Existence, stability and controllability results of a Volterra integro-dynamic system with non-instantaneous impulses on time scales

Malik, Muslim; Kumar, Vipin

doi:10.1093/imamci/dnz001

Cited by 12 publications

(7 citation statements)

References 22 publications

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“…From the year 2000 till now, a rapidly growing attention has been paid in the subject of various studies on time-scales; for instance, the reader may consult the papers [13][14][15][16][17][18][19][20][21][22][23] and references cited therein. On the other hand, solutions of the various types of the first-order dynamic equation on time scale have been discussed by several authors; for instance, the literature.…”

Section: Figure 1 Life Span Of Pharaoh Cicadamentioning

confidence: 99%

A generalized delta derivative on time scale with applications

Negi

Abbas

Malik

2020

Math Methods in App Sciences

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Inspired by a well‐known “Hilger‐derivative” on time scale introduced by Stephan Hilger, we introduce a new derivative on the time scale, called “black delta (or in notation ▴)‐derivative.” It is an adaptive unified approach of derivative of continuous and discrete calculus. Several important nontrivial results concerning this derivative are obtained. Also, a necessary and sufficient condition for the existence of this new derivative and its connection with Hilger‐derivative are discussed. In addition, a new general a‐exponential function (for fixed a ≥ 1) is introduced to obtain the unique solution of the initial value problem on time scales with this derivative. Furthermore, a relation between the a‐exponential function with the existing exponential function on the time scale is obtained. We elucidate the results by considering some interesting examples. We expect that this derivative will be applicable for future investigations on various studies in the field of mathematical modeling, engineering applications.

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Section: Figure 1 Life Span Of Pharaoh Cicadamentioning

confidence: 99%

A generalized delta derivative on time scale with applications

Negi

Abbas

Malik

2020

Math Methods in App Sciences

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“…In recent years, few authors studied the finite-dimensional dynamic systems on time scales and investigated the existence of solutions, Ulam–Hyers type stability, and controllability results, see for instance (András and Mészáros 2013 ; Bohner and Wintz 2012 ; Davis et al. 2009 ; Lupulescu and Younus 2011 ; Malik and Kumar 2020 ; Shen and Li 2019 ; Zada et al. 2017 ; Shah and Zada 2019 , 2022 ; Ben Nasser et al.…”

Section: Introductionmentioning

confidence: 99%

“…2009 ) to the time-varying systems with instantaneous impulses on time scales. In (Malik and Kumar 2020 ), the authors established the existence of a unique solution, Ulam–Hyers stability, and controllability results for a Volterra integro-dynamic system with non-instantaneous impulses on time scales. In (Pervaiz et al.…”

Section: Introductionmentioning

confidence: 99%

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

2022

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We establish several qualitative properties of a neutral switched impulsive evolution system on an arbitrary time domain by using the theory of time scales. This is the first attempt for switched evolution systems with impulses in abstract spaces. First, we investigate the existence of a unique solution and Ulam’s type stability results. After that, we establish the total controllability results, i.e., controllability not only with respect to the endpoint of the interval but also on the impulsive points. We transform the controllability problem into a solvability problem of an operator equation. We used the Banach fixed point theory and evolution operator theory to establish these results. To illustrate the effectiveness and implications of the developed results, we provide theoretical and simulated numerical examples for different time domains.

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“…In Xie and Wang [16], the authors extended the results of Wei and Song [15] to multiple time‐delayed cases. Further, the controllability results of dynamic systems on time scales is a relatively newer area and only a few works have been reported [19–26] and references therein. Particularly, in Davis et al [19], the authors considered a non‐singular dynamical system on time scales

\begin{matrix} alignleft \\ align-1 x^{Δ} (t) & align-2 = A x (t) + B u (t), \\ align-1 x (t_{0}) & align-2 = x_{0} \end{matrix}

and established the controllability and observability results.…”

Section: Introductionmentioning

confidence: 99%

Controllability of singular dynamic systems on time scales

Malik

Sajid²,

Kumar

2021

Asian Journal of Control

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In this work, we investigate the controllability of singular dynamic systems on time scales. First, we decompose the consider systems into a slow subsystem and a fast subsystem. After that, we use the Laplace transform and convolution theorem to derive the state response of these two subsystems. Finally, we established some necessary and sufficient conditions for the controllability of the slow subsystem and fast subsystem. At last, we provide an example to illustrate the obtained analytical results.

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Existence, stability and controllability results of a Volterra integro-dynamic system with non-instantaneous impulses on time scales

Cited by 12 publications

References 22 publications

A generalized delta derivative on time scale with applications

A generalized delta derivative on time scale with applications

Existence, stability and controllability results for a class of switched evolution system with impulses over arbitrary time domain

Controllability of singular dynamic systems on time scales

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