The purpose of Part 1 of this paper is to provide a review of recent results from 1991 through 2003 in the area of theoretical aspects of statistical and equivalent linearization in the analysis of structural and mechanical nonlinear stochastic dynamic systems. First, a discussion about misunderstandings appearing in the literature in derivation of linearization coefficients for mean-square linearization criterion is presented. In Secs. 3–6 new theoretical results, including new types of criteria, nonlinearities, and excitations in the context of linearization methods, are reviewed. In particular, moment criteria called energy criteria, linearization criteria in the space of power spectral density functions and probability density functions are discussed. A survey of a wide class of so-called nonlinearization techniques, including equivalent quadratization and equivalent cubicization methods, is given in Sec. 7. New linearization techniques for nonlinear stochastic systems with parametric Gaussian excitations and external non-Gaussian excitations are discussed in Secs. 8 and 9, respectively. In the last sections, four surveys of papers where stochastic linearization is used as a mathematical tool in other theoretical approaches, namely, models of dynamic systems with hysteresis, finite element method, and control of nonlinear stochastic systems and linearization with sensitivity analysis, are given. A discussion of the accuracy analysis of linearization techniques and some general conclusions close this paper. There are 217 references cited in this revised article.
Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces
