Richard E. Bellman

Infobox_Systems scientist H
region = Control Theory
era = 20th century
color = #B0C4DE

image_caption =


name = Richard E. Bellman
birth = birth date|1920|8|26|df=y New York City, New York
death = death date and age|1984|3|19|1920|8|26|df=y
school_tradition = Princeton University
University of Wisconsin-Madison
Brooklyn College
main_interests = Mathematics and Control theory
influences =
influenced =
notable_ideas = Dynamic programming |

Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics.

Biography

Bellman was born in 1920 in New York City, where his father John James Bellman ran a small grocery store on Bergen Street near Prospect Park in Brooklyn. Bellman completed high school at the Abraham Lincoln High School in1937 Salvador Sanabria. [http://www-math.cudenver.edu/~wcherowi/courses/m4010/s05/sanabria.pdf Richard Bellman's Biography] . Paper at www-math.cudenver.edu. Retrieved 3 Okt 2008.] , and studied mathematics at Brooklyn College where he received a BA in 1941, and later a MA from the University of Wisconsin-Madison. During World War II he worked for a Theoretical Physics Division group in Los Alamos. In 1946 he received his Ph.D. at Princeton under the supervision of Solomon Lefschetz [Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/id.php?id=12968] .

He was a professor at the University of Southern California, a Fellow in the American Academy of Arts and Sciences (1975), and a member of the National Academy of Engineering (1977).

He was awarded the IEEE Medal of Honor in 1979, "For contributions to decision processes and control system theory, particularly the creation and application of dynamic programming". His key work is the Bellman-Equation.

Work

Bellman equation

A Bellman equation also known as a "dynamic programming equation", is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. Almost any problem which can be solved using optimal control theory can also be solved by analyzing the appropriate Bellman equation. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory.

Hamilton-Jacobi-Bellman

The Hamilton-Jacobi-Bellman equation (HJB) equation is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the 'value function', which gives the optimal cost-to-go for a given dynamical system with an associated cost function. Classical variational problems, for example, the brachistochrone problem can be solved using this method as well.

The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton-Jacobi equation by William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Curse of dimensionality

The Curse of dimensionality", is a term coined by Bellman to describe the problem caused by the exponential increase in volume associated with adding extra dimensions to a (mathematical) space. One implication of the curse of dimensionality is that some methods for numerical solution of the Bellman equation require vastly more computer time when there are more state variables in the value function.

For example, 100 evenly-spaced sample points suffice to sample a unit interval with no more than 0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice with a spacing of 0.01 between adjacent points would require 10²⁰ sample points: thus, in some sense, the 10-dimensional hypercube can be said to be a factor of 10¹⁸ "larger" than the unit interval. (Adapted from an example by R. E. Bellman, see below.)

Bellman–Ford algorithm

The Bellman-Ford algorithm sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a weighted digraph (where some of the edge weights may be negative). Dijkstra's algorithm accomplishes the same problem with a lower running time, but requires edge weights to be non-negative. Thus, Bellman–Ford is usually used only when there are negative edge weights.

Publications

Over the course of his career he published 619 papers and 39 books. During the last 11 years of his life he published over 100 papers despite suffering from crippling complications of a brain surgery (Dreyfus, 2003). A selection:
* 1959. "Asymptotic Behavior of Solutions of Differential Equations"
* 1961. "An Introduction to Inequalities"
* 1961. "Adaptive Control Process"
* 1962. "Applied Dynamic Programming"
* 1967. "Introduction to the Mathematic Theory of Control Process"
* 1970. "Algorithms, Graphs and computers"
* 1972. "Dynamic Programming and Partial Differential Equations"
* 1982. "Mathematical Aspects of Scheduling and Applications"
* 1983. "Mathematical Methods in Medicine"
* 1984. "Partial differential Equations"
* 1984. "Eye of the Hurricane, an Autobiography," World Scientific Publishing.
* 1985. "Artificial Intelligence"
* 1995. "Modern Elementary Differential Equations"
* 1997. "Introduction to Matrix Analysis"
* 2003. "Dynamic Programming"
* 2003. "Perturbation Techniques in Mathematics, Engineering and Physics"
* 2003. "Stability Theory of differential Equations"

References

Further reading

* J.J. O'Connor and E.F. Robertson (2005). [http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Bellman.html Biography of Richard Bellman] from the MacTutor History of Mathematics.
* Stuart Dreyfus (2002). [http://www.cas.mcmaster.ca/~se3c03/journal_papers/dy_birth.pdf "Richard Bellman on the Birth of Dynamic Programming"] . In: "Operations Research". Vol. 50, No. 1, Jan–Feb 2002, pp. 48–51.
* Stuart Dreyfus (2003) [http://www.blackwell-synergy.com/doi/abs/10.1111/1475-3995.00426 "Richard Ernest Bellman"] . In: "International Transactions in Operational Research". Vol 10, no. 5, Pages 543 - 545
* Salvador Sanabria. [http://www-math.cudenver.edu/~wcherowi/courses/m4010/s05/sanabria.pdf Richard Bellman's Biography] . Paper at www-math.cudenver.edu

External links

* [http://www.ieee.org/organizations/history_center/legacies/bellman.html IEEE History Center - Legacies]
* [http://www.a2c2.org/awards/bellman/index.php Harold J. Kushner's speech when accepting the Richard E. Bellman Control Heritage Award]
* [http://ieeexplore.ieee.org/xpls/abs_all.jsp?tp=&arnumber=1102033&isnumber=24172 IEEE biography]