This book, titled "Liapunov Functions and Stability in Control Theory," is part of the Lecture Notes in Control and Information Sciences series and is edited by M. Thoma and M. Morari. It is authored by Andrea Bacciotti and Lionel Rosier, focusing on the mathematical models of input systems described by continuous-time, finite-dimensional ordinary differential equations. The primary aim is to survey mathematical definitions of internal and external stability in a nonlinear context and discuss their characterizations using Liapunov functions. The book also explores methods to achieve more desirable stability behaviors through feedback laws. It covers topics such as the existence of Liapunov functions, stability and stabilizability of nonlinear systems, and the relationship between internal and external behaviors. The authors introduce the concept of "stabilizable" systems and discuss the use of discontinuous functions in certain contexts. The book is structured into six chapters, each addressing specific aspects of stability and stabilization, including time-invariant and time-varying systems, differential inclusions, and additional properties of Liapunov functions.This book, titled "Liapunov Functions and Stability in Control Theory," is part of the Lecture Notes in Control and Information Sciences series and is edited by M. Thoma and M. Morari. It is authored by Andrea Bacciotti and Lionel Rosier, focusing on the mathematical models of input systems described by continuous-time, finite-dimensional ordinary differential equations. The primary aim is to survey mathematical definitions of internal and external stability in a nonlinear context and discuss their characterizations using Liapunov functions. The book also explores methods to achieve more desirable stability behaviors through feedback laws. It covers topics such as the existence of Liapunov functions, stability and stabilizability of nonlinear systems, and the relationship between internal and external behaviors. The authors introduce the concept of "stabilizable" systems and discuss the use of discontinuous functions in certain contexts. The book is structured into six chapters, each addressing specific aspects of stability and stabilization, including time-invariant and time-varying systems, differential inclusions, and additional properties of Liapunov functions.