Exponential stability of linear discrete systems with constant coefficients and single delay

Diblı́k, Josef; Khusainov, D. Ya.; Baštinec, Jaromír; Sirenko, A S

doi:10.1016/j.aml.2015.07.008

Cited by 70 publications

(42 citation statements)

References 6 publications

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“…Under the assumptions that A and B are permutation matrices, Khusainov & Shuklin give a representation of a solution of a linear homogeneous system with delay by introducing the concept of delayed matrix exponential

e_{h}^{B t}

corresponding to delay h and matrix B :

e_{h}^{B t} : = \{\begin{matrix} normalΘ, & - \infty < t \leq - h, \\ I, & - h < t \leq 0, \\ I + B t + B^{2} \frac{{((), t - h)}^{2}}{2!} + . . . + B^{k} \frac{{((), t - ((), k - 1) h)}^{k}}{k!}, & ((), k - 1) h < t \leq k h . \end{matrix}

They proved that fundamental matrix of linear delay system (delayed perturbation of exponential matrix e A t ) can be given by

e^{A t} e_{h}^{B_{1} ((), t - h)}, B_{1} = e^{- A h} B .

Notice that the fractional analogue of the same problem was considered by Li and Wang in the case A = Θ. For more recent contributions on oscillating system with pure delay, relative controllability of system with pure delay, asymptotic stability of nonlinear multidelay differential equations, finite time stability of differential equations, one can refer to previous studies and reference therein.…”

Section: Introductionmentioning

confidence: 99%

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

Mahmudov

2018

Math Methods in App Sciences

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In this paper, we propose a delayed perturbation of Mittag‐Leffler type matrix function, which is an extension of the classical Mittag‐Leffler type matrix function and delayed Mittag‐Leffler type matrix function. With the help of the delayed perturbation of Mittag‐Leffler type matrix function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations.

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e_{h}^{B t}

corresponding to delay h and matrix B :

e_{h}^{B t} : = \{\begin{matrix} normalΘ, & - \infty < t \leq - h, \\ I, & - h < t \leq 0, \\ I + B t + B^{2} \frac{{((), t - h)}^{2}}{2!} + . . . + B^{k} \frac{{((), t - ((), k - 1) h)}^{k}}{k!}, & ((), k - 1) h < t \leq k h . \end{matrix}

They proved that fundamental matrix of linear delay system (delayed perturbation of exponential matrix e A t ) can be given by

e^{A t} e_{h}^{B_{1} ((), t - h)}, B_{1} = e^{- A h} B .

Section: Introductionmentioning

confidence: 99%

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

Mahmudov

2018

Math Methods in App Sciences

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“…, . Theorem 3 generalizes Theorem 1 in [4]. Some results can detect asymptotic stability but do not provide estimates of solutions, often necessary for computational purposes (in spite of the fact that, for special classes of equations, they can provide criteria on asymptotic stability, depending on delay, of the type "if on only if").…”

Section: A Discussion Of the Results Obtained Comparisons And Examplesmentioning

confidence: 99%

“…A similar discussion applies with respect to the assumptions of Theorem 2. In establishing the stability or exponential stability of linear systems with constant coefficients and a single delay, [5] utilizes a different set of sufficient conditions (independent of our results). The main result [5,Theorem 2] has the form of the following theorem.…”

Section: A Discussion Of the Results Obtained Comparisons And Examplesmentioning

confidence: 99%

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Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

Diblı́k

2017

Discrete Dynamics in Nature and Society

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The paper investigates the exponential stability of a linear system of difference equations with variable delays ( + 1) = ( ) + ∑ =1 ( ) ( − ( )), = 0, 1, . . ., where ∈ N, is a constant square matrix, ( ) are square matrices, ( ) ∈ N ∪ {0}, and ( ) ≤ for an ∈ N. New criteria for exponential stability are derived using the method of Lyapunov functions and formulated in terms of the norms of matrices of linear terms and matrices solving an auxiliary Lyapunov equation. An exponential-type estimate of the norm of solutions is given as well. The efficiency of the derived criteria is numerically demonstrated by examples and their relations to the well-known results are discussed.

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“…To exhibition the effectiveness of the WDOPs‐based approach proposed in this paper, in Section IV we will provide two numerical examples. Their simulation results indicate that our approach can arrive the larger decay rates than ones in .…”

Section: Introductionmentioning

confidence: 89%

WDOP‐based Summation Inequality and its Application to Exponential Stability of Linear Delay Difference Systems

Zhang

Shang

Wang

2017

Asian Journal of Control

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This paper is concerned with the exponential stability analysis of linear delay difference systems. Firstly, a set of weighted discrete orthogonal polynomials (WDOPs) is established by using the Gram‐Schmidt orthogonalization process, and then two WDOPs‐based summation inequalities, including some existing summation inequalities as special cases, are developed. Secondly, these WDOPs‐based summation inequalities are applied to investigate the exponential stability criteria and explicit exponential estimates of solutions of linear delay difference systems. Finally, two numerical examples indicate that the proposed WDOPs‐based approach can derive the exponential stability condition with larger decay rate than the existing ones.

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Exponential stability of linear discrete systems with constant coefficients and single delay

Cited by 70 publications

References 6 publications

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

Delayed perturbation of Mittag‐Leffler functions and their applications to fractional linear delay differential equations

Exponential Stability of Linear Discrete Systems with Variable Delays via Lyapunov Second Method

WDOP‐based Summation Inequality and its Application to Exponential Stability of Linear Delay Difference Systems

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