Professor Richard Ernest Bellman

An American applied mathematician, celebrated for his invention of dynamic programming in 1953, and important contributions in other fields of mathematics, he was born in New York City, where his father John James Bellman ran a small grocery store on Bergen Street near Prospect Park in Brooklyn. He completed his studies at Abraham Lincoln High School in 1937, and studied Mathematics at Brooklyn College where he received a BA in 1941. He later earned an 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. From 1949 he worked for many years at RAND Corporation and it was during this time that he developed dynamic programming. 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; also known as a Dynamic Programming Equation, which 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. This was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory. 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. 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. The "Curse of dimensionality", is a term he coined 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. 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. 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...the three most recent being: (in 2003) ?Dynamic Programming?, ?Perturbation Techniques in Mathematics, Engineering and Physics?; and ?Stability Theory of Differential Equations?.