## Obituaries: Leonard D. Berkovitz

**January 9, 2010**

Leonard D. Berkovitz, 1924–2009

Len Berkovitz grew up in Chicago. He began his college studies in 1941 as a chemistry major at the University of Chicago but joined the military when the U.S. entered World War II. After completing training programs in meteorology, he served as a weather officer in the U.S. Army Air Corps. He returned to the University of Chicago after the war, receiving a BS in meteorology in 1946 and a PhD in mathematics in 1951. His PhD thesis, written under the supervision of Antoni Zygmund, was in the area of double trigonometric series. He was an Atomic Energy Commission postdoctoral fellow at Stanford in 1951–52. As a research fellow at Caltech from 1952 to 1954, he conducted research on asymptotic expansions.

In 1954, Len joined the Mathematics Division of the RAND Corporation in Santa Monica, California, one of the oldest and most successful among organizations called "think tanks." During the 1950s, RAND actively encouraged work in such new areas of applied mathematics as linear programming, game theory, and nonlinear optimization, which were not yet established as academic research areas in the U.S. G. Dantzig, R. Isaacs, and R. Bellman were among the senior staff at RAND, and other prominent mathematicians were RAND consultants. The U.S.–USSR "space race" began in the late 1950s, and Mathematical Theory of Optimal Processes by L.S. Pontryagin et al. appeared a short time later, influencing the rapid growth of the field of nonlinear control theory during the 1960s. It was in this exciting atmosphere that Len's long-term research interests in control and differential games began. During his time at RAND, Len also worked on a variety of applied problems of interest to the U.S. Air Force. He helped introduce the novel idea (now common practice) of using simulation methods for determining the outcome of tactical engagements.

By the early 1960s, RAND had de-emphasized basic research in applied mathematics, and many of the mathematicians left for university positions. Len joined the faculty at Purdue University as a professor of mathematics in 1962 and remained at Purdue until his retirement in 2003.

Len made many significant contributions to control theory and the calculus of variations during the 1960s and 1970s, including his influential book

*Optimal Control Theory*(1974). One facet of his work concerned methods for deriving necessary conditions for optimality. Pontryagin-type control problems are closely related to Bolza-type problems in the calculus of variations, in which additional inequality constraints are imposed. Len gave definitive answers to questions concerning the relation between the Pontryagin maximum principle and classical necessary conditions for Bolza-type problems. He was also very active in the development of a theory for the existence of optimal controls. Many of his definitive results on existence theorems are included in the 1974 book.

In a different research direction, Len and Harry Pollard analyzed some variational problems that arise from min–max filter applications. Their work stimulated considerable interest among mathematicians, statisticians, and engineers who were concerned with related non-classical problems in the calculus of variations.

The study of two-player, zero-sum differential games began in 1951 with the work of Isaacs on pursuit–evasion games. Isaacs's method for finding the differential game value and optimal play was based on the method of characteristics for what is now called the Isaacs partial differential equation. There remained the difficult task of finding a satisfactory mathematical theory of differential games. Len was among those who made significant contributions to this endeavor during the thirty years between 1957 and 1987. Interesting new applications of differential games continue to arise today, including nonlinear

*H*-infinity control theory, in which they play a key role. Recently, methods based on differential games gave new importance to sampling techniques for estimating probabilities of rare events.

A first step toward a mathematical theory of differential games was to find conditions under which Isaacs's method of characteristics could be rigorously shown to provide optimal feedback controls for the two game players. Len's work during the period from 1957 to 1971 gave such conditions, which involved piecewise smoothness of value functions and other restrictions.

A second stage of differential game theory concerned rigorous definitions for upper and lower game value functions under much less restrictive assumptions. Len's 1985 and 1986 papers in

*SIAM Journal on Control and Optimization*present definitive work in this direction. These papers are related to earlier work of A. Friedman and Krasovskii–Subbotin defining differential game value functions using time-discretization methods.

The Crandall–Lions theory of viscosity solutions in the 1980s provided an important simplification in the theory of differential games. Theorems about uniqueness of viscosity solutions imply that various earlier definitions of upper and lower game values are equivalent. If the Isaacs minimax condition holds, then upper and lower values are equal and are called the game value.

Independently of viscosity solution theory, Subbotin introduced another approach based on Dini derivatives of value functions. In a 1988 paper in Applied Mathematics and Optimization, Len characterized the Subbotin game value as a viscosity solution to the Isaacs PDE and gave simpler proofs of Subbotin's results.

At Purdue, Len played a guiding role in developing the strength of the Mathematics Department in both pure and applied mathematics. Colleagues recall him as an excellent department head, from 1975 to 1980 and (acting) in 1989–90. He was resolute in advocating the department's interests in negotiations with the university administration. In one of his favorite stories, he informed the dean that the department wanted to make simultaneous offers to four outstanding candidates. The dean pointed out that if the offers were all accepted, there would be no money left for mathematics faculty raises. Len responded that in that unlikely event, he would gladly step down in a blaze of glory. (Unfortunately, not all the offers were accepted.) As both colleague and department head, Len's interactions with administrators, colleagues, staff, and students were marked by his always reasonable and civil manner. His ability to keep the conversation focused, calm, and rational, even when tempers were frayed and opinions divergent, served both him and the department well. He could defuse a tense situation with a well-placed amusing anecdote. He was probably unique among department heads in not making enemies. It is hard to remember his ever being without a disarming smile.

During his 41-year career at Purdue, Len was a highly esteemed teacher at all levels, from large (400+) lectures in freshman calculus to graduate seminars. He took his teaching seriously and expected his students to take their studies seriously. He was approachable and respected for his fairness. As a PhD adviser, he was a beloved mentor dedicated to the professional growth and well-being of his students. He preferred to lead students toward a research direction instead of just giving them specific research problems. Len encouraged students to learn things beyond his own expertise and to work with other scholars in order to broaden their knowledge as well as to establish relationships with other mathematicians in the early stages of their careers. As an indication of his integrity and genuine concern, he avoided working on the same problems as his students and was never listed as co-author on his students' papers. Len also provided lifelong friendship and encouragement to former PhD students, and wise counsel when asked. An example of his kindness and concern can be found in the story of one of his former students who, wishing to return to the U.S. in order to teach and continue his research, asked Len's help. Len made all necessary arrangements, including an airline ticket as well as initial financial support.

From 1967 to 1991, as associate editor, editor, and managing editor of

*SIAM Journal on Control and Optimization*, Len played a major role in maintaining its reputation as one of the premier journals in the field. He also served for many years on the editorial boards of the

*Journal of Optimization Theory and Applications*and the

*Journal of Mathematical Analysis and Applications*. From 1985 to 1991, he was a member of the Editorial Committee of

*Mathematical Reviews*.

Len is survived by his wife, Anna, sons Daniel and Kenneth, and five grandchildren. He was admired and respected by all who knew him for his kindness, integrity, and positive outlook. He is sorely missed by family, students, colleagues, and friends.---

*W.J. Browning, Applied Mathematics, Inc., and W.H. Fleming, Brown University.*