The stability cone for a delay differential matrix equation

Khokhlova, T. N.; Kipnis, M. M.; Malygina, V. V.

doi:10.1016/j.aml.2010.12.020

Cited by 20 publications

(7 citation statements)

References 2 publications

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“…(29) (II) Now we prove that assumptions (23)-(25) imply (13) and (16). Due to equivalence of (13) and (14) with (23), it remains to be shown that (23)-(25) imply (15) and (16).…”

Section: Necessary and Sufficient Conditions Determining Weaklymentioning

confidence: 80%

“…The weakly delayed systems analyzed in this paper can be simplified and their solutions can be found in explicit analytical forms (results obtained generalize those in [1,2]). Consequently, analytical forms of solutions can be used directly to solve several problems for weakly delayed systems, for example, problems of asymptotical behavior of their solutions, boundary-value problems, or some problems of control theory (using different methods, such problems have recently been investigated e.g., in [10][11][12][13][14][15][16][17][18]). For an alternative approach to differentialdifference equations using the variational iteration method and new analytical and asymptotic methods see, for example, [19][20][21].…”

Section: Discussionmentioning

confidence: 99%

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General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

Diblı́k

Halfarová

2014

Abstract and Applied Analysis

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Planar linear discrete systems with constant coefficients and delaysx(k+1)=Ax(k)+∑l=1n‍Blxl(k-ml)are considered wherek∈ℤ0∞:={0,1,…,∞},m1,m2,…,mnare constant integer delays,0<m1<m2<⋯<mn,A,B1,…,Bnare constant2×2matrices, andx:ℤ-mn∞→ℝ2. It is assumed that the considered system is weakly delayed. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension2(mn+1)is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained.

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“…(29) (II) Now we prove that assumptions (23)-(25) imply (13) and (16). Due to equivalence of (13) and (14) with (23), it remains to be shown that (23)-(25) imply (15) and (16).…”

Section: Necessary and Sufficient Conditions Determining Weaklymentioning

confidence: 80%

Section: Discussionmentioning

confidence: 99%

General Explicit Solution of Planar Weakly Delayed Linear Discrete Systems and Pasting Its Solutions

Diblı́k

Halfarová

2014

Abstract and Applied Analysis

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“…Будем использовать метод D-разбиения [16]. [18], сразу получаем, что если точка (Re µ, Im µ, λ) принадлежит "конусу" (находится в центре на рис. 2), включая границы, то функция g не имеет нулей справа от мнимой оси.…”

Section: признаки разрешимости для уравнения с сосредоточенным запаздunclassified

“…Значит, учитывая однозначную разрешимость уравнения (15) в паре пространств (B α , L β ), по теореме 3 решение уравнения (??) в паре пространств (B α , L β ) представимо в виде (18).…”

Section: признак однозначной разрешимости для уравнения с распределенunclassified

Solvability of inhomogeneous autonomous differential equation with aftereffect on negative half-axis

Balandin¹,

Sabatilina²

2019

Iz. VUZ. Matematika

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“…Stability of these equation has a wide range of applications in science and engineering. Recently, These equations have been investigated by many authors; for example, Matsunaga [1], Cooke and van den Driessche [2], Kuang [3] , Cooke and Grossman [4], Ruan and Wei [5], Hale and Lunel [6], Khokhlovaa, Kipnis and Malygina [8], Cermák and Jánsky [9], Hrabalova [10], Nakajima [11], Hara and Sakata [12], Smith [13], Freedman and Kuang [14] and Bellman and Cooke [15] which have studied the asymptotic stability of linear retarded differential equations. In this paper, we give some new necessary and sufficient conditions for the asymptotic stability of the following system…”

Section: Introductionmentioning

confidence: 99%