he should remain at Stanford where he was working on the topic he loved most, namely analytic number theory, or whether he should take up a position at RAND in Los Angeles. He wrote [1]:-

I was intrigued by dynamic programming. It was clear to me that there was a good deal of good analysis there. Furthermore, I could see many applications. It was a clear choice. I could either be a traditional intellectual, or a modern intellectual using the results of my research for the problems of contemporary society. This was a dangerous path. Either I could do too much research and too little application, or too little research and too much application. I had confidence that I could do this delicate activity, pie a la mode.The decision made, Bellman left Stanford in 1952 and took up the position of Research Mathematician at RAND. As Dreyfus writes in [6]:-

Assured of a successful conventional academic career, Bellman ... cast his lot instead with the kind of applied mathematics later to be known as operations research. In those days applied practitioners were regarded as distinctly second-class citizens of the mathematical fraternity. Always one to enjoy controversy, when invited to speak at various university mathematics department seminars, Bellman delighted in justifying his choice of applied over pure mathematics as being motivated by the real world's greater challenges and mathematical demands.Bellman's first publication on dynamic programming appeared in 1952 and his first book on the topic

*An introduction to the theory of dynamic programming*was published by the RAND Corporation in 1953. To get an idea of what the topic was about we quote a typical problem studied in the book. Bellman writes:-

We are informed that a particle is in either state 0 or 1, and we are given initially the probability x that it is in state 1. Use of the operation A will reduce this probability to ax, where a is some positive constant less than 1, whereas operation L, which consists in observing the particle, will tell us definitely which state it is in. If it is desired to transform the particle into state 0 in a minimum time, what is the optimal procedure?Bellman's diverse applications of these tools continued to solve a whole range of problems. As he wrote [1]:-

... as of 1954 or so I had stumbled into some important types of problems and had been pushed, willy-nilly, into answering some significant kinds of questions. I could handle deterministic control processes to some extent and stochastic decision process in economics and operations research as well.At this point he made some very definite decisions on the direction his research should take, going back at looking at the original assumptions underlying his mathematical model and trying to introduce more realistic assumptions. He went on to introduce Markovian decision problems in 1957 and in 1958 he published his first paper on stochastic control processes where he introduced what is today called the Bellman equation.

The number of papers and books which Bellman wrote is quite amazing. For example the paper [3] list 621 papers, 41 books and 21 translations of books authored (or co-authored) by Bellman. Although it is impossible to give more than a flavour in an article such as this we give an indication of their scope by listing the titles of a few of his books. These include, in addition to those already mentioned:

*A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations*(1949);

*A survey of the mathematical theory of time-lag, retarded control, and hereditary processes*(1954);

*Dynamic programming of continuous processes*(1954);

*Dynamic programming*(1957);

*Some aspects of the mathematical theory of control processes*(1958);

*Introduction to matrix analysis*(1960);

*A brief introduction to theta functions*(1961);

*An introduction to inequalities*(1961);

*Adaptive control processes: A guided tour*(1961);

*Inequalities*(1961);

*Applied dynamic programming*(1962);

*Differential-difference equations*(1963);

*Perturbation techniques in mathematics, physics, and engineering*(1964); and

*Dynamic programming and modern control theory*(1965).

We have stopped at 1965 in giving a list of Bellman's books since it ws at this time that he left RAND and accepted an appointment as Professor of Mathematics, Electrical Engineering, and Medicine at the University of Southern California. He continued his remarkable research and publication record. His approach was summed up in a course of lectures he gave at the University of Kentucky in 1966 in which he aimed:-

... to describe some of the ways that the problems of the modern world provide interesting mathematical questions and open up entirely new domains of mathematics. ... [I believe that] the growth of vital mathematics depends crucially on continuing interaction with the real world.His interests turned towards computers as a tool in mathematical research and he wrote such books as

*Algorithms, graphs and computers*(1970). In 1973 tragedy struck when he was diagnosed with a brain tumour. An operation to remove the tumour was successful but following the operation complications set in which left him an almost total cripple. Remarkably, however, he remained extremely active in his mathematical research despite his physical problems and in the remaining 10 years of his life he wrote around 100 papers. Among the books he published during these years we should mention the particularly important

*An introduction to invariant imbedding*(1975) written jointly with G M Wing. However he also wrote

*Analytic number theory*(1980),

*Mathematical methods in medicine*(1983), and

*The Laplace transform*(1984).

