##
**Local stabilizability of nonlinear control systems.**
*(English)*
Zbl 0757.93061

Series on Advances in Mathematics for Applied Sciences. 8. Singapore etc.: World Scientific. viii, 202 p. (1992).

The stabilizability problem can be considered a version of the classical stability problem for equilibrium positions of ordinary differential equations. For the case of linear systems a complete and satisfactor theory is available. When nonlinearities are involved, the problem is considerably more difficult. Systematic studies started at the end of 70’s.

The main purpose of this lecture notes is to survey some recent advances, which are known at the beginning of the 90’s, concerning the nonlinear stabilizability. Besides the “Introduction and some notation”, this book consists of four chapters. Chapter I contains the statement of the problem and a number of basic results, such as the relationship between controllability and stabilizability, the stabilizability of linear systems and the more version of the linearization approach etc. Chapter II is devoted to the extension of the Lyapunov direct method to nonlinear control systems. The affine systems, the Artstein-Sontag Theorem and results of the Jurdjevic-Quinn type are included in this chapter. In Chapter III the author collects a number of results which have been obtained in different contexts. Instead of investigating directly the given problem, the author modifies the system by simplifying its structural equations. Certain extensions of the linearization approach, the center manifold approach, the cascade systems and the existence of suitable canonical forms are considered in this chapter. The purpose of Chapter IV is to review some recent results about stabilizability of low dimensional systems, that is, the state space is \(R^ n\), with \(n\leq 3\).

Four appendices are included. They are respectively devoted to basic concepts of classical stability theory, greater spaces, center manifold theory and normal form expansions. There is a bibliography of 152 items, it consists of two parts one contains papers especially devoted to stabilizability of nonlinear systems and one contains occasional references and titles of general interest in stability theory and control of systems. Most of them are appeared in the last dozen of years.

The main purpose of this lecture notes is to survey some recent advances, which are known at the beginning of the 90’s, concerning the nonlinear stabilizability. Besides the “Introduction and some notation”, this book consists of four chapters. Chapter I contains the statement of the problem and a number of basic results, such as the relationship between controllability and stabilizability, the stabilizability of linear systems and the more version of the linearization approach etc. Chapter II is devoted to the extension of the Lyapunov direct method to nonlinear control systems. The affine systems, the Artstein-Sontag Theorem and results of the Jurdjevic-Quinn type are included in this chapter. In Chapter III the author collects a number of results which have been obtained in different contexts. Instead of investigating directly the given problem, the author modifies the system by simplifying its structural equations. Certain extensions of the linearization approach, the center manifold approach, the cascade systems and the existence of suitable canonical forms are considered in this chapter. The purpose of Chapter IV is to review some recent results about stabilizability of low dimensional systems, that is, the state space is \(R^ n\), with \(n\leq 3\).

Four appendices are included. They are respectively devoted to basic concepts of classical stability theory, greater spaces, center manifold theory and normal form expansions. There is a bibliography of 152 items, it consists of two parts one contains papers especially devoted to stabilizability of nonlinear systems and one contains occasional references and titles of general interest in stability theory and control of systems. Most of them are appeared in the last dozen of years.

Reviewer: He Chungyou (Nanjing)

### MSC:

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

93C10 | Nonlinear systems in control theory |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |