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International Journal of Trend in Scientific Research and Development (IJTSRD) International Open Access Journal ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume - 2 | Issue – 3

Time Delay and nd Mean Square St Stochastic Differential tial Equations in n Impetuous Stabilization G. Pramila Department of Mathematics, Vivekanandha College of Arts and Sciences for Women (Autonomous), Elayampalayam, yampalayam, Thiruchengode,, Namakkal, Tamil Nadu, India

S. Ramadevi Department of Mathematics, Vivekanandha College of Arts and Sciences for Women (Autonomous), Elayampalayam, Thiruchengode,, Namakkal, Tamil Nadu, India

ABSTRACT This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed. By projecting a delay differential inequality and using the stochastic analysis technique, a few present stage for mean square exponential stabilization are survived. It is express that an unstable stochastic delay system can be achieved some stability by impetuous. This example is also argued to derived the efficiency of the obtained results. Keywords: Impetuous stabilization, mean square stochastic delay differential equation, differential inequality 1. INTRODUCTION Impetuous control and stability tability has been extensively the variety of areas has found many application to be studied over the past centuries,, such as orbital transfer of satellite [11],, ecosystems management [12,13], dosage supply in pharmacokinetics [14], and stability and synchronization in chaotic secure comm communication systems and other chaotic systems [15,16]. At present, many significant results esults for the impetuous control and stabilization of delay differential systems. Systematically, the input states of systems not only get the disturbance of delay phenomena, but also get the disturbance of mean square stochastic phenomena in many practical problems. In current years, the stability investigation of impetuous stochastic delay differential systems is catching to many authors, and a

large number of stability criteria of these systems have been reported [1-6]. ]. However, these criteria are all needed so that the corresponding mean square stochastic delay differential systems without impetuous must be stable itself, which implies that the impetuous are only to inhibit the stability of mean square stochastic delay y differential systems. Although,, it is known that ther is no general rule for guiding how a proper Lyapunov functional can be constructed for a given n model. Therefore, it is very impetuous to projecting a new approach to studying the impetuous stabilization on of mean square stochastic functional differential equations without resorting to the Lyapunov functional. Without motivation from the above discussions, our main aim in this paper is to the study the time delay and mean square stochastic differential equations eq in impetuous stabilization. By projecting a delay differential inequality and using the stochastic analysis technique, some sufficient conditions for mean square exponential stability are obtained. 2. Model and Preliminaries In this paper, unless otherwise specified, we adopt the standard notation used previously by Yang [8]. Let E denote the n-dimensional dimensional unit matrix, |. | denote the Euclidean norm, đ?’Š ≜ {1,2, ‌ , đ?‘›}, â„• ≜ {1,2, ‌ }, â„? = [0, ∞), â„? = (0, ∞). for đ??´ ∈ â„? Ă— , đ?œ† (đ??´)đ?‘Žđ?‘›đ?‘‘ đ?œ† (đ??´) are

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr Apr 2018

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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 largest and smallest singular values of A, respectively. ‖đ??´â€– = đ?œ† (đ??´ đ??´). B[X,Y] denotes the space of continuous mappings from the topological space Y. In particular, let đ??ľ ≜ đ??ľ[[−đ?œ?, 0], â„? ], where đ?œ? > 0: PB[đ?• , â„? ] = {Ďˆ: đ?• → â„? âˆŁ Ďˆ(s)is continuous for all but at most countable ponit s ∈ đ?• and at these points ∈ đ?• , Ďˆ(s )andĎˆ(s ) exist and Ďˆ(s) = Ďˆ(s )},

đ??´(đ?&#x2018;Ą) = â&#x201E;&#x17D;(đ?&#x2018;Ą)đ??´, â&#x201E;&#x17D;(đ?&#x2018;Ą) â&#x2C6;&#x2C6; â&#x201E;&#x2039;, đ??´ â&#x2C6;&#x2C6; â&#x201E;? Ă&#x2014; . đ??ź â&#x2C6;&#x2C6; â&#x201E;? Ă&#x2014; , 0 â&#x2030;¤ đ?&#x153;?(đ?&#x2018;Ą) â&#x2030;¤ đ?&#x153;?, đ?&#x153;? đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;Ą, and fixed impetuous moment đ?&#x2018;Ą (đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;) satisfy đ?&#x2018;Ą < đ?&#x2018;Ą < â&#x2039;Ż and lim â&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;Ą = â&#x2C6;&#x17E;. đ?&#x2018;&#x201C;: â&#x201E;? Ă&#x2014; â&#x201E;? â&#x2020;&#x2019; â&#x201E;? đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x153;&#x17D;: â&#x201E;? Ă&#x2014; â&#x201E;? Ă&#x2014; â&#x201E;? â&#x2020;&#x2019; â&#x201E;? Ă&#x2014; are piecewise continuous vector-valued functions with đ?&#x2018;&#x201C;(đ?&#x2018;Ą, 0) = đ?&#x153;&#x17D;(đ?&#x2018;Ą, 0,0) â&#x2030;Ą 0, đ?&#x2018;Ą â&#x2C6;&#x2C6; â&#x201E;? , and ensuring the existence and uniqueness of (1). One may refer to[9,10] for the results on the existence and uniqueness of solutions of impetuous stochastic system.