After his death in 1984 his books continued to be published such as

*Partial differential equations*(1985),

*Selective computation*(1985),

*Methods in approximation*(1986), and

*Wave propagation: An invariant imbedding approach*(1986).

Bellman received a great many honours for his outstanding contributions to the applications of mathematics. In 1970 he was awarded the first Norbert Wiener Prize in Applied Mathematics, this being a joint prize of the American Mathematical Society and the Society for Industrial and Applied Mathematics. Also in 1970 Carnegie-Mellon University awarded Bellman the first Dickson Prize and three years later he was appointed to the ALZA Distinguished Lectureship by the Biomedical Engineering Society.

He was elected to Fellowship in the American Academy of Arts and Sciences in 1975 and, in the following years, he received the John von Neumann Theory Award, another joint award this time by the Institute of Management Sciences and the Operations Research Society of America. In 1977 be was elected to the National Academy of Engineering. He was awarded the IEEE Gold Medal of Honor in 1979:-

For contributions to decision processes and control system theory, particularly the creation and application of dynamic programming.Bellman was elected a Fellow of the Society for Mathematical Biology in 1980 and in 1983 he was elected to the National Academy of Sciences (United States). In 1983 he received the Heritage Medal from the American Council for Control.

The 1979 IEEE Awards Reception Brochure states:-

Richard Bellman is a towering figure among the contributors to modern control theory and systems analysis. His invention of dynamic programming marked the beginning of a new era in the analysis and optimization of large-scale systems and opened a way for the application of sophisticated computer-oriented techniques in a wide variety of problem-areas ranging from the design of guidance systems for space vehicles to pest control and network optimization.After listing his many honours, as we have above, the brochure then pays him this personal tribute:-

But what matters most is that Richard Bellman won more than fame - he won the admiration and affection of all who know him for his outstanding courage and greatness as a human being.

### References (show)

- R Bellman,
*Eye of the hurricane : An autobiography*(World Scientific Publishing Co., Singapore, 1984). - G Adomian, Obituary: the contributions of Richard Bellman to computing,
*Kybernetes*13 (4) (1984), 253-254. - G Adomian and E S Lee, The research constributions of Richard Bellman,
*Comput. Math. Appl. Ser. A***12**(6) (1986), 633-651. - K L Bellman and C Landauer, Towards an integration science : the influence of Richard Bellman on our research,
*J. Math. Anal. Appl.***249**(1) (2000), 3-31. - Richard Bellman (Russian),
*Avtomat. i Telemekh.*(3) (1985), 175-176. - S Dreyfus, Richard Bellman on the birth of dynamic programming,
*Oper. Res.***50**(1) (2002), 48-51. - S Dreyfus, IFORS' Operational Research Hall of Fame : Richard Bellman,
*Intl. Trans. in Op. Res.***10**(2003), 543-545. - A O Esogbue, Professor Richard Bellman: academician, scholar, practitioner and patron of fuzzy sets,
*Fuzzy Sets and Systems***18**(3) (1986), 201-204. - J Gani, Obituary: Richard Bellman,
*J. Appl. Probab.***21**(4) (1984), 935-936. - J A Jacquez, Richard Bellman,
*Math. Biosci.***77**(1-2) (1985), 1-4. - H Kagiwada and R E Kalaba, The work of Richard Bellman. I. Optimization,
*Comput. Math. Appl. Ser. A***12**(6) (1986), 785-790. - C T Leondes, An appreciation of Professor Richard Bellman,
*J. Optim. Theory Appl.***32**(4) (1980), 399-406. - A Lew, Richard Bellman's contributions to computer science.
*J. Math. Anal. Appl.***119**(1-2) (1986), 90-96.- List of publications: Richard Bellman,
*IEEE Trans. Automat. Control***26**(5) (1981), 1213-1223. - H Panossian, Richard Bellman and stochastic control systems,
*Comput. Math. Appl. Ser. A***12**(6) (1986), 825-829. - L A Zadeh, In memoriam: Richard E Bellman (1920-1984),
*IEEE Trans. Automat. Control***29**(11) (1984), 961. - L A Zadeh, Richard E Bellman (1920-1984),
*Fuzzy Sets and Systems***14**(2) (1984), 97-98.

### Additional Resources (show)

Other websites about Richard Bellman:

### Honours (show)

Honours awarded to Richard Bellman

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update December 2005

Last Update December 2005