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Über dieses Buch

The book discusses set-valued differential equations defined in terms of the Hukuhara derivative. Focusing on equations with uncertainty, i.e., including an unknown parameter, it introduces a regularlization method to handle them. The main tools for qualitative analysis are the principle of comparison of Chaplygin – Wazhewsky, developed for the scalar, vector and matrix-valued Lyapunov functions and the method of nonlinear integral inequalities, which are used to establish existence, stability or boundedness.

Driven by the question of how to model real processes using a set-valued of differential equations, the book lays the theoretical foundations for further study in this area. It is intended for experts working in the field of qualitative analysis of differential and other types of equations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. General Properties of Set-Valued Equations

Abstract
The chapter discusses the general properties of equations with a set of trajectories. Here a regularization procedure for the set of uncertain equations is proposed.
Anatoly A. Martynyuk

Chapter 2. Analysis of Continuous Equations

Abstract
In this chapter the application of the comparison principle and the direct Lyapunov method in terms of auxiliary matrix-valued functions is proposed for solution of the problems under consideration.
Anatoly A. Martynyuk

Chapter 3. Discrete-Time Systems with Switching

Abstract
In the present chapter we set out a general approach to stability analysis problem for a set of trajectories of difference equations with uncertain parameter values.
Anatoly A. Martynyuk

Chapter 4. Qualitative Analysis of Impulsive Equations

Abstract
In recent years, the method of matrix Lyapunov-like functions, which is a generalization of the classical Lyapunov direct method based on matrix-valued functions, has been significantly developed (see, for example, Martynyuk [64, 66, 69] and the references therein). Parallel to the development of the method for different classes of new equations, the structure of the matrix-valued Lyapunov functions remains of great importance and attracts an increasing attention. It is well known that the components of the matrix-valued functions depend on the system of equations under consideration, as well as on compositions of its subsystems. A natural subject for the investigation by means of the multi-component Lyapunov-like functions with different components is the class of impulsive systems.
Anatoly A. Martynyuk

Chapter 5. Stability of Systems with Aftereffect

Abstract
In this chapter we consider a set of equations with aftereffect and uncertain parameters. As a result of regularization of the family of equations according to the scheme adopted in the book, a set of equations with aftereffect are obtained, for which the solution existence conditions are established, an estimate of the distance between the extreme solution sets is obtained, and stability conditions for the set of stationary solutions on a finite time interval are found as well as the attenuation conditions for the set of trajectories.
Anatoly A. Martynyuk

Chapter 6. Impulsive Systems with Aftereffect

Abstract
In the first part of the chapter a family of differential equations with aftereffect under impulsive perturbations is considered. For such equations, some results of set trajectories analysis based on the matrix-valued function defined on the product of spaces are given. In the second part of the chapter, for the first time, uncertain sets of equations under impulse perturbations are investigated. Estimates of the distance between extremal sets of trajectories are derived for the systems under consideration. In addition, conditions for the global existence of the sets of solutions regularized with respect to the uncertainty parameter are proved.
Anatoly A. Martynyuk

Chapter 7. Dynamics of Systems with Causal Operator

Abstract
This chapter presents the results of dynamical analysis of the equations and the set of equations with robust causal operator. The conditions for the local and global existence of solutions to the regularized equation are established, the estimate of the funnel containing a set of trajectories is given, and the stability conditions for a set of stationary solutions are found. In this case, the direct Lyapunov method and the principle of comparison with the matrix Lyapunov function are applied.
Anatoly A. Martynyuk

Backmatter

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