Where đ?&#x2022; â&#x160;&#x201A; â&#x201E;? is an interval, Ď&#x2C6;(s )andĎ&#x2C6;(s ) denot the right-hand and left-hand limits of function Ď&#x2C6;(s) at time s, respectively. In particular, let đ?&#x2018;&#x192;đ??ľ â&#x2030;&#x153; đ?&#x2018;&#x192;đ??ľ [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? :

The solution đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ?&#x2018;Ľ (đ?&#x2018;Ą), â&#x20AC;Ś , đ?&#x2018;Ľ (đ?&#x2018;Ą) is a PB â&#x20AC;&#x201C; valued stochastic process. đ?&#x2018;¤(đ?&#x2018;Ą) = đ?&#x2018;¤ (đ?&#x2018;Ą), â&#x20AC;Ś , đ?&#x2018;¤ (đ?&#x2018;Ą) is an m-dimensional Brownian motion defined (Ί, â&#x201E;ą, {â&#x201E;ą } , đ?&#x2018;&#x192;).

â&#x201E;&#x2039; = {â&#x201E;&#x17D;(đ?&#x2018; ): â&#x201E;? â&#x2020;&#x2019; â&#x201E;? â&#x2C6;Ł â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2013;đ?&#x2018;  đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x2018;&#x2019;đ?&#x2018;&#x201D;đ?&#x2018;&#x;đ?&#x2018;&#x17D;đ?&#x2018;&#x2122; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x201C;đ?&#x2018;&#x2013;đ?&#x2018;&#x2019;đ?&#x2018;  sup

Definition 2.1

đ??ť < â&#x2C6;&#x17E;, đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; lim

â&#x2020;&#x2019;â&#x2C6;&#x17E; â&#x2C6;Ť

â&#x2C6;Ť

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  =

â&#x2C6;&#x17E;, đ?&#x2018;¤â&#x201E;&#x17D;đ?&#x2018;&#x2019;đ?&#x2018;&#x;đ?&#x2018;&#x2019; đ??ť đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x17D; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018; đ?&#x2018;&#x2013;đ?&#x2018;Ąđ?&#x2018;&#x2013;đ?&#x2018;Łđ?&#x2018;&#x2019; đ?&#x2018;?đ?&#x2018;&#x153;đ?&#x2018;&#x203A;đ?&#x2018; đ?&#x2018;Ąđ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;Ą}.

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  = The trivial solution of (1) is said to be globally mean square asymptotically stable if for any given initial condition đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľâ&#x201E;ą [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? such that

For đ?&#x153;&#x2018; â&#x2C6;&#x2C6; đ??ľ[đ?&#x2022; , â&#x201E;? ] đ?&#x2018;&#x153;đ?&#x2018;&#x; đ?&#x153;&#x2018; â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľ[đ?&#x2022; , â&#x201E;? ], we define[đ?&#x153;&#x2018;(đ?&#x2018;Ą)] = ([đ?&#x153;&#x2018; (đ?&#x2018;Ą)] , â&#x20AC;Ś , [đ?&#x153;&#x2018; (đ?&#x2018;Ą)] ) , [đ?&#x153;&#x2018; (đ?&#x2018;Ą)] = sup {đ?&#x153;&#x2018; (đ?&#x2018;Ą + đ?&#x2018; )}, đ?&#x2018;&#x2013; â&#x2C6;&#x2C6; đ?&#x2019;Š, And đ??ˇ đ?&#x153;&#x2018;(đ?&#x2018;Ą) denotes the upper right-hand derivative of đ?&#x153;&#x2018;(đ?&#x2018;Ą) at time t. Let (Ί, â&#x201E;ą, {â&#x201E;ą } , đ?&#x2018;&#x192;) be a complete probability space with a filtration {â&#x201E;ą } satisfying the usual conditions ( i.e., it is right continuous and â&#x201E;ą contains all p-null sets ). Let đ?&#x2018;&#x192;đ??ľâ&#x201E;ą [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? (đ?&#x2018;&#x192;đ??ľâ&#x201E;ą [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? ) denote the family of all bounded â&#x201E;ą (â&#x201E;ą )-measurable, đ?&#x2018;&#x192;đ??ľ [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? -valued random variable đ?&#x153;&#x2122;, satisfying â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013; = sup đ?&#x201D;ź|đ?&#x153;&#x2122;(đ?&#x153;&#x192;)| < â&#x2C6;&#x17E;, where đ?&#x201D;ź denotes the expectation of stochastic process. Let â&#x201E;&#x2019; denote the well-known â&#x201E;&#x2019;-operator given by the Ito formula. In this paper, we consider the following Ito impetuous stochastic delay systems: â&#x17D;§ đ?&#x2018;&#x2018;đ?&#x2018;Ľ(đ?&#x2018;Ą) = đ??´(đ?&#x2018;Ą)đ?&#x2018;Ľ(đ?&#x2018;Ą) + đ?&#x2018;&#x201C; đ?&#x2018;Ą, đ?&#x2018;Ľ đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x153;?(đ?&#x2018;Ą) đ?&#x2018;&#x2018;đ?&#x2018;Ą + â&#x17D;Ş đ?&#x153;&#x17D; đ?&#x2018;Ą, đ?&#x2018;Ľ(đ?&#x2018;Ą), đ?&#x2018;Ľ đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x153;?(đ?&#x2018;Ą) đ?&#x2018;&#x2018;đ?&#x2018;¤(đ?&#x2018;Ą), đ?&#x2018;Ą â&#x2030;  đ?&#x2018;Ą , đ?&#x2018;Ą â&#x2030;Ľ đ?&#x2018;Ą ,   â&#x17D;¨â&#x2C6;&#x2020;đ?&#x2018;Ľ = đ?&#x2018;Ľ(đ?&#x2018;Ą ) â&#x2C6;&#x2019; đ?&#x2018;Ľ(đ?&#x2018;Ą ) = đ??ź đ?&#x2018;Ľ(đ?&#x2018;Ą ), â&#x17D;Ş â&#x17D;Šđ?&#x2018;Ľ(đ?&#x2018;Ą + đ?&#x2018; ) = đ?&#x153;&#x2122;(đ?&#x2018; ), đ?&#x2018;  â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;đ?&#x153;?, 0],

