Nonlinear mathematics. (English) Zbl 0198.00102

Saaty, Thomas L.; Bram, Joseph

International Series in Pure and Applied Mathematics. Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd. 400 p. (1964).

The rather unrevealing title “Nonlinear mathematics” refers to the following topics: approximate solution of nonlinear equations, nonlinear programming, stability of nonlinear ordinary differential equations, automatic control, and prediction theory.
The first chapter is a short introduction to functional analysis, including the contraction principle and fixed-point theorems. To some extent this is window-dressing. The book is designed to be accessible to a rather wide audience and for most of it the reader needs little more than standard advanced calculus and linear algebra. Under the heading nonlinear algebraic and transcendental equations comes a thorough discussion of Newton’s method, the method of steepest descent, and various generalizations of them.
The chapter on nonlinear programming begins with some basic facts about extrema of convex functions. The saddle-point condition and theorems about duality are proved. Then eight different algorithms for nonlinear programming are described. The authors have brought together a great deal of material scattered through the literature, much of it less than five years old.
The chapter on differential equations begins with the usual existence and uniqueness theorem for first-order systems, followed by a review of linear, constant coefficient systems. Then stability under small perturbations, geometric theory (for two equations), Floquet theory, and periodic solutions of nonlinear systems with periodic coefficients are discussed. An introduction to Lyapunov stability is given, with applications to control systems. Pontryagin’s maximum principle is stated, in the general formulation of Berkovitz, as well as the functional analysis approach of Kulikowski in optimal control theory.
The last chapter introduces the linear theory of filtering and prediction. Then the general estimation problem is discussed, following Balakrishnan. Finally, a maximum likelihood technique related to the method of Kalman and Bucy is described.
The style is lucid and the choice of topics excellent. The bibliography is extensive. The book should prove valuable to student and researcher alike.

Reviewer: W. H. Fleming (M. R. 29 # 2103)

MSC:

00A05	Mathematics in general
34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
62-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
90-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
34A34	Nonlinear ordinary differential equations and systems
34Dxx	Stability theory for ordinary differential equations
49J15	Existence theories for optimal control problems involving ordinary differential equations
62M20	Inference from stochastic processes and prediction
93E11	Filtering in stochastic control theory
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
90C30	Nonlinear programming

Keywords:

approximate solution of nonlinear equations; nonlinear programming; stability of nonlinear ordinary differential equations; automatic control; prediction theory