Stability in the First Approximation of Random-Structure Diffusion Systems with Aftereffect and External Markov Switchin
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STABILITY IN THE FIRST APPROXIMATION OF RANDOM-STRUCTURE DIFFUSION SYSTEMS WITH AFTEREFFECT AND EXTERNAL MARKOV SWITCHINGS UDC 519.217; 519.718; 519.837
V. K. Yasinsky
Abstract. The second Lyapunov–Krasovskii method is used to derive the sufficient conditions for the stability and global asymptotic stability in different interpretations for the Ito random-structure stochastic diffusion dynamic system with finite aftereffect with Markov switchings. Keywords: dynamic system of random structure, system with aftereffect, Markov switchings. INTRODUCTION Analyzing the mathematical model of control of dynamic systems both deterministic (nonrandom) and stochastic (random) is extremely important in the modern mathematical science. In turn, developing the qualitative theory is impossible without a fundamental stability analysis of such controlled systems. The qualitative theory of stability became a real tool in answering the question about the stability or instability of the solutions of the system under study, asymptotic stability or instability, and analysis of the problem of global stability. This is discussed in the monographs by Lyapunov, Krasovskii, Malkin [8], Merkin [9], and many others [1–3, 14, 19–25]. The Influence of Markov perturbations on the stability of an impulse dynamic system is considered in the monographs by Korolyuk [22], Skorokhod [12, 13], Korenevskii [15], Khas’minskii [10], Kats [11], Tsar’kov [4, 6, 16], Shurenkov [24], and many others. In the present paper, we will analyze the first-approximation stability of stochastic diffusion dynamic systems where the original system is a linear stochastic dynamic system. We will use the second Lyapunov method in the analysis. 1. PROBLEM STATEMENT On a probability basis (W , Á, F, P ) , F º {Á t , t ³ 0}, Á t Ì Á, consider a diffusion system of random structure (DSRS) with aftereffect dx( t ) = [ a 0 ( t , x t , x ( t )) + a1 ( t , x t , x ( t ))]dt + b0 ( t , x t , x ( t ))dw 0 ( t ) + b1 ( t , x t , x ( t ))dw1 ( t ) ,
(1)
Dx( t )|tk = g 0 ( t k , x tk - , x ( t k - ), h k ) + g 1 ( t k , x tk - , x ( t k - ), h k ) ,
(2)
with external Markov switchings
and with the initial conditions x t0 = j Î Dm , x( t 0 ) = y, h k0 = h,
(3)
where x (t ) º x (t , w): [ 0, ¥ ) ´ W ® Rm; xt º {x (t + q ), q Î [ - t , 0 ]} Î Dm , t > 0; ai : R+ ´ Dm ´ Y ® Rm; bi : R + ´ Dm ´ Y ® R m ; Yu. Fedkovych Chernivtsi National University, Chernivtsi, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2014, pp. 99–111. Original article submitted March 6, 2013. 248
1060-0396/14/5002-0248
©
2014 Springer Science+Business Media New York
and g i : R + ´ Dm ´ Y´ H ® R m , i = 0, 1 , are mappings measurable in their variables and satisfying the global Lipschitz property | a i ( t , x t(1) , y ) - a i ( t , x t( 2 ) , y )| + | bi ( t , x t(1) , y ) - bi ( t , x t( 2 ) , y )| + | g i ( t , x t(1) , y, h ) - g i ( t , x t( 2 ) , y, h )| £ L || x t(1) - x t( 2 ) ||
(4)
and the uniform boundedness condition with respect to t | a i ( t ,
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