A Stochastic Tikhonov Theorem in Infinite Dimensions

A Stochastic Tikhonov Theorem in Infinite Dimensions The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a "slow" and a "fast" variable; the system is strongly coupled and driven by linear unbounded operators generating a C 0 -semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A Stochastic Tikhonov Theorem in Infinite Dimensions

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Publisher
Springer-Verlag
Copyright
Copyright © 2006 by Springer
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-005-0845-y
Publisher site
See Article on Publisher Site

Abstract

The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a "slow" and a "fast" variable; the system is strongly coupled and driven by linear unbounded operators generating a C 0 -semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 1, 2006

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