Relevance of the research qualitative behaviour of nonlinear control systems due, above all, the importance of their practical application in various fields of knowledge, control theory, the theory of stability of motion, mechanics, radio engineering, ecology, etc.
The purpose of the dissertation is to develop ideas of the Lyapunov’s direct method in relation to the qualitative analysis of complex dynamic systems of a special form.
The dissertation consists of an introduction and six parts.
The introduction substantiates the relevance of the topic of the dissertation, formulates the purpose and tasks of the research, determines the scientific novelty and the significance of the obtained results, give a brief overview of the current state of the problems that are being studied in the work.
The first section is devoted to the review of literary sources on the topic of the dissertation and the main results obtained over the last decades, analysed the state of the scientific problem. The present state of the problem of the interval stability of solutions of dynamical systems is analysed, an overview on the absolute stability of dynamic systems is carried out. The peculiarities of applying the direct Lyapunov method to the study of dynamical systems with the aftereffect are considered.
In the second and third sections we obtain the conditions for absolute interval stability of nonlinear control systems with time-delay argument and interval uncertainty. In the study, we use the direct Lyapunov method: the approach of finite-dimensional Lyapunov functions and the approach of the Lyapunov-Krasovskii functionals. The results are developed and generalized on the case of systems, which described by neutral type equations. Subsequently, in part four, nonlinear control systems, which are described by the difference delay equations are investigated. Using the analogues Lyapunov-Krasovskii functionals in Lur’e-Postnikov type, in which instead of the quadratic form of the integral stand the sum of quadratic forms of the prehistory, similar results are obtained.
The fifth section is devoted to the problems of stabilization in Lur’e-type control problem. Namely, by using the additionally introduced feedback control to the linear part of the system, could built constructive conditions for the stabilization of the the investigated systems to the state of absolute stability within the framework of the selected classes of Lyapunov’s function or functional.
Thesis is devoted to the development of the general conception of the qualitative analysis of Lur’e-type control systems which described in terms of functional differential equations with deviating argument, and interval uncertainty of the linear part through the use of Lyapunov's direct method .
Lyapunov-Krasovskii functionals and finite dimensional Lyapunov functions approaches which applied to the study of stability and constructing of systems solutions are developed.
Sufficient conditions for the absolute, absolute interval stability are proved. The value of the "critical" time-delay argument is calculates. The estimates of solutions exponential decay for continuous systems of direct and indirect control with delay and neutral type are constructed. Results are extended to the corresponding discrete systems.
It is known that the main theorem of Lyapunov stability and asymptotic stability has necessary and sufficient character. That is, if the zero solution of the system is asymptotically stable, then the corresponding function always exists. But the central and unresolved problem is the actual construction of this function. The problem is simplified if a function is searched in a parametrically given in advance class of functions, for example, in the class of quadratic functions. Or, as in the case of the Lur’e problem, - in the class of quadratic forms plus the integral from nonlinearity, and others, so to speak, derived from this type form. In this case, the problem of finding the "best" Lyapunov function can be reduced to the problem of convex programming. New optimization problems of constructing Lyapunov functions and Lyapunov-Krassovskii functionals in such predefined classes, with aim to solve the absolute stability problem of continuous Lur’e type control systems with deviating argument are formulated and solved in final part of dissertation thesis.
