Avrachenkov K.E., Filar J.A., Howlett P.G. Analytic Perturbation Theory and Its Applications

Philadelphia: SIAM, 2013. - 373p.Mathematical models are often used to describe complex phenomena such as climate change dynamics, stock market fluctuations, and the Internet. These models typically depend on estimated values of key parameters that determine system behavior. Hence it is important to know what happens when these values are changed. The study of single-parameter deviations provides a natural starting point for this analysis in many special settings in the sciences, engineering, and economics. The difference between the actual and nominal values of the perturbation parameter is small but unknown, and it is important to understand the asymptotic behavior of the system as the perturbation tends to zero. This is particularly true in applications with an apparent discontinuity in the limiting behavior - the so-called singularly perturbed problems.
Analytic Perturbation Theory and Its Applications includes a comprehensive treatment of analytic perturbations of matrices, linear operators, and polynomial systems, particularly the singular perturbation of inverses and generalized inverses. It also offers original applications in Markov chains, Markov decision processes, optimization, and applications to Google PageRank and the Hamiltonian cycle problem as well as input retrieval in linear control systems and a problem section in every chapter to aid in course preparation.
Audience: This text is appropriate for mathematicians and engineers interested in systems and control. It is also suitable for advanced undergraduate, first-year graduate, and advanced, one-semester, graduate classes covering perturbation theory in various mathematical areas.Introduction and MotivationFinite Dimensional PerturbationsInversion of Analytically Perturbed Matrices
Perturbation of Null Spaces, Eigenvectors, and Generalized Inverses
Polynomial Perturbation of Algebraic Nonlinear SystemsApplications to Optimization and Markov ProcessApplications to Optimization
Applications to Markov Chains
Applications to Markov Decision ProcessesInfinite Dimensional PerturbationsAnalytic Perturbation of Linear Operators
Background on Hilbert Spaces and Fourier Analysis