The Method of Lyapunov-Razumikhin Functions in Stability Analysis of Systems with Delay

Andreev, Aleksandr; Sedova, N. O.

doi:10.1134/s0005117919070014

Cited by 11 publications

(3 citation statements)

References 74 publications

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“…Studies involving different time delays can be found in ship stabilization, control processes for pressure, and heat transfer regulation, but they are sometimes deliberately introduced into feedback systems to improve system performances see [1,5] and references therein for details. See also [6,7,8], and [9] for more details on system performances in the presence of time delays.…”

Section: Original Research Articlementioning

confidence: 99%

Stability and Feedback Control of Volterra Type Systems with Time Delay

Davies

Ebiwareme

Ekesiobi

et al. 2022

ARJOM

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New results for stability and feedback control of time delay systems were proposed. These results were obtained by using Lyapunov Razumikhin method to approximate the stability of the uncontrolled Volterra type system with delay and designing a state feedback controller using a model transformation technique, the Lyapunov matrix equation and the Razumikhin approach for the stabilization of the controlled Volterra type system with delay. Examples are given to illustrate the effectiveness of the theoretical results.

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Section: Original Research Articlementioning

confidence: 99%

Stability and Feedback Control of Volterra Type Systems with Time Delay

Davies

Ebiwareme

Ekesiobi

et al. 2022

ARJOM

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show abstract

“…Consequently, it has been encountered in a deluge number of monographs, [8], [9], [12]- [14], [16], [18], [19], [21],23], [25], [28] and research papers, see the review papers [2], [3], [15], [20] and further references therein, which develop stability criteria for some types of system with time delay. Advancements in this area were achieved primarily through extensions of the Lyapunov methodology to systems with delays that were initially commenced by N. Krasovskii [24] and Razumikhin [33], and also through the applications of Halanay's inequality [17], see also [12], 13] and additional references therein, and Bohl-Perron theorem, [4], [5], [14] and some other techniques.…”

Section: Introductionmentioning

confidence: 99%

Application of a Novel Model Reduction Technique to the Assessment of Boundedness/Stability of Some Delay Time-Varying Vector Nonlinear Systems

Pinsky

2024

Preprint

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This paper develops a new approach to the assessment of the boundedness/stability of some vector nonlinear systems with delays and variable coefficients. The approach rests on the development of scalar counterparts to the original vector systems. We show that the solutions to these scalar auxiliary nonlinear equations with delay and variable coefficients bound from the above the norms of solutions to the original equations with the matched history functions. This prompts the assessment of the boundedness/stability traits of the vector systems through the abridged evaluation of the dynamics of their scalar counterparts. The latter task is achieved in effortless simulations or through the application of simplified analytical inferences. Consequently, we convey some novel boundedness/ stability criteria and estimate the radiuses of the balls imbedded in the boundedness/stability regions. Lastly, we authenticate our inferences in representative simulations that also measure their accuracy. This prompts the evaluation of the boundedness/stability properties of nonautonomous VNDS through the assessment of dynamics of their scalar counterparts that can be triggered in effortless simulations or abridged analytical reasoning. Consequently, we derive some novel boundedness/stability criteria and estimate the radiuses of the balls that are immersed in the boundedness/stability regions of the original systems. Lastly, we authenticate the developed approach in representative simulations that also measure the accuracy of these techniques. This paper is organized as follows. The next section outlines our notation, some preliminary statements, and defines the underlined system. Section 3 ascribes the reduction technique and its applications to the evaluation of the boundedness/stability of some VNDS with variable coefficients. Section 4 presents a closed-form stability criterion. Section 5 discusses the applications of linearized auxiliary equations. Section 6 highlights the results of our simulations, and Section 7 concludes this study and outlines some directions for subsequent research.

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Section: Introductionmentioning

confidence: 99%

Optimal Dichotomy of Temporal Scales and Boundedness/ Stability of Time – Varying Multidimensional Nonlinear Systems

Pinsky¹

2022

Preprint

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This paper develops a new approach to the estimation of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients – an essential task in various engineering and natural science applications. Known approaches to assessing the stability of such systems rest on the utility of Lyapunov functions and Lyapunov first approximation methodologies, typically providing conservative and computationally elaborate criteria for multidimensional systems of this category. Adequate criteria of boundedness of solutions to nonhomogeneous systems of this kind are rare in the contemporary literature. Lately, we develop a new approach to these problems which rests on bounding the evolution of the norms of solutions to initial systems by matching solutions of a scalar auxiliary equation we introduced in [1], [2] and [3]. Still, the technique advanced in [3] rests on the assumption that the average of the linear components of the underlying system is defined by a stable matrix of general position. The current paper substantially amplifies the application domain of this approach. It is merely assumed that the time – dependent linear block of the underlying system can be split into slow and fast varying components by application of any smoothing technique. This dichotomy of temporal scales is determined by the optimal criterion reducing the conservatism of our estimates. In turn, we transform the linear subsystem with slow-varying matrix in a diagonally dominant form by successive applications of the Lyapunov transforms. This prompts the development of novel scalar auxiliary equations embracing the estimation of the norms of solutions to our initial systems. Next, we formulate boundedness/ stability criteria and estimate the relevant regions of the underlying systems using analytical and abridged numerical reasoning. Lastly, we authenticate the developed methodology in inclusive simulations.

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The Method of Lyapunov-Razumikhin Functions in Stability Analysis of Systems with Delay

Cited by 11 publications

References 74 publications

Stability and Feedback Control of Volterra Type Systems with Time Delay

Stability and Feedback Control of Volterra Type Systems with Time Delay

Application of a Novel Model Reduction Technique to the Assessment of Boundedness/Stability of Some Delay Time-Varying Vector Nonlinear Systems

Optimal Dichotomy of Temporal Scales and Boundedness/ Stability of Time – Varying Multidimensional Nonlinear Systems

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