lim đ?&#x201D;ź|đ?&#x2018;Ľ(đ?&#x2018;Ą, đ?&#x2018;Ą , đ?&#x153;&#x2122;)| = 0. â&#x2020;&#x2019;â&#x2C6;&#x17E;

Definition 2.2 The trivial solution of (1) is said to be globally mean square exponentially stable if there exist constants đ?&#x153;&#x2020; > 0 and đ?&#x2018;&#x2DC; > 1 such that for any solution x(t) with the initial condition đ?&#x153;&#x2122; â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľâ&#x201E;ą [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? , đ?&#x201D;ź|đ?&#x2018;Ľ(đ?&#x2018;Ą)| â&#x2030;¤ đ?&#x2018;&#x2DC;â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013; đ?&#x2018;&#x2019;

(

)

, đ?&#x2018;Ąâ&#x2030;Ľđ?&#x2018;Ą .

3. Main results In this section, we will first projecting an impetuous differential inequality with delays and then investigate the mean square stability of (1) by employing the obtained impetuous differential inequality, and stochastic analysis technique. It is shown that an unstable stochastic delays system can be successfully stabilized by impetuous. Theorem 3.1 Let đ?&#x2018;&#x17E; â&#x2030;Ľ 0, and đ?&#x2018;&#x2039;(đ?&#x2018;Ą), đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľ[[đ?&#x2018;Ą , â&#x2C6;&#x17E;), â&#x201E;?] be a solution of the following impetuous delay differential inequality with the initial condition đ?&#x2018;&#x2039;(đ?&#x2018;Ą), đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľ, â&#x2C6;&#x2019;đ?&#x153;? â&#x2030;¤ đ?&#x2018; â&#x2030;¤ đ?&#x2018;Ą ,

(1) Where

the

đ?&#x153;&#x2122;(đ?&#x2018; ) = đ?&#x153;&#x2122; (đ?&#x2018; ), â&#x20AC;Ś , đ?&#x153;&#x2122; (đ?&#x2018; )

initial

function

â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľâ&#x201E;ą [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? .

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume â&#x20AC;&#x201C; 2 | Issue â&#x20AC;&#x201C; 3 | Mar-Apr 2018

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International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 đ??ˇ đ?&#x2018;&#x2039;(đ?&#x2018;Ą) â&#x2030;¤ â&#x201E;&#x17D;(đ?&#x2018;Ą)[â&#x2C6;&#x2019;đ?&#x2018;&#x192;đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x17E;[đ?&#x2018;&#x2039;(đ?&#x2018;Ą)] ], đ?&#x2018;Ą â&#x2030; đ?&#x2018;Ą , đ?&#x2018;Ą â&#x2030;Ľ đ?&#x2018;Ą   đ?&#x2018;&#x2039;(đ?&#x2018;Ą ) â&#x2030;¤ đ?&#x203A;ź đ?&#x2018;&#x2039;(đ?&#x2018;ĄĚ&#x2026; ), 0<đ?&#x203A;ź < 1, đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, đ?&#x2018;&#x2039;(đ?&#x2018;Ą + đ?&#x2018; ) = đ?&#x153;&#x2122;(đ?&#x2018; ) â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľ [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? â&#x2C6;&#x2019;đ?&#x153;? â&#x2030;¤đ?&#x2018;  â&#x2030;¤0 +

â&#x2C6;&#x2019;2đ?&#x2018;? + đ?&#x203A;žđ?&#x2018;&#x17E;đ?&#x2018;&#x2019; â&#x201E;&#x2022;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;.

