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Necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations. (English) Zbl 0883.39005
Summary: Many processes in automatic regulation, physics, mechanics, biology, economy, ecology, etc. can be modelled by hereditary systems. One of the main problems for the theory of such systems and their applications is connected with stability. Many stability results were obtained by the construction of appropriate Lyapunov functionals. At present, a method is proposed which allows us, in some sense, to formalize the procedure of constructing the corresponding Lyapunov functionals. In this work, by virtue of the proposed procedure, necessary and sufficient conditions of asymptotic mean square stability for stochastic linear difference equations are obtained.

MSC:
39A11 Stability of difference equations (MSC2000)
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[1] Kolmanovskii, V.B.; Shaikhet, L.E., Control of systems with aftereffect, () · Zbl 1123.34323
[2] Kolmanovskii, V.B.; Myshkis, A.D., Applied theory of functional differential equations, (1992), Kluwer Academic Publishers Boston · Zbl 0907.39012
[3] Kolmanovskii, V.B.; Nosov, V.R., Stability and periodical regimes of regulating hereditary systems, (1981), Nauka Moscow
[4] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press New York · Zbl 0593.34070
[5] Kolmanovskii, V.B., On stability of some hereditary systems, Avtomatika i telemekhanika, 11, 45-59, (1993) · Zbl 0279.93049
[6] Kolmanovskii, V.B.; Shaikhet, L.E., Stability of stochastic hereditary systems, Avtomatika i telemekhanika, 7, 66-85, (1993)
[7] Kolmanovskii, V.B.; Shaikhet, L.E., On one method of Lyapunov functional construction for stochastic hereditary systems, Differentsialniye uravneniya, 11, 1909-1920, (1993) · Zbl 0815.34068
[8] Kolmanovskii, V.B.; Shaikhet, L.E., New results in stability theory for stochastic functional differential equations (SFDEs) and their applications, (), 167-171 · Zbl 0811.34062
[9] Kolmanovskii, V.B.; Shaikhet, L.E., General method of Lyapunov functionals construction for stability investigations of stochastic difference equations, (), 397-439, Singapore · Zbl 0846.93083
[10] Shaikhet, L.E., Stability in probability of nonlinear stochastic hereditary systems, Dynamic systems and applications, 4, 2, 199-204, (1995) · Zbl 0831.60075
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