Two-Scale Stochastic Systems: Asymptotic Analysis and Control: 49 - Rilegato

Kabanov, Yuri; Pergamenshchikov, S.

Rilegato

ISBN 10: 3540653325 ISBN 13: 9783540653325

Casa editrice: Springer Nature, 2002

Riassunto

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.

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Dalla quarta di copertina

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 description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.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 description "Stochastic Tikhonov-Levinson theory and its applications." The book provides a mathematical apparatus designed to analyze the dynamic behaviour of a randomly perturbed system with fast and slow variables. In contrast to the deterministic Tikhonov-Levinson theory, the basic model is described in a more realistic way by stochastic differential equations. This leads to a number of new theoretical questions but simultaneously allows us to treat in a unified way a surprisingly wide spectrum of applications like fast modulations, approximate filtering, and stochastic approximation.

Dettagli dell'edizione

• Editore: Springer Nature
• Data di pubblicazione: 2002
• Lingua: Inglese
• ISBN 10: 3540653325
• ISBN 13: 9783540653325
• Rilegatura: Copertina rigida
• Numero di pagine: 250
• Contatto del produttore: Elevation; https://www.also.com/ec/cms5/en_6000/6000/index.jsp; ; Friedrich-Karl-Str. 9; Hamburg; 22391; Germania