+ đ?&#x203A;żđ?&#x2018;&#x17E;đ?&#x2018;&#x2019;

< â&#x2C6;&#x2019;đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź (7)

â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź

,đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6;

And

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  < đ??ˇ đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤ â&#x201E;&#x17D;(đ?&#x2018;Ą)[â&#x2C6;&#x2019;đ?&#x2018;&#x192;đ?&#x2018;&#x152;(đ?&#x2018;Ą) + đ?&#x2018;&#x17E;[đ?&#x2018;&#x152;(đ?&#x2018;Ą)] ], đ?&#x2018;Ą â&#x2030;  đ?&#x2018;Ą , đ?&#x2018;Ą â&#x2030;Ľ đ?&#x2018;Ą (đ?&#x153;&#x201A; + đ?&#x153;&#x2020;) â&#x2C6;Ť â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  + (đ?&#x153;&#x152; + đ?&#x153;&#x2021;) â&#x2C6;Ť   â&#x2C6;&#x2019;đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź â&#x2030;¤ đ??źđ?&#x2018;&#x203A;đ?&#x203A;ž + đ??źđ?&#x2018;&#x203A;đ?&#x203A;ż (8) đ?&#x2018;&#x152;(đ?&#x2018;Ą ) â&#x2030;¤ đ?&#x203A;ź đ?&#x2018;&#x152;(đ?&#x2018;ĄĚ&#x2026; ), 0<đ?&#x203A;ź < 1, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, đ?&#x2018;&#x152;(đ?&#x2018;Ą + đ?&#x2018; ) = đ?&#x153;&#x2122;(đ?&#x2018; ) â&#x2C6;&#x2C6; đ?&#x2018;&#x192;đ??ľ [â&#x2C6;&#x2019;đ?&#x153;?, 0], â&#x201E;? â&#x2C6;&#x2019; đ?&#x153;? â&#x2030;¤ đ?&#x2018;  â&#x2030;¤ 0 From (8) we can choose đ?&#x2018;&#x20AC; â&#x2030;Ľ 1, đ??ş â&#x2030;Ľ 1 such that (2) Where â&#x201E;&#x17D;(đ?&#x2018;Ą) â&#x2C6;&#x2C6; â&#x201E;&#x2039;. Assume that the following condition is satisfied â&#x2C6;&#x2019;đ?&#x2018;? +

â&#x2C6;Ť

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  +

â&#x2C6;&#x2019;đ?&#x2018;? +

â&#x2C6;Ť

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  < â&#x2C6;&#x2019;đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź

đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;.

1<đ?&#x2018;&#x2019;

(

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

đ?&#x203A;žđ?&#x2018;&#x2019;

(

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

đ?&#x203A;żđ?&#x2018;&#x2019;

(

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

+đ?&#x2018;&#x2019; đ?&#x2018;&#x2019;

â&#x201E;&#x2022;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;.

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019;

, đ?&#x2018;Ąâ&#x2030;Ľđ?&#x2018;Ą

â&#x2C6;Ť

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  + 2đ?&#x153;&#x2021; â&#x2C6;&#x2019; đ?&#x2018;? +

â&#x201E;&#x17D;(đ?&#x2018; )đ?&#x2018;&#x2018;đ?&#x2018;  < â&#x2C6;&#x2019;đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź

â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź

, đ?&#x2018;&#x2DC;â&#x2C6;&#x2C6;

(5)

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

+ . (9)

â&#x201E;&#x17D;( )

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x2030;¤ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

â&#x201E;&#x17D;( )

(10)

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; [đ?&#x2018;Ą , đ?&#x2018;Ą ).

â&#x2C6;Ť â&#x201E;&#x17D;( )

â&#x201E;&#x17D;( )

â&#x201E;&#x17D;( )

â&#x2C6;Ť â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; â&#x2C6;Ť + â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019; [đ?&#x2018;Ą , đ?&#x2018;Ą ), đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022; & đ?&#x2018;Ą â&#x2C6;&#x2C6; [đ?&#x2018;Ą , đ?&#x2018;Ą ), đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; (6) â&#x201E;&#x2022;.

,đ?&#x2018;Ą â&#x2C6;&#x2C6;

By using the continuity, from (3) we know that there must there must exist a positive constants đ?&#x153;&#x2020; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x153;&#x2021; such that for all (5) holds. â&#x2C6;&#x2C6;â&#x201E;&#x2022; {

â&#x2C6;Ť â&#x201E;&#x17D;( )

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019;

, đ?&#x2018;Ąâ&#x2C6;&#x2C6;

(11)

To do this, we only to prove that đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

(12)

} â&#x2030;Ľ 1 and đ?&#x203A;ż =

} â&#x2030;Ľ 1 in (5), then we can select constant

đ?&#x153;&#x201A; > 0, đ?&#x153;&#x152; > 0 such that for all đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;, đ?&#x2018;&#x203A; â&#x2C6;&#x2C6; â&#x201E;&#x2022;.

