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Journal of Convex Analysis 23 (2016), No. 1, 3--4
Copyright Heldermann Verlag 2016

Foreword to Volume 23 (2016)

Charles Castaing
IMAG, Université Montpellier 2, Place E. Bataillon, 34095 Montpellier, France

Lionel Thibault
IMAG, Université Montpellier 2, Place E. Bataillon, 34095 Montpellier, France

Michel Valadier
IMAG, Université Montpellier 2, Place E. Bataillon, 34095 Montpellier, France

Jean Jacques Moreau, one of the pioneers in nonregular mechanics and modern convex analysis, died in Montpellier on January 9, 2014 at age 91.

Moreau was born in Blaye (France) on July 31, 1923. In 1949 he earned his Thèse de Doctorat d'État at the Faculté des Sciences de Paris, with a dissertation entitled: Bilan Dynamique d'un Écoulement Rotationnel (Dynamical Balance of a Rotational Flow). He started his professorship at the Université de Poitiers and moved to the Université de Montpellier in 1952.

Moreau's first research was concerned with studies of the mechanics of fluid flows, cavitation, plasticity and related topics. In the early sixties, with his investigation of the extension of the Gauss principle to certain mechanical problems with unilateral (this word is central in Moreau's thought) constraints and on the study of the cavitation in hydrodynamics, he observed that such problems required mathematical concepts taking non-regularity into account. On page 178 of his article "On unilateral constraints, friction and plasticity", the lectures that he delivered in June 1973 in Bressanone, he wrote: "The study of dynamical problems for systems of finite or infinite freedom with unilateral constraints (e.g. the inception of cavitation in a perfect incompressible fluid) initially motivated the part taken by the author in the development of convexity theory". A few lines above he also wrote: "About convexity, ..., Mechanics was probably the first physical domain to make use of this concept".

Moreau probably started working in the field of convex analysis in 1961 in the Department of Mathematics of the Faculté des Sciences de Montpellier. He created around him a small group of researchers in this domain, and initiated a research seminar titled Séminaire d'Analyse Unilatérale de Montpellier. Their works are collected in two volumes edited in 1968 and 1969. Some years later, in the early seventies, with Charles Castaing, Bernard Lemaire and Michel Valadier as new professors, the group expanded, and the weekly seminar Séminaire d'Analyse Convexe de Montpellier was born. In several talks in this seminar Moreau presented his fundamental works on what he called "sweeping process". This is confirmed by the number of Moreau's articles in the famous annual publication Travaux du Séminaire d'Analyse Convexe de Montpellier. Manuel D. P. Monteiro Marques, who regularly attented the seminar in the period preceding the 1980's, gives a comprehensive presentation of sweeping process in his book "Differential Inclusions in Nonsmooth Mechanical Problems", Basel 1993.

From the eighties to the end of his life, Moreau's interests were essentially turned to mechanics. He then went beyond the first order sweeping process to concentrate his investigations on "unilateral dynamics". He produced significant mechanical and mathematical results on this problem, and provided the "contact dynamics algorithm". He realized various numerical computations with that scheme; he also collaborated extensively with Michel Jean on this subject.

The impact of Moreau's ideas in nonregular mechanics and analysis is, and will remain, immense. The four numbers of Volume 2016 of the Journal of Convex Analysis are dedicated to the memory of this Mechanician / Mathematician to whom, among a few leaders such as Werner Fenchel and Ralph Tyrrell Rockafellar, we trace the foundations of modern convex analysis. The reader is invited to consult the tribute to Jean Jacques Moreau's scientific career paid in 2014 by the third author [Jean Jacques Moreau, Gazette des Mathématiciens 140 (2014) 75--77].

The subjects covered by the volume are almost all related to Moreau's research fields, and can be classified in nine groups: 12 articles about general problems related to convexity; 4 articles about the calculus of variations and optimal control; 3 articles about dynamical systems and viability theory; 6 articles about convex optimization algorithms; 6 articles about monotone operator theory; 3 articles about generalized duality and polarity; 6 articles about infimal convolution; 3 articles about the Moreau envelope; 7 articles about the Moreau sweeping process.

We express our grateful thanks to the authors and referees for the homage paid to Jean Jacques Moreau in their contributions. We are sure that the volume will enjoy a great success and will be an important reference.