Vector Lyapunov functions and stability analysis of nonlinear systems.

*(English)*Zbl 0721.34054This is a fine book for all who are interested in the actual state of the art in a domain which has been substantially promoted by the (mutually independent) contributions of the first and second author over the last thirty years. From the authors’ preface: “The year 1992 marks the century after publication of the original pioneering work of Lyapunov and therefore it is desirable to examine the current state of his method and the accomplishments thus far achieved. It is in this spirit, we present in this book, a systematic account of the main trends of the basic theory of the method of vector Lyapunov functions, describe the current status of this approach, offer some new directions and provide a unified general structure applicable to a variety of nonlinear systems.”

The book consists of four chapters: Chapter 1. Why several Lyapunov functions? (Trends in basic Lyapunov theory. Definitions of stability and boundedness. Fundamental comparison of results. Refinements of basic Lyapunov theorems. Boundedness and practical stability. Method of vector Lyapunov functions. Stability concepts in terms of two measures. Practical stability and boundedness. Global results). Chapter 2. Refinements. (Perturbed systems. Large scale dynamic systems. A technique in perturbation theory. Quasisolutions. Analysis of comparison systems. Matrix Lyapunov functions. Cone-valued Lyapunov functions. Higher derivatives of Lyapunov functions. New directions. Existence and stability of stationary points). Chapter 3. Extensions (Differential equations with infinite delay. Integro-differential equations of Volterra type. Difference equations. Impulsive differential equations. Reaction- diffusion systems. Control systems. Decentralized control systems. Optimal controllability. Set-valued differential inequalities. Stability criteria). Chapter 4. Applications (Models from economics. Motion of an aircraft. Models in immunology. Models from neural networks. Population models. Models from chemical kinetics). Each chapters is opened by introductory remarks and closed by “notes” which give references to the bibliography (118 items). In an appendix certain comparison theorems for differential inequalities are presented.

The book consists of four chapters: Chapter 1. Why several Lyapunov functions? (Trends in basic Lyapunov theory. Definitions of stability and boundedness. Fundamental comparison of results. Refinements of basic Lyapunov theorems. Boundedness and practical stability. Method of vector Lyapunov functions. Stability concepts in terms of two measures. Practical stability and boundedness. Global results). Chapter 2. Refinements. (Perturbed systems. Large scale dynamic systems. A technique in perturbation theory. Quasisolutions. Analysis of comparison systems. Matrix Lyapunov functions. Cone-valued Lyapunov functions. Higher derivatives of Lyapunov functions. New directions. Existence and stability of stationary points). Chapter 3. Extensions (Differential equations with infinite delay. Integro-differential equations of Volterra type. Difference equations. Impulsive differential equations. Reaction- diffusion systems. Control systems. Decentralized control systems. Optimal controllability. Set-valued differential inequalities. Stability criteria). Chapter 4. Applications (Models from economics. Motion of an aircraft. Models in immunology. Models from neural networks. Population models. Models from chemical kinetics). Each chapters is opened by introductory remarks and closed by “notes” which give references to the bibliography (118 items). In an appendix certain comparison theorems for differential inequalities are presented.

Reviewer: W.Müller (Berlin)

##### MSC:

34D20 | Stability of solutions to ordinary differential equations |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93D30 | Lyapunov and storage functions |