Contents:General Description of Impulsive Differential SystemsLinear SystemsStability of SolutionsPeriodic and Almost Periodic Impulsive SystemsIntegral Sets of Impulsive SystemsOptimum Control in Impulsive SystemsAsymptotic Study of Oscillations in Impulsive SystemsA Periodic and Almost Periodic Impulsive SystemsBibliographySubject Index Readership: Researchers in nonlinear science. keywords:Differential Equations with Impulses;Linear Systems;Stability;Periodic and Quasi-Periodic Solutions;Integral Sets;Optimal Control “… lucid … the book … will benefit all who are interested in IDE…” Mathematics Abstracts

Significant interest in the investigation of systems with discontinuous trajectories is explained by the development of equipment in which significant role is played by impulsive control systems and impulsive computing systems. Impulsive systems are also encountered in numerous problems of natural sciences described by mathematical models with conditions reflecting the impulsive action of external forces with pulses whose duration can be neglected. Differential equations with set-valued right-hand side arise in the investigation of evolution processes in the case of measurement errors, inaccuracy or incompleteness of information, action of bounded perturbations, violation of unique solvability conditions, etc. Differential inclusions also allow one to describe the dynamics of controlled processes and are widely used in the theory of optimal control. This monograph is devoted to the investigation of impulsive differential equations with set-valued and discontinuous right-hand sides. It is intended for researchers, lecturers, postgraduate students, and students of higher schools specialized in the field of the theory of differential equations, the theory of optimal control, and their applications.

Many evolution processes are characterized by the fact that at certain moments of time they experience a change of state abruptly. These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics and frequency modulated systems, do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems.

This book is devoted to impulsive differential equations with a small parameter. It consists of three chapters. Chapter One serves as an introduction. In Chapter Two, regularly perturbed impulsive differential equations are considered. Modifications of the method of small parameter, the averaging method, and the method of integral manifolds are proposed. In Chapter Three, singularly perturbed differential equations are considered. A modification of the method of boundary functions is proposed, and asymptotic expansions along the powers of the small parameters of the solutions of the initial value problem, the periodic problem, and some boundary value problems are found. Numerous nonstandard applications to the theory of optimal control are made. The application of some other methods to impulsive singularly perturbed equations is illustrated, such as the numerical-analytical method for finding periodic solutions, the method of differential inequalities and the averaging method.The book is written clearly, strictly, and understandably. It is intended for mathematicians, physicists, chemists, biologists and economists, as well as for senior students of these specialities.

The book presents qualitative results for different classes of fractional equations, including fractional functional differential equations, fractional impulsive differential equations, and fractional impulsive functional differential equations, which have not been covered by other books. It manifests different constructive methods by demonstrating how these techniques can be applied to investigate qualitative properties of the solutions of fractional systems. Since many applications have been included, the demonstrated techniques and models can be used in training students in mathematical modeling and in the study and development of fractional-order models.

Equipment to be installed in electric power-transmission and distribution systems must pass acceptance tests with standardized high-voltage or high-current test impulses which simulate the stress on the insulation caused by external lightning discharges and switching operations in the grid. High impulse voltages and currents are also used in many other fields of science and engineering for various applications. Therefore, precise impulse-measurement techniques are necessary, either to prevent an over- or understressing of the insulation or to guarantee the effectiveness and quality of the application. The target audience primarily comprises engineers and technicians but the book may also be beneficial for graduate students of high-voltage engineering and electrical power supply systems.