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Journal of Applied Nonlinear Dynamics 13(1) (2024) 65--81 | DOI:10.5890/JAND.2024.03.006

A. Sreenivasulu$^{1}$, B. V. Appa Rao$^2$

$^1$ Department of Science and Humanities, MLR Institute of Technology, Dundigal, Hyderabad- 500043, Telangana, India

$^2$ Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, 522302, Andhra Pradesh, India

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Abstract

In this paper, we establish the asymptotic stability and boundedness of the Volterra integro-dynamic matrix Sylvester impulsive system on time scales. First, we convert the linear Volterra integro-dynamic matrix Sylvester impulsive system on time scale to an equivalent Kronecker product Volterra integro-dynamic impulsive system on time scales using vectorization operator. Then, we obtaine the results for asymptotic stability and boundedness of a time-varying Volterra integro-dynamic matrix Sylvester impulsive system on time scales in which the coefficient matrix is not necessarily stable. We generalize to a time scale some known properties concerning the asymptotic stability and boundedness from the continuous version. Finally, we’ve given some numerical and theoretical examples of the way those advanced analytical consequences may be applied. } [\hfill Asymptotic behavior\par \hfill Impulsive integro-dynamic system\par \hfill Boundedness\par \hfill Time scales\par][Sajad Jafari][16 May 2022][27 July 2022][1 January 2024][2024 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} The dynamics of several evolving processes are subject to sudden changes, such as shocks, harvesting, and natural disasters. These phenomena include short-term perturbations in the development of continuous and smooth dynamics, the duration of which is insignificant compared to an entire evolution. When such perturbations are included in models, they assume they act instantly or like impulses. Thus, impulsive differential equations have been developed to model impulsive problems in physics, ecology, population dynamics, biotechnology, biological systems, industrial robotics, optimal control, pharmacokinetics, and so on. \cite{bb1,bb5,bb9}. Basic qualitative and quantitative results about impulsive Volterra integro differential equations were studied in the literature \cite{bb6,bb13}. The works of burton \cite{bb10} are notable exceptions that disregarded the stability condition on the coefficient matrix. In \cite{bb19}, the asymptotic behavior of the solution to an integro-differential equation is discussed in which the coefficient matrix may not be stable. This area of dynamic equations on time scales is a new, modern and progressive component of applied analysis. It acts as a framework for effectively describing processes with both continuous and discrete elements .The author Mahfoud \cite{bb18} was studied sufficient conditions of Boundedness properties in the Volterra integro-differential system are uniformly bounded. In 1988, Hilger introduced \cite{bb11,bb12} the calculus on time scales in his Ph.D. thesis. The study of dynamic equations encompasses both continuous analyses as well as discrete analyses of the system for more details about the calculus on time scales. Volterra type equations (both integral and integro-dynamic) on time scales become a new field of interest in\cite{bb7,bb8}. The solution of Volterra integro-dynamic equations on time scales using variation of parameter obtained in \cite{bb15}. In \cite{bb17}, Kulik and Tisdell obtained basic qualitative and quantitative results for Volterra integral equations. Furthermore, \cite{bb16} Karpuz studied a solution's existence and uniqueness to generalized Volterra integral equations. In \cite{bb21}, the authors presented a theory for linear impulsive dynamic systems on time scales. Recently, the author Ionescu, Adela Ionescu \cite{ia} studied qualitatively the analysis of dynamical systems, particularly stability, a powerful tool with several connected appliances. In both natural and technological systems. There are presented in \cite{xi1} some new Razumikhin conditions for determining the uniform stability of impulsive functional differential equations with finite or infinite delay. In \cite{xi2} there have been several Lyapunov-based finite-time stability theorems involving stabilizing impulses and destabilizing impulses, derived through impulsive control theory. A system with state-dependent delay under impulse control under sufficient conditions was developed in \cite{xi3}. In \cite{rk1} discussed with a plot of bifurcation diagrams with a variety of initiation methods illustrates the multistability of the oscillator. In \cite{rk2} investigated a variety of bifurcation diagrams along with its Lyapunov exponents. They examined the fractional order synchronous reluctance motor system as well as its bifurcation plots. In \cite{rk3}, The results indicate that the proposed chaotic oscillator possesses new dynamical characteristics, including double-scroll chaotic attraction, four-scroll chaotic attraction, and coexisting attraction. The authors \cite{bb4,bb20} demonstrated the existence of $\psi$-bounded solutions for a system of linear dynamic equations on time scales. The matrix differential systems are the generalizations