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ISSN 2075-9827 http://www.journals.pu.if.ua/index.php/cmp

Carpathian Math. Publ. 2014, 6 (1), 96103 i . . 2014, .6, 1, .96103

doi:10.15330/cmp.6.1.96-103

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ii i (, F, F {Ft F, t 0}, P) [4] i i-i i () (1) dx(t) = a(t, (t), x(t), x(t r))dt + b(t, (t), x(t), x(t r))dw(t), i ii x(tk ) = x(tk ) x(tk ) = g(tk, (tk ), k, x(tk )), (2) tk S {tn, n N}, 519.217; 519.718 2010 Mathematics Subject Classication: 60J10, 60J27, 60H10, 68U20.

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III. i x(t).

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P {x() [z, z + dz]/(t 0) = x} = pij (, z/x)dz + o(dz).

i, i (1), i i () (t) [5] (3) ( ii (2)).

, , (t) i i i, Y = {y1, y2,..., yk } ( II), i i ii (t), i i .

i ii t [ s,, ), (t) = yi Y, i, ii (1), t [ s, ) dx(t) = a(t, yi, x(t), x(t r))dt + b(t, yi, x(t), x(t r))dw(t), (7) x(t s) = x(), [ s r, s]; (t s) = yi.

i, (t 0) = yi () = y j = yi, ii i () = y j i (7) y j i yi.

, , i (1) .

ii ii i i i i i (1) (3).

1. i i (t) x(t) i iii 1, i i x( 0) = x().

2. 0 i i , i i ( 0) = yi () = y j = yi.

, x() = ij (x( 0)), i = j, ij C(Rm ), ij (0) = 0.

3. i i, i (1) yi y j i i

x() x(, ) Rm, R+, i (1):

P {x() [z, z + dz]/x( 0) = x} = pij (, z/x)dz + (dz), pij (, z/x) ii m-i i.

ii (2), i {k, k 0}, ii i (1) i i .

3 II II I

   

i tk S {t0 }, k N {0}, D, y Y, h H i C D, Y, G H, x(tk, y, h, ) (1), (2) (3).

1. (lvk )(y, h, ) ii i ii vk (y, h, ) : Y H D R1, k N {0}, (1) i ii (2) ii [3]

   

i i i ii, sup i , ii i- i ii i i i i i-I.

(5), (6),

   

ᒺ i i ii, i ii (9).

1. :

l) 0 |tk+1 tk |, k 0, 0;

2) i (5);

3) i i ii ii - vk (y, h, ) i ak (y, h, ), k 0, ii (1)(3) ii

   

i (1), (3) i ii (2) i i.

. Utk ii -, i ii (t) i t [t0, tk ] i n n k. i i [3]

   

i y Y, h H, D i T t0 0, , ii i (1)(3). .

i 1, .

2. :

1) 1), 2) 1;

2) ii (1)(3) ii ii - {vk, k 0} ii (lvk )(y, h, ) 0 k 0, y Y, h H, D.

i i (1)(3) i ii i.

   

[1] Basak G.K., Bisi A., Ghosh M.K. Stability of a random diffusion with linear drift. J. Math. Anal. Appl. 1996, 202 (2), 604622. doi:10.1006/jmaa.1996.0336 [2] Gikhman I.I., Skorokhod A.V. Stochastic Differential Equations and its Applications. Naukova dumka, Kyiv, 1982. (in Russian) [3] Dynkin E.B. Markov processes. Phyzmatgiz, Moscow, 1969. (in Russian) [4] Doob J. Stochastic processes. Phyzmatgiz, Moscow, 1963. (in Russian) [5] Katz I.Ya. Lyapunov functions method for the stability and stabilization problems of the systems with random structure. UGAPS, Ekaterinburg, 1998. (in Russian) [6] Mao X. Stability of stochastic differential equations with Markovian switching. Stochastic Process. Appl. 1999, 79 (1), 4567. doi:10.1016/S0304-4149(98)00070-2 [7] Mao X., Matasov A., Piunovskiy A.B. Stochastic differential delay equations with Markovian switching. Bernoulli 2000, 6 (1), 7390. doi:10.2307/3318634 [8] Mariton M. Jump Linear Systems in Automatic Control. Marcel-Dekker, New York, 1990.

[9] Lukashiv T.O., Yurchenko I.V., Yasynskyy V.K. Lyapunov functions method for the investigation of the stability of the stochastic Itos systems with random structure and impulse Markov switchings. I. General theorem on stability of the impulse stochastic systems. Cybernet. Systems Anal. 2009, 45 (2), 281290. doi:10.1007/s10559-009-9102-8 (translation of Kibernet. Sistem. Anal. 2009, 2, 135145. (in Russian)) [10] Sverdan M.L., Tsarkov Ye.F. Stability of the stochastic impulse systems. RTU, Riga, 1994. (in Russian) [11] Shaikhet L. Stability of stochastic hereditary systems with Markov switching. Theory Stoch. Process. 1996, 2, 180 184.

[12] Skorokhod A.V. Asymptotic methods of the theory of the stochastic differential equations. Naukova dumka, Kyiv, 1987. (in Russian) [13] Tsarkov Ye.F. Random disturbances of the functional differential equations. Zinatne, Riga, 1989. (in Russian) [14] Yasynskyy V.K., Yasynskyy Ye.V. Problems of the stability and stabilization of the dynamical systems with nite aftereffect. TViMS, Kyiv, 2005. (in Ukrainian) i 22.02.2014 Lukashiv T.O. Asymptotic stochastic stability of the stochastic dynamical systems of the random structure with constant delay. Carpathian Math. Publ. 2014, 6 (1), 96103.

The method of Lyapunov-Krasovskyy functionals is used for the researching of the asymptotic stochastic stability in whole of the stochastic diffusion dynamical systems of the random structure with constant delay, which is under the inuence of external impulse disturbances of the Markovs chain type.

Key words and phrases: Lyapunov-Krasovskyy functionals method, dynamical system of the random structure, external Markov switchings, asymptotic stochastic stability in the whole.

.. // . . 2014.

.6, 1. C. 96103.

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