â&#x201E;&#x17D;( )

â&#x2C6;Ť

đ?&#x2018;&#x2039;(đ?&#x2018;Ą)Ě&#x2026; + đ?&#x2018;&#x152;(đ?&#x2018;Ą)Ě&#x2026; > â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;

đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤

Let đ?&#x203A;ž = đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?

đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤

If (12) is not true, by (10), then there exists some đ?&#x2018;ĄĚ&#x2026; â&#x2C6;&#x2C6; (đ?&#x2018;Ą , đ?&#x2018;Ą ) such that

We shall show that

â&#x2C6;&#x2C6;â&#x201E;&#x2022; {

(

We first prove that â&#x2C6;Ť â&#x201E;&#x17D;( )

Proof

đ?&#x2018; đ?&#x2018;˘đ?&#x2018;?

â&#x201E;&#x17D;( )

â&#x2C6;Ť

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013; < â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019;

|đ?&#x153;&#x2122;(đ?&#x2018; )|, and the constants Where â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013; = sup đ?&#x153;&#x2020; > 0, đ?&#x153;&#x2021; > 0 satisfies the following inequality

â&#x2C6;Ť

)â&#x2C6;Ť

â&#x2030;¤đ?&#x2018;&#x20AC;+đ??ş â&#x2030;¤

,

đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤

2đ?&#x153;&#x2020; â&#x2C6;&#x2019; đ?&#x2018;? +

(

đ?&#x2018;&#x2019;

Then there exist a constant đ?&#x2018;&#x20AC; â&#x2030;Ľ 1, đ??ş â&#x2030;Ľ 1 and a small enough number đ?&#x153;&#x2020; > 0, đ?&#x153;&#x2021; > 0 such that

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; (4)

â&#x201E;&#x17D;( )

)â&#x2C6;Ť

It then follows that â&#x2C6;&#x2019; đ??źđ?&#x2018;&#x203A;đ?&#x203A;ź (3)

â&#x2C6;Ť â&#x201E;&#x17D;( )

(

â&#x2C6;Ť

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019; â&#x2C6;Ť

â&#x201E;&#x17D;( )

â&#x201E;&#x17D;( )

â&#x2C6;Ť

â&#x201E;&#x17D;( )

â&#x2030;Ľ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019; + â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019; > â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013; â&#x2030;Ľ đ?&#x2018;˘(đ?&#x2018;Ą + đ?&#x2018; ), đ?&#x2018;  â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;đ?&#x153;?, 0] (13)

Which implies that there exists some đ?&#x2018;Ą â&#x2C6;&#x2014; â&#x2C6;&#x2C6; (đ?&#x2018;Ą , đ?&#x2018;ĄĚ&#x2026;) such that đ?&#x2018;&#x2039;(đ?&#x2018;Ą â&#x2C6;&#x2014; ) + đ?&#x2018;&#x152;(đ?&#x2018;Ą â&#x2C6;&#x2014; ) = â&#x201E;&#x17D;( )

â&#x201E;&#x17D;( )

â&#x2C6;Ť â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; â&#x2C6;Ť + â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019; , â&#x2C6;&#x2014;) â&#x2C6;&#x2014; ), đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą) â&#x2030;¤ đ?&#x2018;&#x2039;(đ?&#x2018;Ą + đ?&#x2018;&#x152;(đ?&#x2018;Ą đ?&#x2018;Ą â&#x2C6;&#x2C6; [đ?&#x2018;Ą â&#x2C6;&#x2019; đ?&#x153;?, đ?&#x2018;Ą â&#x2C6;&#x2014; ] (14)

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume â&#x20AC;&#x201C; 2 | Issue â&#x20AC;&#x201C; 3 | Mar-Apr 2018

Page: 629


International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 And there exists 𝑡 ∗∗ ∈ [𝑡 , 𝑡 ∗ ) such that 𝑋(𝑡 ∗∗ ) + 𝑌(𝑡 ∗∗ ) = ‖𝜙‖, 𝑎𝑛𝑑 𝑋(𝑡 ∗∗ ) + 𝑌(𝑡 ∗∗ ) ≤ 𝑋(𝑡) + 𝑌(𝑡) ≤ 𝑋(𝑡 ∗ ) + 𝑌(𝑡 ∗ ), 𝑡 ∈ [𝑡 ∗∗ , 𝑡 ∗ ]. (15)

For the reasons of contradiction, suppose (20) is not true. Then we define 𝑡̅ = inf{𝑡 ∈ [𝑡 , 𝑡 ‖𝜙‖𝑀𝑒

Hence, we get for any 𝑠 ∈ [−𝜏, 0],

)|𝑋(𝑡) + 𝑌(𝑡) >

∫ ℎ( )

+ ‖𝜙‖𝐺𝑒

∫ ℎ( )

(21)

𝑋(𝑡 + 𝑠) + 𝑌(𝑡 + 𝑠) ≤ ‖𝜙‖𝑀𝑒

ℎ( )

≤ ‖𝜙‖𝛾𝑒

(

)∫

ℎ( )

(

)∫

ℎ( )

+ ‖𝜙‖𝛿𝑒 = 𝛾𝑒 𝑋(𝑡 ∗∗ ) + 𝛿𝑒 𝑡 ∈ [𝑡 ∗∗ , 𝑡 ∗ ].

