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Book review

Author(s) Marques, M.D.P.Monteiro
Title Differential inclusions in nonsmooth mechanical problems: shocks and dry friction
Publisher Birkhäuser
Year of publication 1993
Reviewed by Radu Precup

The book is devoted to some evolution problems arising in the dynamics of mechanical systems involving unilateral contraints possibly in the presence of dry friction. In this context the velocity function is not necessarily absolutely continuous and so the mathematical model expressed by a differential inclusion is not a traditional one. This book is mainly concerned with existence and approximation of the solutions to such nonstandard differential inclusions. For instance, Chapters 1 and 2 deal with Moreau's sweeping process: <formula>, where <formula> is a convex moving set of a real Hilbert space H and the right-hand side is the outward normal cone to C(t) at point u(t). By a solution one means a right-continuous function of bounded variation <formula> such that <formula> for all <formula> and there exists a positive measure <formula> on I relative to which the Stieltjes measure du of u has a density <formula>, i.e., <formula> and <formula> for <formula>-almost every t in I. To treat such nonstandard differential inclusions, several typical mathematical tools are needed. Mainly they concern functions of bounded variation defined in real intervals (derivation of Stieltjes measures, compactness results, approximation and convergence in the sense of graphs) and geometrical inequalities. These are presented in Chapter 0: Preliminaries. Chapter 3: Inelastic shocks with or without friction: existence results, is concerned with a problem of the dynamics of a mechanical system with a finite number of degrees of freedom, subject to a unique unilateral constraint and experiencing inelastic shocks. The case of inelastic shocks with friction is also considered. Chapter 4: Externally induced dissipative collisions, deals with a system with finite number of degrees of freedom subject to unilateral constraints which can produce dissipative or inelastic collisions. Chapter 5 reviews some directions of research that can be linked to the subject of this book.

The book is concluded by a good Bibliography, an Index and an Index of notation.

Most of the material is based on author's results and on those of J.J. Moreau and of the Montpellier School on Evolution problems in mechanics.

The book is well-written and presupposes only a moderate background in functional analysis. The author is both a good researcher and and expositor.

We recommend this book to all interested in nonlinear evolution equations, set-valued analysis and mathematical models in mechanics.