Partial Stability and Control

Unlike the conventional research for the general theory of stability, this mono­ graph deals with problems on stability and stabilization of dynamic systems with respect not to all but just to a given part of the variables characterizing these systems. Such problems are often referred to as the problems of partial stability (stabilization). They naturally arise in applications either from the requirement of proper performance of a system or in assessing system capa­ bility. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: • "Lotka-Volterra ecological principle of extinction;" • focusing and acceleration of particles in electromagnetic fields; • "drift" of the gyroscope axis; • stabilization of a spacecraft by specially arranged relative motion of rotors connected to it. Also very effective is the approach to the problem of stability (stabilization) with respect to all the variables based on preliminary analysis of partial sta­ bility (stabilization). A. M. Lyapunov, the founder of the modern theory of stability, was the first to formulate the problem of partial stability. Later, works by V. V. Rumyan­ tsev drew the attention of many mathematicians and mechanicians around the world to this problem, which resulted in its being intensively worked out. The method of Lyapunov functions became the key investigative method which turned out to be very effective in analyzing both theoretic and applied problems. Unlike the conventional research for the general theory of stability, this monoƯ graph deals with problems on stability and stabilization of dynamic systems with respect not to all but just to a given part of the variables characterizing these systems. Such problems are often referred to as the problems of partial stability (stabilization). They naturally arise in applications either from the requirement of proper performance of a system or in assessing system capaƯ bility. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: " "Lotka-Volterra ecological principle of extinction;" " focusing and acceleration of particles in electromagnetic fields; " "drift" of the gyroscope axis; " stabilization of a spacecraft by specially arranged relative motion of rotors connected to it. Also very effective is the approach to the problem of stability (stabilization) with respect to all the variables based on preliminary analysis of partial staƯ bility (stabilization). A.M. Lyapunov, the founder of the modern theory of stability, was the first to formulate the problem of partial stability. Later, works by V.V. RumyanƯ tsev drew the attention of many mathematicians and mechanicians around the world to this problem, which resulted in its being intensively worked out. The method of Lyapunov functions became the key investigative method which turned out to be very effective in analyzing both theoretic and applied problems Front Matter....Pages i-xii Introduction: Stability, Stabilization, and Control with Respect to Part of the Variables: Objectives and Methods....Pages 1-50 Linear Problems of Stability, Stabilization, and Control with Respect to Part of the Variables....Pages 51-118 Nonlinear Problems of Stability with Respect to Part of the Variables in the First (Linear and Nonlinear) Approximation....Pages 119-166 “Essentially” Nonlinear Problems of Stability with Respect to Part of the Variables....Pages 167-202 Nonlinear Problems of Stabilization and Control with Respect to Part of the Variables....Pages 203-288 Nonlinear Game-Theoretic Problems of Control with Respect to Part of the Variables under Uncontrollable Interference....Pages 289-346 Stability and Stabilization of Functional-Differential Equations with Respect to Part of the Variables....Pages 347-378 Stability and Stabilization of Stochastic Systems with Respect to Part of the Variables....Pages 379-398 Back Matter....Pages 399-430 Partial Stability and Control develops a new, efficient method of analysis and of control synthesis for problems on partial stability and control in dynamic systems described by ordinary differential equations, including delay, stochastic, and uncertain systems. The method is based on efficient procedures of transformation on initial systems or their subsystems, for controlled systems, and allows the solutions to be simplified. In addition, the method also allows many linear and nonlinear problems to be solved that cannot be easily done with available methods. Ample attention is given to nonlinear game-theoretical problems of reorientation of an asymmetric solid. This book will be a valuable reference for advanced graduates and professionals in applied mathematics, mechanics and control, and control engineering who use stability theory and control methods.