**Lamberto Cesari** studied at the Scuola Normale Superiore in Pisa. He undertook research there in the Mathematical Seminar run by Leonida Tonelli. He published the paper *Sulle serie doppie* in 1932 and in it he acknowledged his debt to the Pisa Mathematical Seminar. He was awarded his laurea in 1933 after submitting his thesis *Sulle condizioni sufficienti per le successioni di Fourier* Ⓣ. He then went to Munich where he studied with Constantin Carathéodory during the academic year 1934-35. While he extending his mathematical experience in Munich he met Isotta Hornauer, the daughter of Franziska Brandl and Ludwig Hornauer, who had been born in Marktredwitz, Bavaria in 1913. They were married and formed a strong partnership throughout their lives.

After spending this year in Munich, Cesari returned to the Scuola Normale Superiore in Pisa where he taught for the academic year 1935-36 before going to Rome where he spent the two years 1936-38 at the Istituto Nazionale per le Applicazioni del Calcolo. This Institute had been founded by the mathematician Mauro Picone in 1927 and is today named after him. Influenced by Picone, Cesari looked at some new research areas while in Rome and he published the significant paper *Sulla risoluzione dei sistemi di equazioni lineari per approssimazioni successive* Ⓣ in 1937. This paper on iterative methods contains his general theory of stationary iterations given in terms of matrix splittings. The paper also contains applications of his general methods to more specialised problems, solutions to which had been found by Carl Jacobi and Richard von Mises. It was in applying his theory to the method discovered von Mises that Cesari was led to the idea of 'polynomial preconditioning'. His important discoveries during this period are discussed in [2]:-

In1937, Cesari proposed the concept of L-integrable functions f(x, y)of bounded variation, and these functions were later called BVC functions. He proved that a suitably generalized area of the discontinuous surface z = f(x, y)is finite if and only if f is BVC, a difficult extension to the discontinuous case of Tonelli's analogous statement for the continuous case. Cesari also proved that the double Fourier series of any BVC function f(x, y)converges almost everywhere to f(x, y), a sharp result.

While in Rome his wife, Isotta, worked as a language instructor and translated Italian literature into German. Cesari had published several other works during these years, for example *Sulle funzioni a variazione limitata* Ⓣ (1936) and *Sulle serie di Fourier delle funzioni lipschitziane di piť variabili* Ⓣ (1938). In 1938 the Cesaris left Rome and returned to Pisa where Lamberto was appointed as 'professore incaricato' (assistant professor) at the University of Pisa. They spent the first years of World War II in Pisa and gained great respect for the humanitarian work they carried out at this time. During this period he studied surfaces given by parametric equations, in particular the Lebesgue area of such a surface. He succeeded in giving necessary and sufficient conditions for such an area to be finite, a result which many mathematicians had tried unsuccessfully to produce.

The Cesaris left Pisa in 1942 when Lamberto was appointed to the University of Bologna. Again during these war years spent in Bologna they carried out humanitarian work. He was appointed as Professor of Mathematical Analysis at the University of Bologna in 1947 after winning a competition for the chair. However, his work on the Lebesgue area of surfaces which we mentioned above, and other variational problems he had studied, had given him an international reputation and he was invited to spend time at the Institute for Advanced Study at Princeton in the United States. At the Institute he met Tibor Radó who, like Cesari, had been undertaking research on area theory. Radó and Cesari were invited to address the International Congress of Mathematicians in Cambridge, Massachusetts in September 1950 and they presented their paper *Applications of area theory in analysis*. In the lecture, R G Helsel writes, they describe:-

... the two-dimensional concepts of bounded variation and absolute continuity devised by L Cesari, T Radó, and P V Reichelderfer and ... show applications of these concepts in the transformation of double integrals, calculus of variations, and Lebesgue area theory.

After a spell as a visiting member of the Institute where he arrived in 1949, he was invited as a Visiting Professor to the University of California at Berkeley. A further invitation saw him occupy a similar role at the University of Wisconsin, Madison, in 1950. Wendell Fleming writes:-

Lamberto Cesari visited from Italy in the second semester. His course on calculus of variations and seminar fostered my interest in that area. Soon after, Cesari moved to Purdue University, and I joined his group there in1955.

As Fleming suggests, Cesari was offered a professorship of mathematics at Purdue University, Lafayette, Indiana in 1952. Again Cesari was invited to address the International Congress of Mathematicians, this time in Amsterdam in 1954. He gave the lecture *Retraction, homotopy, integral* which, Tibor Radó writes:-

... is an excellent expository paper on surface area theory and its application in various fields of analysis. ...[The]paper furnishes a valuable account of many of the principal lines of research in this general field, with particular emphasis upon topics where the author and his associates made outstanding contributions. These include concepts of bounded variation, absolute continuity, and generalized Jacobians for continuous mappings T, and the principal theorems relating to these concepts; tangential properties of general continuous surfaces; the theory of a general Weierstrass-type double integral; extremely general forms of the Gauss-Green and Stokes theorems; theorems on homotopy and retraction for general continuous surfaces; and far-reaching results in calculus of variations for double integrals.

In 1956 Princeton University Press published Cesari's monograph *Surface area*. Laurence Chisholm Young writes:-

Since[Radó's 'Length and area'(1948)]has inspired many researches during the intervening years, it has rendered Cesari's task all the more arduous, by the wealth of new material to be presented as well by the high standard of its proved excellence. ... Cesari's "Surface area'' ... constitutes, like[Radó's 'Length and area'], a new departure in area theory, likely to have a profound influence on future developments.

Three years later, in 1959, Cesari published the monograph *Asymptotic behavior and stability problems in ordinary differential equations*. Cesari writes in the Introduction that:-

.. the purpose of the present volume is to present many viewpoints and questions in a readable short report...

J A Nohel, in a review, explains that "this objective has been accomplished admirably" and writes of the author's "clarity of presentation".

In 1960 Cesari was called to the chair of mathematics at the University of Michigan, Ann Arbor. He was appointed to the R L Wilder Distinguished Professorship of Mathematics in 1975 and was named the Henry Russel Lecturer for 1976. He retired in 1980. In [4] his research contributions are summarised:-

One of Cesari's abiding interests was the study of problems in the calculus of variations, and he also did a great deal of work in optimal control. Particularly noted for his study of the existence theorems for optimal solutions for both single- and multi-dimensional systems, he also contributed to the theory of necessary conditions and the analysis of Pareto problems.

In fact much of this appears in his book *Optimization - theory and applications: Problems with ordinary differential equations* published in 1983. We will return to say more about this book after giving further details of his research contributions from [4]:-

In the last twenty years, much of his attention was devoted to the study of questions arising in nonlinear analysis and its applications to differential equations. He continued his work, begun in the1950s, on the Alternative Method, especially as applied to problems with large nonlinearities. He also investigated the existence of solutions to certain quasi-linear hyperbolic systems. In recent years, he continued his study of existence theorems, analysis various problems, including those which arise in the theory of plasticity and whose optimal trajectories may have jump discontinuities. During the last few years of his life, Cesari worked on the theory of functions of bounded variation, a field that he himself had pioneered, and its applications to the theory of hyperbolic systems of conservation laws.

We promised to return to his book *Optimization - theory and applications: Problems with ordinary differential equations* published three years after he retired. Thomas Angell writes in a review:-

This book is an encyclopedic treatment dedicated to the exposition of both the classical and modern theories. ... It is devoted to problems in one independent variable, and in particular to nonparametric problems. ... this work is unique both in scope and scholarly approach. ... This book is very carefully written, quite detailed, and for the most part, self-contained. ... Some of these technical sections will require close attention, but the rewards will be great.

Cesari lays great stress on the connection of theory to applications and devotes two chapters to illustrative examples from geometry, mechanics, aerospace science, economics and other fields. In addition, he provides a large number of examples and counterexamples to illustrate the power and the limitations of various theorems. These very welcome features of Cesari's book distinguish it from most other texts on optimal control. Aside from Young's book[L C Young, Lectures on the calculus of variations and optimal control theory(1969)], Cesari's text is perhaps the only one attempting to bridge the gap between the calculus of variations and optimal control theory. This is especially true in existence theory in which the ideas of Tonelli are merged with those of Filippov and of Cesari himself to construct a largely unified framework.

E O Roxin also reviewed the book writing in [5]:-

... this book is an excellent addition to the existing literature. As a reference manual for the subject, it is very useful, as it presents different kinds of approaches and their relations, which are often quite confusing. But this book is not addressed only to the specialists. It could be used very well for a graduate level course, provided a judicious selection is made on which sections to cover.

Finally we note that, in collaboration with his wife, Cesari published *Mathematics in the Mediterranean: today's view* in 1990. Cesari and his wife took much pleasure from their home and its garden in Ann Arbor. They planted trees and kept a garden that was a pleasure to visit providing a peaceful haven in the middle of a busy city. Perhaps Isotta Cesari's poem which was published in the *American Mathematical Monthly* gives an idea of part of their home life.

See Isotta Cesari's Residues of Ideals

Cesari received many honours including election to the Accademia dei Lincei in Rome and the Academies of Bologna, of Modena and of Milan. In 1976 he was awarded an honorary degree by the University of Perugia. The University of Texas organised a conference in his honour in 1980 and, two years later, the University of Bologna organised an international conference in his honour.

**Article by:** *J J O'Connor* and *E F Robertson*