𝑒

ℎ( )

+ ‖𝜙‖𝐺𝑒 (

)∫

ℎ( )

(

)∫

ℎ( )

𝑒 𝑌 ≤ 𝛾𝑒

𝑋(𝑡) + 𝛿𝑒 (16)

By (8) and (19), we have 𝑋(𝑡 ) + 𝑌(𝑡 ) 𝑌(𝑡),

𝐷 𝑋(𝑡) ≤ ℎ(𝑡)[−𝑝𝑋(𝑡) + 𝑞[𝑋(𝑡)] ] ≤ ℎ(𝑡) −𝑝 + 𝛾𝑞𝑒 ]𝑋(𝑡) ≤ ℎ(𝑡)(𝜂 − 𝜆)𝑋(𝑡), 𝑡 ∈ [𝑡 ∗∗ , 𝑡 ∗ ] + 𝐷 𝑌(𝑡) ≤ ℎ(𝑡)[−𝑝𝑌(𝑡) + 𝑞[𝑌(𝑡)] ] ≤ ℎ(𝑡)[−𝑝 + 𝛿𝑞𝑒 ]𝑌(𝑡) ≤ ℎ(𝑡)(𝜌 − 𝜇)𝑌(𝑡), 𝑡 ∈ [𝑡 ∗∗ , 𝑡 ∗ ] (17)

𝑌(𝑡 ∗∗ )𝑒 ( ‖𝜙‖𝑒

‖𝜙‖𝐺𝑒

ℎ( ) ∫

)

≤ ‖𝜙‖𝑀𝑒

ℎ( )

+

ℎ( )

ℎ( )

+

+

= 𝑋(𝑡 ∗ ) + 𝑌(𝑡 ∗ )

Which is contradiction. Hence (12) holds and then (6) is true for 𝑘 = 1. Now we assume that (6) holds for 𝑘 = 1,2, … , 𝑚(𝑚 ∈ ℕ, 𝑚 ≥ 1), 𝑋(𝑡) + 𝑌(𝑡) ≤ ‖𝜙‖𝑀𝑒 [𝑡 , 𝑡 ),

+ ‖𝜙‖𝐺𝑒

+𝛼 𝑒

ℎ( )

∫ ℎ( )

,𝑡 ∈

𝑘 = 1,2, … , 𝑚.

∫ ℎ( )

< ‖𝜙‖𝑀𝑒

∫ ℎ( )

‖𝜙‖𝑀𝑒 [𝑡 , 𝑡 ).

+ ‖𝜙‖𝐺𝑒

,

(22)

And so 𝑡̅ ≠ 𝑡 . By utilizing the continuity of ), we get 𝑋(𝑡) + 𝑌(𝑡) in the interval [𝑡 , 𝑡 𝑋(𝑡)̅ + 𝑌(𝑡)̅ = ℎ( )

, 𝑎𝑛𝑑 𝑋(𝑡) + (23)

From (22), we know that there exist some 𝑡 ∗ ∈ (𝑡 , 𝑡̅) such that

𝛼 𝑒

ℎ( )

𝛼 𝑒

ℎ( )

(20)

∫ ℎ( )

‖𝜙‖𝐺𝑒

+ ,

On the other hand, for any 𝑡 ∈ [𝑡 ∗ , 𝑡]̅ , 𝑠 ∈ [−𝜏, 0], either 𝑡 + 𝑠 ∈ [𝑡 − 𝜏, 𝑡 )𝑜𝑟 𝑡 + 𝑠 ∈ [𝑡 , 𝑡̅]. Two cases will be discussed as follows. If 𝑡 + 𝑠 ∈ [𝑡 − 𝜏, 𝑡 ), from (19),we obtain 𝑋(𝑡 + 𝑠) + ‖𝜙‖𝐺𝑒

, 𝑡∈

∫ ℎ( )

‖𝜙‖𝑀𝑒

𝑎𝑛𝑑 𝑋(𝑡 ∗ ) + 𝑌(𝑡 ∗ ) ≤ 𝑋(𝑡) + 𝑌(𝑡) ≤ 𝑋(𝑡)̅ + 𝑌(𝑡)̅ , 𝑡 ∈ [𝑡 , 𝑡̅]. (24)

∫ ℎ( )

Next, To find equation (6) holds for 𝑘 = 𝑚 + 1, (𝑡) + 𝑌(𝑡) ≤ ∫ ℎ( )

∫ ℎ( )

‖𝜙‖𝐺𝑒

∫ ℎ( )

𝑌(𝑡 + 𝑠) ≤ ‖𝜙‖𝑀𝑒

(19)

∫ ℎ( )

∫ ℎ( )

‖𝜙‖𝑀𝑒

𝑋(𝑡 ∗ ) + 𝑌(𝑡 ∗ ) =

(18)

∫ ℎ( )

ℎ( )

∫ ‖𝜙‖𝑀𝑒 ∫ + ‖𝜙‖𝐺𝑒 𝑌(𝑡) ≤ 𝑋(𝑡)̅ + 𝑌(𝑡)̅ , 𝑡 ∈ [𝑡 , 𝑡̅].

