On the theory of stability of motion of a nonlinear system on a time scale

Martynyuk-Chernienko, Yu. A.

doi:10.1007/s11253-008-0103-y

Cited by 4 publications

(5 citation statements)

References 7 publications

(7 reference statements)

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“…for any t 0 ; t 2 T : Using relation (14) and properties of the exponential function, we obtain the representation which, by virtue of assumption .H 3 / and the last estimate, proves estimate (9). We now present a theorem [10] that plays a fundamental role in the investigation of the stability of solutions of dynamic equations using functions from the class of functions discontinuous in t 2 T [10].…”

Section: Resultsmentioning

confidence: 99%

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Stability of solutions of one class of nonlinear dynamic equations

Babenko¹,

Слынько²

2011

Nonlinear Oscill

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We study the stability of the zero solution of a nonlinear dynamic equation on a time scale under certain assumptions on the right-hand side of this equation. In addition to conditions for the existence and uniqueness of a solution of the Cauchy problem, we also assume that the exponential function of the linear approximation is bounded, and the norms of the nonlinear part and its derivatives with respect to the components of the space variable are majorized by power functions of the norm of the space variable. Using the generalized method of Lyapunov functions, we obtain sufficient conditions for the stability of the zero solution of the nonlinear equation under consideration.

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Section: Resultsmentioning

confidence: 99%

“…In [9], the stability of the zero solution of a nonlinear dynamic equation of the general form was studied using matrix-valued Lyapunov functions, and sufficient conditions for its stability, uniform stability, asymptotic stability, and uniform asymptotic stability were obtained.…”

Section: Introductionmentioning

confidence: 99%

Stability of solutions of one class of nonlinear dynamic equations

Babenko¹,

Слынько²

2011

Nonlinear Oscill

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“…(2) Lyapunov's second method based on scalar, vector, or matrix-valued functions [26,33,34,35,36,90];…”

Section: Methods Of Stability Analysis Of Systems On a Time Scalementioning

confidence: 99%

“…Now we will make the following assumptions on system (1.1). The basic theorems of Lyapunov's second method for the dynamic equations (1.1) were proved in [26,33,34,90] in terms of the existence of function (1.16) and its total D-derivative (1.18). Here is one of these theorems.…”

Section: Generalized Lyapunov's Second Methodmentioning

confidence: 99%

Elements of the Theory of Stability of Hybrid Systems (Review)

Martynyuk

2015

Int Appl Mech

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The sufficient conditions for different types of stability of three classes of hybrid systems modeled by dynamic equations on a time scale, impulsive hereditary systems, and equations in the Banach space are discussed. Some general results are illustrated by examples and applications in mechanics and theory of neural networks Keywords: DE-model of hybrid system, hybrid impulsive hereditary systems, hybrid systems in metric spaces, matrix Lyapunov function method, stability, boundedness, periodicityIntroduction. The classical theory of stability of motion unites methods and approaches that allow analyzing an equilibrium state for stability in a mathematical model or a real process. Such models are, as a rule, systems of ordinary differential equations or partial differential equations. There are monographs and books [10, 13-15, 67, 77, 117] that discuss basic approaches to this problem.To describe modern engineering and process facilities and systems, use is made of mathematical models that differ from those mentioned above and require adequate methods of qualitative analysis. One of the first models of a hybrid system was Witsenhausen's model [115].In this model, the state of the system has continuous and discrete-time components. The continuous state of the system is described by a system of ordinary differential equations with the right-hand side depending on the discrete state. The discrete state changes once the continuous state has fallen within a certain domain in the state space.A mathematical model of a real system is hybrid when its behavior is described by equations of different types. Examples of such systems are -continuous systems with changing phase (bouncing ball, walking robot, growing and dividing cell); -continuous systems controlled by discrete automata (thermostat, discretely accelerated chemical process, autopilot); -coordinated processes (take-off and landing of aircraft at an airport terminal, highway traffic control). Moreover, the term "hybrid system" is used to describe the dynamics of objects containing neural networks, various fuzzy-logic devices, electric and mechanical components in complex systems, and in many other situations. An important example of hybrid systems is systems consisting of a digital control device and a continuous component describing the model of the process under consideration. Traditionally, an analysis of such systems involves the discretization of the mathematical model of the continuous component, resulting in a system of difference equations to be analyzed. Such an approach may not be applicable to the modern theories of robust control where both continuous and discrete components play an important role in their natural form.Hence, an important task at the current stage of development of this research area is to set up a theory of stability of hybrid systems.Here we will consider the following classes of hybrid systems: -systems on a time scale described by dynamic equations (DE-model) (not to be confused with dynamic systems in the sense of Nemytskii and St...

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“…Lemmas 4 and 5 of the present paper enable one to extend the range of applicability of integral inequalities on a time scale in the process of analysis of solutions of dynamic equations. Their applications may be promising in combination with the method of Lyapunov functions for dynamic equations (see [7]). …”

Section: Remarkmentioning

confidence: 99%