Control theory and optimization. I: Homogeneous spaces and the Riccati equation in the calculus of variations. Transl. from the Russian by S. A. Vakhrameev
This book is devoted to the development of geometric methods for studying and revealing geometric aspects of the theory of differential equations with quadratic right-hand sides (Riccati-type equations), witch are closely related to the calculus of variations and optimal control theory. The book contains the following three parts: 1. the classical calculus of variations and the geometric theory of the Riccati equation (Ch. 1-5), 2. complex Riccati equations as flows on Cartan-Siegel homogeneity domains (Ch. 6), and 3. the minimization problem for multiple integrals and Riccati partial differential equations (Ch. 7 and 8). Chapters 1-4 are mainly auxiliary and the author has organized the presentation so that it is not obscured by formal and technical details and, at the same time, is sufficiently precise. Other chapters contain the author's results concerning the matrix double ratio, complex Riccati equations, and also the Riccati partial differential equation, which arises in the minimization problem for a multiple integral. In the last two decades has been an intensive development of geometric methods for studying the Riccati equation. In particular, it was recently observed that Riccati equations define a smooth flow on Lagrange-Grassmann manifolds. However, these ideas, which are mainly presented in journals, have not yet found a sufficient reflection in monographs; this book is intended to fill this gap. The book is based on a course of lectures given at the Department of Mechanics and Mathematics of Moscow State University during several years and is addressed to undergraduate and graduate students, scientific researchers and all specialists interested in the problems of geometry, the calculus of variations, and differential equations. See also the Russian original cited above.
scientific article; zbMATH DE number 1406735