) ∫ ∗∗ ℎ( )

< ‖𝜙‖𝑒

ℎ( )

+ 𝛼 ‖𝜙‖𝐺𝑒

ℎ( )

It follows from (10),(14) and (15) that 𝑋(𝑡 ∗ ) + 𝑌(𝑡 ∗ ) ≤ 𝑋(𝑡 ∗∗ )𝑒 (

<𝛼 𝑒

+ ‖𝜙‖𝐺𝑒

ℎ( )

≤ 𝛼 ‖𝜙‖𝑀𝑒

Thus, by (7) and (16), we have

∗ ) ∫ ∗∗ ℎ(

}.

ℎ( )

𝑒

‖𝜙‖𝑀𝑒 ‖𝜙‖𝐺𝑒

∫ ℎ( )

≤ ‖𝜙‖𝑒

ℎ( )

𝑒

+

∫ ℎ( )

𝑒

+

𝑒 𝑒

ℎ( )

𝑀𝑒

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 3 | Mar-Apr 2018

∫ ℎ( )

+ Page: 630


International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019;

â&#x201E;&#x17D;( )

â&#x2C6;Ť

đ?&#x2018;&#x2019;

đ??şđ?&#x2018;&#x2019;

â&#x2C6;Ť â&#x201E;&#x17D;( )

(25)

if đ?&#x2018;Ą + đ?&#x2018; â&#x2C6;&#x2C6; [đ?&#x2018;Ą , đ?&#x2018;ĄĚ&#x2026;], form (23), then đ?&#x2018;&#x2039;(đ?&#x2018;Ą + đ?&#x2018; ) + đ?&#x2018;&#x152;(đ?&#x2018;Ą + đ?&#x2018; ) = â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; â&#x2C6;Ť â&#x201E;&#x17D;( )

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019; đ?&#x2018;&#x2019;

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019;

đ?&#x2018;&#x2019;

â&#x201E;&#x17D;( )

â&#x2C6;Ť

+

â&#x2030;¤ â&#x201E;&#x17D;( )

â&#x2C6;Ť

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x2019;

â&#x2C6;Ť â&#x201E;&#x17D;( )

đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019; đ??şđ?&#x2018;&#x2019;

â&#x2C6;Ť â&#x201E;&#x17D;( )

+

â&#x2C6;Ť â&#x201E;&#x17D;( )

(26)

In any case, from (24)-(26), we all have for any đ?&#x2018; â&#x2C6;&#x2C6; [â&#x2C6;&#x2019;đ?&#x153;?, 0], đ?&#x2018;?[đ?&#x2018;&#x2039;(đ?&#x2018;Ą + đ?&#x2018; ) + đ?&#x2018;&#x152;(đ?&#x2018;Ą + đ?&#x2018; )] â&#x2030;¤ đ?&#x203A;žđ?&#x2018;&#x2019; đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x203A;żđ?&#x2018;&#x2019; (27)

đ?&#x2018;&#x2039;(đ?&#x2018;Ą â&#x2C6;&#x2014; ) +

đ?&#x2018;&#x152;(đ?&#x2018;Ą â&#x2C6;&#x2014; ) â&#x2030;¤

đ?&#x2018;&#x152;(đ?&#x2018;Ą), đ?&#x2018;Ą â&#x2C6;&#x2C6; [đ?&#x2018;Ą â&#x2C6;&#x2014; , đ?&#x2018;ĄĚ&#x2026;] ,

Finally, by (7) and (27), we all have đ??ˇ [đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x2018;&#x152;(đ?&#x2018;Ą)] â&#x2030;¤ â&#x201E;&#x17D;(đ?&#x2018;Ą)[(đ?&#x153;&#x201A; â&#x2C6;&#x2019; đ?&#x153;&#x2020;)đ?&#x2018;&#x2039;(đ?&#x2018;Ą) + đ?&#x153;&#x152;â&#x2C6;&#x2019;đ?&#x153;&#x2021;đ?&#x2018;&#x152;đ?&#x2018;Ą. (28) It follows from (8), (23) and (24) that đ?&#x2018;&#x2039;(đ?&#x2018;Ą)Ě&#x2026; + đ?&#x2018;&#x152;(đ?&#x2018;ĄĚ&#x2026;) â&#x2030;¤ đ?&#x2018;&#x2039;(đ?&#x2018;Ą â&#x2C6;&#x2014; )đ?&#x2018;&#x2019; (