of the system of differential equations. The fundamental matrix of a system consists of linearly independent solutions. The system's transition matrix was obtained from the fundamental matrix. Murty et al \cite{bb14,bb25} studied matrix Lyapunov systems. In \cite{bb3}, the authors establish the necessary and sufficient conditions for controllability and observability of the Sylvester matrix dynamical system on time scales. In \cite{bb22,bb23,bb24} established for sufficient and necessary conditions for Ulam’s type stability, uniform stability, asymptotic stability, exponential stability, and strong stability, controllability of with impulsive and without impulsive matrix Sylvester Volterra integro-dynamic system. With this motivation in this paper, we deal with the asymptotic stability and boundedness of Volterra integro-dynamic matrix Sylvester impulsive on time scales. \begin{equation}\label{eq1} \left\{\begin{split} X^{\Delta}(t)=&P(t)X(t)+X(t)Q(t)+\mu(t)P(t)X(t)Q(t)\\ &+\int_{t_{0}}^{t}(L_{1}(t,s)X(s)+X(s)L_{2}(t,s))\Delta s +T(t),t\in \mathbb{T}_{0}\backslash\{t_k\}_{k=1}^\infty \\ X(t_k^+)=&(I+D_k)X(t_k),k=1, 2, ...,m\\ X(t_0)=&X_0 \end{split} \right .\end{equation} Where $\mathbb{T}$ is unbounded above time scale with bounded graininess, $\mathbb{T}_0:=[t_0,\infty)\cap\mathbb{T},t_k\in\mathbb{T}_0 $ are right dense, $0\leq t_0\leq t_1 \leq ,...,\leq t_k \leq..., \lim_{k\to\infty}t_k=\infty X(t_k^+)=\lim_{h\to 0^+}X(t_k+h),X(t_k^-)=\lim_{h\to0^-}X(t_k-h)$, $X(t)\in M_{n\times n}(\mathbb{R})$ state variable, and $P(t), Q(t), L_1(t,s), L_2(t,s)$ are dimensions $ n\times n$ matrix respectively. $T(t)\in n\times n,$ and $D_k\in n\times n$, $X^\Delta (t)$ is generalized delta derivative of X, $\mu(t)$ is a graininess function. In section 2,we study some standard properties and also, convert a Matrix-valued system into a Kronecker product system by using Variation of Parameter. In section 3, we establish the asymptotic stability of the solutions of the system (\ref{eq1}), which generalizes the continuous version $\mathbb{T}=\mathbb{R}$. In section 4, we discuss the boundedness of the solution of the system (\ref{eq1}) by constructing a Lyapunov functional. Furthermore, we developed the results for boundedness, uniform boundedness, and stability of solutions. \section{Preliminaries} We recollection of some fundamental definitions, notations and useful lemmas. \begin{definition}\cite{bb7} A nonempty closed subset of $\mathbb{R}$ is called a time scale. It is denoted by $\mathbb{T}$. We define a T interval as $[a,b]_{\mathbb{T}}=\{ t\in \mathbb{T}:a\leq t\leq b\}$ accordingly, we define $(a,b)_\mathbb{T}$,$[a,b)_\mathbb{T}$,$(a,b]_\mathbb{T}$ and so on. Also, we define $\mathbb{T}^k=\mathbb{T}\{max\mathbb{T}\}$ if $max\mathbb{T}$ exists, otherwise the forward jump operator $\sigma:\mathbb{T} \to \mathbb{T}$ is defined by $\sigma(t)=inf\{s\in \mathbb{T}:s>t\}\in \mathbb{T}$ with the substitution $inf\{\emptyset\}=sup\mathbb{T}$. The backward jum operator $\rho:\mathbb{T}\to \mathbb{T}$ is defined $\rho(t)=sup\{s\in\mathbb{T}:S>t\}$ with the substution $sup\{\emptyset\}=inf\mathbb{T}$, We say that t right scattered or left scattered if $\sigma(t)>t$ or $\rho(t)0$, there is a neighbourhood U of $\tau$ such that \begin{equation*} \mid[\psi(\sigma (\tau))-\psi(s)]-\psi^{\Delta}(\tau)[\sigma(\tau)-s]\mid\leq\varepsilon \mid \sigma(\tau)-s\mid,\forall s \in U \end{equation*} \end{definition} \begin{definition}\cite{bb7} If $H: \mathbb{T}^k \to \mathbb{R}$ is know as anti-derivative of $h: \mathbb{T}^k \to \mathbb{R}$ provided $H^\Delta (t)=h(t)$ fulfilled, for all $t\in \mathbb{T}^k$. then \begin{equation*} \int_{a}^{t}h(s)\Delta{s}=H(t)-H(a). \end{equation*} \end{definition} \begin{definition}\cite{bb8} A function $x:\mathbb{T}\to\mathbb{R}$ is known as regressive if $1+\mu(\tau)P(t)\ne 0$ for all $t\in \mathbb{T},$ if the set of all regressive functions are symbolized by $\mathcal{R}$. Also, $x$ is known as positive regressive function if if $1+\mu(\tau)p(t)>0$ for all $t\in\mathbb{T}$ and it is symbolized by $\mathcal{R}^+$. \end{definition} \begin{lemma} \cite{bb8}\label{lemma2.1} Let $r,s\in C_{rd}\mathcal{R}(\mathbb{T},\mathbb{R}).$ then $e_{r\ominus s}^\Delta(.,t_0)=(r-s)\frac{e_r(.,t_0)}{e_s^\sigma(.,t_0)}$ \end{lemma} \begin{theorem}\cite{bb15} \label{thm2.1} let $a,b\in\mathbb{T}$ with $b>a$ assume that $f:\mathbb{T}\times\mathbb{T}\to \mathbb{R}$ is integrable on $\{(t,s)\in\mathbb{T}\times\mathbb{T}:b>t>s\geq a\},$ then \begin{equation*} \int_{a}^{b}\int_{a}^{\eta}f(\eta,\xi)\Delta\xi\Delta\eta=\int_{a}^{b}\int_{\sigma(\xi)}^{b}f(\eta,\xi)\Delta\eta\Delta\xi \end{equation*} \end{theorem} \begin{lemma}\cite{bb8}\label{lemma2.2} Let $\alpha\in \mathbb{R}$ with $\alpha\in C_{rd}^+\mathcal{R}(\mathbb{T},\mathbb{R}).$ then $e_\alpha(t,s)\geq1+\alpha(t-s)$ forall $t\geq s$ \end{lemma} \begin{lemma}\cite{bb8}\label{lemma2.3} If $X\in PC_{rd}(\mathbb{T}, \mathbb{R^+})$ satisfies the inequality condition. Then \begin{equation*} X(t)\leq \alpha+\int_{a}^{t}P(s)x(s)\Delta s+\sum_{a

Acknowledgments

The authors would like to express their sincere thanks to the editor and anonymous reviewers for constructive comments and suggestions to improve the quality of this paper.

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