The Second Lyapunov Method for Time-Delay Systems

Демиденко, Г. В.; Матвеева, И. И.

doi:10.1007/978-981-16-6297-3_11

Cited by 3 publications

(2 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Over several decades of development of this method, many new ideas have been proposed (see [2,3]). Nowadays, the method continues to be applied to different classes of equations, and these classes are expanded (see, for example, works [4][5][6][7][8]). Another well-known method is the Razumikhin function method, the foundations of which are outlined in [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stability Conditions for Linear Semi-Autonomous Delay Differential Equations

Malygina,

Chudinov

2023

Mathematics

View full text Add to dashboard Cite

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods.

show abstract

Section: Introductionmentioning

confidence: 99%

“…It follows from Lemma 2 that h > 0. Suppose x is the solution of the Cauchy problem(8). If there exists t 1 ,t 2 ∈ R such that l + ω ≤ t 1 < t 2 and |x(t 1 )| < |x(t 2 )|, then there exists t 0 ∈ [t 1 − l − ω, t 1 ) such that x(t 0 ) = − 1 h x(t 1 ).…”

mentioning

confidence: 99%

Stability Conditions for Linear Semi-Autonomous Delay Differential Equations

Malygina,

Chudinov

2023

Mathematics

View full text Add to dashboard Cite

show abstract

Estimates of Solutions in a Model of Antiviral Immune Response

Skvortsova

2023

Sib. Adv. Math.

View full text Add to dashboard Cite

On the Partial Stability of Nonlinear Discrete-Time Systems with Delay

Vorotnikov

2023

View full text Add to dashboard Cite

A system of nonlinear discrete (finite-difference) of a general form with a bounded delay is considered. Interest in the tasks of qualitative analysis of such systems has increased significantly in recent years. At the same time, the problem of stability with respect to all variables of the zero equilibrium position, which has a great generality, is mainly analyzed in domestic and foreign literature. The main research method is a discrete-functional analogue of the direct Lyapunov method. In this article, it is assumed that the system under consideration admits a “partial” (in some part of the state variables) zero equilibrium position. The problem of stability of a given equilibrium position is posed, and stability is considered not in all, but only in relation to a part of the variables that determine this equilibrium position. Such a problem belongs to the class of problems of partial stability, which are actively studied for systems of various forms of mathematical description. The proposed statement of the problem complements the scope of the indicated studies in relation to the system under consideration. To solve this problem, a discrete version of the Lyapunov– Krasovskii functionals method is used in the space of discrete functions with appropriate specification of the functional requirements. To expand the capabilities of this method, it is proposed to use two types of additional auxiliary (vector, generally speaking) discrete functions in order to: 1) adjustments of the phase space region of the system in which the Lyapunov–Krasovskii functional is constructed; 2) finding the necessary estimates of the functionals and their differences (increment) due to the system under consideration, on the basis of which conclusions about partial stability are made. The expediency of this approach lies in the fact that as a result, the Lyapunov-Krasovskii functional, as well as its difference due to the system under consideration, can be alternating in the domain that is usually considered when analyzing partial stability. Sufficient conditions of partial stability, partial uniform stability, and partial uniform asymptotic stability of the specified type are obtained. The features of the proposed approach are shown on the example of two classes of nonlinear systems of a given structure, for which partial stability is analyzed in parameter space. Attention is drawn to the expediency of using a one-parameter family of functionals.

show abstract

The Second Lyapunov Method for Time-Delay Systems

Cited by 3 publications

References 49 publications

Stability Conditions for Linear Semi-Autonomous Delay Differential Equations

Stability Conditions for Linear Semi-Autonomous Delay Differential Equations

Estimates of Solutions in a Model of Antiviral Immune Response

On the Partial Stability of Nonlinear Discrete-Time Systems with Delay

Contact Info

Product

Resources

About