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.

Table of Contents:
0.1 Preliminary Remarks.- 0.2 General Situations and Specific Problems Leading to Investigation of Problems of Stability and Stabilization with Respect to Part of the Variables.- 0.3 Formulation of the Problems of Stability and Stabilization with Respect to Part of the Variables. Lines and Stages of Their Research.- 0.4 The Method of Lyapunov Functions in Problems of Stability and Stabilization with Respect to Part of the Variables.- 0.5 The Problem of Stability with Respect to Part of the Variables of Linear Systems, in Linear Approximation, and in Critical Cases.- 0.6 Special Features and Possibilities of the Problem of Stability with Respect to Part of the Variables.- 0.7 The Problem of Control with Respect to Part of the Variables in a Finite Time Interval.- 1 Linear Problems of Stability, Stabilization, and Control with Respect to Part of the Variables.- 1.1 Stability with Respect to Part of the Variables of Linear Systems with Constant Coefficients.- 1.2 Stability with Respect to Part of the Variables of Linear Systems with Periodic Coefficients.- 1.3 Stability with Respect to Part of the Variables of Linear Systems with Continuous Sufficiently Differentiable Coefficients.- 1.4 The Effect of Constantly Acting and Parametric Perturbations on Stability with Respect to Part of the Variables.- 1.5 Stabilization with Respect to Part of the Variables.- 1.6 Control with Respect to Part of the Variables.- 1.7 Overview of References.- 2 Nonlinear Problems of Stability with Respect to Part of the Variables in the First (Linear and Nonlinear) Approximation.- 2.1 Features of the Problem of Stability with Respect to Part of the Variables in a Linear Approximation.- 2.2 A Method of Nonlinear Transformation of Variables in Investigating Stability with Respect to Part of the Variables in a Linear Approximation (1).- 2.3 Damping (with Respect to Part of the Variables) of Angular Motions of an Asymmetric Solid.- 2.4 A Method of Nonlinear Transformation of the Variables in Investigating Stability with Respect to Part of the Variables in a Linear Approximation (2).- 2.5 Stability with Respect to Part of the Variables in a Nonlinear Approximation.- 2.6 Stability with Respect to Part of the Variables in Lyapunov Critical Cases.- 2.7 Overview of References.- 3 “Essentially” Nonlinear Problems of Stability with Respect to Part of the Variables.- 3.1 Using a New Class of Lyapunov Functions.- 3.2 Developing Theorems of the Barbashin—Krasovskii Type.- 3.3 Partial Stability in the Presence of Large Initial Perturbations.- 3.4 Using Differential Inequalities.- 3.5 Instability with Respect to Part of the Variables.- 3.6 Stability with Respect to a Specified Number of the Variables.- 3.7 Overview of References.- 4 Nonlinear Problems of Stabilization and Control with Respect to Part of the Variables.- 4.1 Stabilization with Respect to Part of the Variables.- 4.2 The Auxiliary Function of the Partial Stabilization Problem in Studying Problems of Stabilization with Respect to All the Variables and Polystabilization.- 4.3 Stabilization and Partial Stabilization of Permanent Rotations (of Equilibrium Positions) of a Solid and a Satellite in Orbit.- 4.4 Reorientation of an Asymmetric Solid and Coordinated Control of a System of Solids (Manipulator Model).- 4.5 Finite-Time Control with Respect to Part of the Variables with Constraints on Controls.- 4.6 The Nonlinear Problem of “Passage” of an Asymmetric Solid through a Given Angular Position.- 4.7 Overview of References.- 5 Nonlinear Game-Theoretic Problems of Control with Respect to Part of the Variablesunder Uncontrollable Interference.- 5.1 Guaranteed Conditions for Controllability with Respect to Part of the Variables under Uncontrollable Interference.- 5.2 The Nonlinear Game-Theoretic Problem of “Passage” of an Asymmetric Solid through a Given Angular Position.- 5.3 An Auxiliary Function of the Problem of Control with Respect to Part of the Variables in Game-Theoretic Problems of Control with Respect to All the Variables.- 5.4 The Nonlinear Game-Theoretic Problem of Triaxial Reorientation of an Asymmetric Solid (First Method of Solution).- 5.5 The Nonlinear Game-Theoretic Problem of Triaxial Reorientation of an Asymmetric Solid (Second Method of Solution).- 5.6 The Nonlinear Game-Theoretic Problem of Uniaxial Reorientation of an Asymmetric Solid.- 5.7 Overview of References.- 6 Stability and Stabilization of Functional-Differential Equations with Respect to Part of the Variables.- 6.1 Formulation of the Problem of Stability with Respect to Part of the Variables.- 6.2 Using the Method of Lyapunov-Krasovskii Functionals.- 6.3 Using the Method of Lyapunov Functions.- 6.4 The Stability of Linear Delayed Systems with Respect to Part of the Variables.- 6.5 The Stabilization of Linear Delayed Systems with Respect to Part of the Variables.- 6.6 The Stability of Nonlinear Delayed Systems in the Linear Approximation.- 6.7 Overview of References.- 7 Stability and Stabilization of Stochastic Systems with Respect to Part of the Variables.- 7.1 Formulation of the Problem of Stability with Respect to Part of the Variables.- 7.2 Using the Method of Lyapunov Functions.- 7.3 Damping of Rotational Motion of a Solid (with Respect to Part of the Variables) with Random Interference in Control Channels Taken into Account.- 7.4 The Stability of Linear Systems with Respect to Part ofthe Variables.- 7.5 Stability with Respect to Part of the Variables in a Linear Approximation.- 7.6 The Stabilization of Nonlinear Controlled Systems with Respect to Part of the Variables.- 7.7 Overview of References.- References.

