"Control Theory and Optimization I" is a textbook that focuses on the development of geometric methods for the investigation and uncovering of geometric aspects of the theory of differential equations with quadratic right-hand sides, particularly Riccati equations. These equations are closely related to the calculus of variations and optimal control theory. The book is divided into three main parts: the first part addresses classical calculus of variations and the geometric theory of the Riccati equation, while the second part examines complex Riccati equations as flows on Cartan-Siegel homogeneous spaces. The third part is dedicated to the minimization problem for multiple integrals and Riccati partial differential equations. The initial chapters provide a comprehensive introduction to the necessary foundations to clearly and precisely present the subsequent topics. The presentation is designed to be both informative and accessible, without overwhelming the reader with formal and technical details.