Springer Two-Scale Stochastic Systems: Asymptotic Analysis and Control: 49 (Stochastic Modelling and Applied Probability, 49)

In many complex systems one can distinguish "fast" and "slow" processes with radically di?erent velocities. In mathematical models based on di?er- tialequations,suchtwo-scalesystemscanbedescribedbyintroducingexpl- itly a small parameter?on the left-hand side ofstate equationsfor the "fast" variables,and these equationsare referredto assingularly perturbed. Surpr- ingly, this kind of equation attracted attention relatively recently (the idea of distinguishing "fast" and "slow" movements is, apparently, much older). Robert O'Malley, in comments to his book, attributes the originof the whole historyofsingularperturbationsto the celebratedpaperofPrandtl[79]. This was an extremely short note, the text of his talk at the Third International Mathematical Congress in 1904: the young author believed that it had to be literally identical with his ten-minute long oral presentation. In spite of its length, it had a tremendous impact on the subsequent development. Many famous mathematicians contributed to the discipline, having numerous and important applications. We mention here only the name of A. N. Tikhonov, whodevelopedattheendofthe1940sinhisdoctoralthesisabeautifultheory for non-linear systems where the fast variables can almost reach their eq- librium states while the slow variables still remain near their initial values: the aerodynamics of a winged object like a plane or the "Katiusha" rocket may serve an example of such a system. It is generally accepted that the probabilistic modeling of real-world p- cesses is more adequate than the deterministic modeling. Review From the reviews: "The book presents Tikhonov theorems for stochastic systems with two time scales when a parameter decreases to zero. … The topic of the book is relevant for the research areas of stochastics and of control and system theory. … The strength of the book is in the analysis of Tikhonov theorems … . The text is very well structured, runs very smooth, and the exposition allows detailed scrutiny of all proofs. The book is expected to become a basic reference on two-scale stochastic systems." (J. H. van Schuppen, Nieuw Archief voor Wiskunde, Vol. 6 (2), 2005) "This research monograph needs to be placed on your shelves … . The monograph is organized by seven chapters and a valuable appendix. … The book is written for a graduated mathematically oriented readership who is supposed to be familiar with basic concepts of stochastic differential equations and related issues." (Henri Schurz, Zentralblatt MATH, Vol. 1033 (8), 2004) "This very well-crafted monograph examines the coupled stochastic and singularly perturbed stochastic dynamics … . the monograph displays in a very attractive manner the general scene of singularly perturbed stochastic differential equations. Telling examples are provided along with enlightening explanations, comparisons and detailed background material needed to grasp the theory. … To sum up, a mixture of modern techniques concerning singularly perturbed stochastic differential equations, along with classical arguments, is presented in a rigorous, clear and attractive manner." (Zvi Artstein, Mathematical Reviews, 2004 c) "Two time-scale problems arise in many problems of practical interest. The essential ingredient in these problems is a decomposition of the dynamics into ‘fast’ and ‘slow’ modes. … This book deals, inter alia with ‘averaging methods’ where the coefficients of the limiting slow dynamics are obtained by averaging the fast dynamics. Indeed, the idea of ‘averaging’ has found wide-spread application in related problems. The book is uncompromisingly mathematical but clearly and elegantly written." (G. C. Goodwin, Short Book Reviews, Vol. 23 (3), 2003) From the Back Cover Two-scale systems described by singularly perturbed SDEs have been the subject of ample literature. However, this new monograph develops subjects that were rarely addressed and could be given the collective desc