Stability in impulsive systems with markov perturbations in averaging scheme. 3. Weak convergence of solutions of impuls
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STABILITY IN IMPULSIVE SYSTEMS WITH MARKOV PERTURBATIONS IN AVERAGING SCHEME. 3. WEAK CONVERGENCE OF SOLUTIONS OF IMPULSIVE SYSTEMS1 Ye. F. Tsarkov,a V. K. Yasynsky,b† and I. V. Malykb‡
UDC 519.217; 519.718
Abstract. For a stochastic dynamic system with a small parameter, the uniform boundedness of the p-th moment of the solution (p > 1), the weak convergence of the solution of the system to the solution of Ito stochastic differential equation, and the weak convergence of normalized deviations are proved. The stability of linear systems with a small parameter and Markov perturbations is analyzed. Keywords: impulsive dynamic Markov system, Markov process, averaging method, weak convergence. 1. PROBLEM STATEMENT Let a right-continuous random process x( t ) º x( t , w ) Î R m be specified on a probabilistic basis (W , F , F , P ), F º {F t Ì F , t ³ 0}. For all t Î ( t j -1 , t j ), j Î N, the process satisfies a differential equation (DE) W dx = e f ( x, y( t ), e ), dt and for all t Î {t j º t j ( w ), j Î N } it satisfies a jump condition x( t ) = x( t - ) + eg ( x( t - ), y( t - ), e )
(1)
(2)
with the initial condition x( t )
t=0
= x,
(3)
where y( t ) is a Markov process [7] specified on the phase space and determining jumps in real systems [13]. See [12] for the analysis of deterministic systems with other jump conditions. Assume that the functions f and g are representable as f ( x, y, e ) = f1 ( x, y ) + e f 2 ( x, y ) + e f 3 ( x, y, e ),
(4)
g ( x, y, e ) = g 1 ( x, y ) + e g 2 ( x, y ) + e g 3 ( x, y, e ),
(5)
where f1 and g 1 are continuous functions having the first and second Frechet derivatives [4] with respect to x; f 2 , f 3 , g 2 , and g 3 are continuous functions having continuous bounded Frechet derivative with respect to x, and for all y Î Y and e Î ( 0, 1) we can write (6) || D1 f 3 ( x, y, e ) || + || D1 g 3 ( x, y, e ) || £ b( e ) , where lim b( e ) = 0. Hereinafter, D k is the kth Frechet derivative with respect to x Î R m [4]. e® 0
1
Continued from Cybernetics and Systems Analysis, No. 6 (2010), No. 1 (2011). a
Riga Technical University, Riga, Latvia, [email protected]. bYu. Fed’kovych National University, Chernivtsi, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 127–145, May–June 2011. Original article submitted May 14, 2009. 442
1060-0396/11/4703-0442
©
2011 Springer Science+Business Media, Inc.
Let us consider a DE W
du = b1 ( u ) , dt
(7)
and call it averaged equation for the impulsive system (1)–(3). It can be easily seen that by virtue of assumptions (4)–(6), the solution of DE (7) exists and is unique for any initial condition u( 0) = u . If b1 ( u ) º 0, then the averaged equation (7) carries no information on the concept of the solution of system (1)–(3) on a time interval of order e -1 . In this case, following [15, 22], it is possible to pass to “very slow time” e -2 t. By virtue of Lemma 3 from [17], we may assume that the terms f 3 ( x, y, e ) and g 3 ( x, y, e ) do not influence the asymptotics
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