Control Theory and Optimization I

This book is devoted to geometric methods in the theory of differential equations with quadratic right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. Connections of the calculus of variations and the Riccati equation with the geometry of Lagrange-Grassmann manifolds and classical Cartan-Siegel homogeneity domains in a space of several complex variables are considered. In the study of the minimization problem for a multiple integral, a quadratic partial differential equation that is an analogue of the Riccati equation in the calculus of variations is studied. This book is based on lectures given by the author over a period of several year's in the Department of Mechanics and Mathematics of Moscow State University. The book is addressed to, under- graduate and graduate students, scientific researchers and all specialists interested in the problems of geometry, the calculus of variations, and differential equations.