Review :

"Lucid and very well written.... Very suitable for graduate students of mathematics and engineering, in optimal control theory, as well as research professionals in the area. The book has many worked examples taken entirely from aerospace applications. It would thus also appeal to design engineers who work in aerospace or complex-manufacturing operations provided they have the willingness to accept rigorous mathematical arguments or maturity in functional analysis and differential equations. With nearly 400 references, this book provides a rich resource for further study and should be in the library of every graduate college mathematics and engineering department and every industrial R&D laboratory with interests in the control area." —Applied Mechanics Reviews

"This book deals with problems on stability and stabilization of dynamical systems with respect to a given part of the variables characterizing these systems. The theory goes back to Lyapunov, who was the first to formulate these kinds of problems. Here the author develops a new method based on transformation of original systems in some more convenient systems.... A number of problems of controlling the angular motion of an asymmetric solid in various formulations as well as problems of stabilizing an artificial satellite in circular and geostationary orbits are solved, as an illustration of the efficiency of the method proposed. The monograph is based mainly on the author’s results published in the last two decades. It is a valuable reference for advanced graduates and specialists in applied mathematics and engineering engaged in research involving differential equations, differential games, stability and control." —Applications of Mathematics

"This monograph gives a detailed and comprehensive study of the stability and stabilisation of dynamical systems with respect to part of thevariables.... The monograph begins with an excellent Introduction. This places the topic of the monograph clearly in the historical development of the partial stability of dynamical systems, [and] includes a detailed account of the relevant literature and makes use of the monograph's comprehensive bibliography. Several situations are then described which help motivate and justify the need for a study of partial stability/stabilisation.

This is a well-written book. At well over 400 pages there is much to discover and digest. There are numerous useful examples, both of a textbook style, aimed to clarify a definition or detail in a proof, and those of a more significant application-based style. I found the applications to the control of geo-stationary orbit of satellites most interesting and illuminating. Each chapter is clearly introduced and each concludes with an extensive and impressive overview of the literature." —UK Nonlinear News