Representation of a solution of the Cauchy problem for an oscillating system with pure delay

Khusainov, D. Ya.; Diblı́k, Josef; Růžičková, Miroslava; Lukáčová, J

doi:10.1007/s11072-008-0030-8

Cited by 76 publications

(72 citation statements)

References 2 publications

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“…Motivated by delayed exponential representing a solution of a system of differential or difference equations with one or multiple fixed or variable delays [1][2][3][4][5][6], which has many applications in theory of controllability, asymptotic properties, boundary-value problems, and so forth [3][4][5][7][8][9][10][11][12][13][14][15], we extended representation of a solution of a system of differential equations of second order with delay [1] ( ) = − 2 ( − ) (1) to the case of two delays…”

Section: Introductionmentioning

confidence: 99%

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Diblı́k

Fečkan

Pospíšil

2013

Abstract and Applied Analysis

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Section: Introductionmentioning

confidence: 99%

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Diblı́k

Fečkan

Pospíšil

2013

Abstract and Applied Analysis

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“…Inspired by the pinner work in Khusainov and Shuklin, more and more researchers develop the similar ideas to deal with new classes of delay differential equations. Khusainov et al introduced a notation of delayed matrix cosine/sine of a polynomial of degree and use this new notation to derive a representation of a solution to a second‐order linear differential equations with pure delay. Diblík et al extend 1‐delay to 2‐delay cases.…”

Section: Introductionmentioning

confidence: 99%

“…Khusainov et al introduced a notation of delayed matrix cosine/sine of a polynomial of degree and use this new notation to derive a representation of a solution to a second‐order linear differential equations with pure delay. Diblík et al extend 1‐delay to 2‐delay cases. Li and Wang introduce a concept of delayed Mittag‐Leffler–type matrix function and seek an explicit formula of solution to a fractional‐order homogeneous linear differential equation with delay.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by Khusainov and Diblík, Diblík et al, and Liang et al, we develop the similar idea to consider finite‐time stability of an oscillating system with 2 delays as follows:

({, \begin{matrix} right x^{''} (t) & left = - {normalΩ}_{1}^{2} x (t - τ_{1}) - {normalΩ}_{2}^{2} x (t - τ_{2}) + g (t), t \in [0, T], & right \\ right x (t) & left = ρ (t), x^{'} (t) = ρ^{'} (t), t \in [- τ, 0), τ = \max {τ_{1}, τ_{2}}, \end{matrix})

where τ 1 , τ 2 >0 is 2 delays; ∞ > T ≥ l τ 1 + m τ 2 >0 is fixed;

l, m \in double-struckN, k \in double-struckN, Ω_{1}, Ω_{2}

are permutable n × n matrices, ie,

Ω_{1} Ω_{2} = Ω_{2} Ω_{1}, g \in C false(false[0, T false], R^{n} false)

; and

ρ \in C^{1} false(false[- τ, 0 false], R^{n} false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Observing the expression of and , one can find that Q , R depend on 2 delays τ 1 , τ 2 and 2 indices i , j , which are more complex than the formulation of delayed matrix cosine/sine of a polynomial of degree, Definitions 1, 2 depending only 1 delay and 1 index. Thus, it also brings the challenge to deal with finite‐time stability of by using the representation of solution .…”

Section: Introductionmentioning

confidence: 99%

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Finite‐time stability of a class of oscillating systems with two delays

Cao

Wang

2018

Math Methods in App Sciences

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In this paper, we study finite‐time stability of an oscillating system with 2 delays. To derive a bounded of state vector, we use a representation of explicit solution involving 2‐delayed matrix polynomial of 2 indices after deriving some fundamental estimates for such delayed matrix polynomial of 2 indices. A sufficient condition is given. Finally, an example is given to demonstrate the application of the main result.

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Finite time stability and relative controllability of Riemann‐Liouville fractional delay differential equations

Wang

2019

Math Methods in App Sciences

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This paper firstly deals with finite time stability (FTS) of Riemann-Liouville fractional delay differential equations via giving a series of properties of delayed matrix function of Mittag-Leffler. We secondly study relative controllability of such type-controlled system. With the help of the representation of solution, both Gram-like type matrix and rank criterion are derived, which extend the corresponding results for linear systems. KEYWORDSdelayed matrix function of Mittag-Leffler, finite time stability, fractional delay differential equations, relative controllability INTRODUCTIONRecently, some authors apply delayed exponential matrix function to find the explicit solution formula of linear delay equations. 1-3 Further, there are some continued work on finding the representation of solutions to delay continuous and discrete systems; see, for example, previous studies. [4][5][6][7][8][9][10][11] Fractional differential equations have been used in solving the practical problems in the fields of natural science. 12,13 In particular, FTS of linear fractional delay differential equations with controls was offered in Lazarević, 14 and an inequality of Gronwall type is used to derive the results. For other related contribution, one can refer to previous works. [15][16][17][18][19][20][21][22][23][24][25] However, an initial singular condition for the Riemann-Liouville fractional differential equations is much different from the standard initial condition for a Caputo fractional differential equations. The main results of the standard initial condition for a Caputo fractional delay differential equation were obtained in Li and Wang. 11 Relative controllability and its related problems of linear systems represented by different type delay systems have been studied in literature. [26][27][28][29][30][31][32][33] In particular, rank and Kalman criteria for relative controllability are studied extensively.Very recently, Li and Wang 12 studiedHere, D − + x denotes Riemann-Liouville fractional derivative, I 1− − + x denotes Riemann-Liouville fractional integral, T = k * < ∞ for a fixed k * ∈ {1, 2, … }, is a constant, ∈ C([− , T], R n ), and (D − + )( ) exists. We consider the Riemann-Liouville fractional derivative D − + , where the lower limit is − + (− is a singular point). To keep the "consistence" with the lower limit in Riemann-Liouville derivative, we keep the initial condition at − + .

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Representation of a solution of the Cauchy problem for an oscillating system with pure delay

Cited by 76 publications

References 2 publications

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Finite‐time stability of a class of oscillating systems with two delays

Finite time stability and relative controllability of Riemann‐Liouville fractional delay differential equations

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