) â&#x2C6;Ť â&#x2C6;&#x2014; â&#x201E;&#x17D;( )

+ đ?&#x2018;&#x152;(đ?&#x2018;Ą â&#x2C6;&#x2014; )đ?&#x2018;&#x2019; (

) â&#x2C6;Ť â&#x2C6;&#x2014; â&#x201E;&#x17D;( )

(

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

(

)â&#x2C6;Ť

â&#x201E;&#x17D;( )

<đ?&#x2018;&#x2019; +đ?&#x2018;&#x2019;

< â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;

â&#x2C6;Ť â&#x201E;&#x17D;( )

đ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

đ?&#x2018;&#x2019;

â&#x2C6;Ť

â&#x201E;&#x17D;( )

+ â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??şđ?&#x2018;&#x2019;

= đ?&#x2018;&#x2039;(đ?&#x2018;Ą)Ě&#x2026; + đ?&#x2018;&#x152;(đ?&#x2018;ĄĚ&#x2026;)

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ?&#x2018;&#x20AC;đ?&#x2018;&#x2019;

â&#x2C6;Ť â&#x201E;&#x17D;( )

â&#x20AC;&#x2013;đ?&#x153;&#x2122;â&#x20AC;&#x2013;đ??ş â&#x2C6;Ť â&#x201E;&#x17D;( )

(29)

Which is a contradiction. This implies that the assumption is not true, and hence (6) holds for đ?&#x2018;&#x2DC; = đ?&#x2018;&#x161; + 1. Therefore, by some simple mathematical induction, we can obtain that (6) holds for any đ?&#x2018;&#x2DC; â&#x2C6;&#x2C6; â&#x201E;&#x2022;. This completes the proof of theorem. References 1) Z.G. Yang, D.Y.Xu, L.Xiang, Exponential pstability of impulsive stochastic differential equations with delays, Phys. Lett. A 359 (2006) 129-137. 2) L.G.Xu, D.Y.Xu, p-attracting and p-invariant sets for a class of impulsive stochastic functional differential equations, comput. Math. Appl. 57 (2009) 54-61.

3) L.G.Xu, D.Y.Xu, Exponential p-stability of impulsive stochastic neural networks with mixed delays, Chaos Solitons Fractals 41(2009) 263272. 4) L.G.Xu, Exponential p-stability of singularly perturbed impulsive stochastic delay differential systems, Int. J. Control Autom. 9 (2011) 966-972. 5) L.G.Xu, D.H.He, Mean square exponential stability analysis of impulsive stochastic switched systems with mixed delays, comput. Math. Appl. (26) 62 (2011) 109-117. 6) J.Liu, X.Z.Liu, W.C.Xie, Impulsive stabilization of stochastic functional differential equations, Appl. Math. Lett. 24 (2011) 264-269. 7) J.Zhou, Q.J.Wu, L.Xiang, et al., Impulsive synchronization seeking in general complex delayed dynamical networks, Nonlinear Anal. Hybrid Sys. 5 (2011) 513-524. 8) Z.G. Yang, Z.C. Yang, Dissipativity in mean square of non- autonomous impulsive stochastic neural networks with delays, Lecture Notes in Comput. Sci. 6063 (2010) 735-744. 9) B.Liu, X.Z.Liu, X.X.Liao, Existence and uniqueness and stability of solutions for stochastic impulsive systems, J.Syst. Sci. Complex. 20 (2007) 149-158. 10) M.S.Alwan, X.Z.Liu, W.Xie, Existence, continuation and uniqueness problems of stochastic impulsive systems with time delay, J.Franklin Inst. 347 (2010) 1317-1333. 11) X.Z.Liu, A.R.Willms, impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Math. Probl. Eng. 2 (1996) 277-299. 12) D.Bainov, P.S.Simeonov, Systems with impulse Effect: Stability, Theory and Applications, Ellis Horwood Limited, Chichester, 1989. 13) X.Z.Liu, K.Rohlf, Impulsive control of a LotkaVolterra system, IMAJ. Math. Control Inform. 15 (1998) 269-284. 14) V.Lakshmikantham, D.D.Bainov, P.S.Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. 15) A. Khadra, X.Z.Liu, X.M.Shen, Application of impulsive synchronization to communication security, IEEE Trans. Circuits Syst. I 50 (2003) 341-351. 16) J.T.Sun, Impulsive control of a new chaotic system, Math. Comput. Simul. 64 (2004) 669-677.

@ IJTSRD | Available Online @ www.ijtsrd.com | Volume â&#x20AC;&#x201C; 2 | Issue â&#x20AC;&#x201C; 3 | Mar-Apr 2018

Page: 631

Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization  

This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed....

Time Delay and Mean Square Stochastic Differential Equations in Impetuous Stabilization  

This paper specially exhibits about the time delay and mean square stochastic differential equations in impetuous stabilization is analyzed....

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