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Construction of Lyapunov functionals for stochastic hereditary systems: A survey of some recent results. (English) Zbl 1029.93057
Stochastic discrete-time systems with multiple delays are considered. A procedure for Lyapunov function construction is given for such systems. This procedure is applied to linear systems with constant coefficients, with monotone coefficients, with time-varying delays, and with discrete Volterra equations of the second type. Then, continuous-time versions of these results are presented. Many illustrative examples are also incorporated into the paper.

MSC:
93D30 Lyapunov and storage functions
93E15 Stochastic stability in control theory
93C55 Discrete-time control/observation systems
93C23 Control/observation systems governed by functional-differential equations
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[1] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, (1986), Academic Press New York · Zbl 0593.34070
[2] Kolmanovskii, V.B.; Myshkis, A.D., Applied theory of functional differential equations, (1992), Kluwer Academic Dordrecht · Zbl 0907.39012
[3] Kolmanovskii, V.B.; Shaikhet, L.E., Control of systems with aftereffect, () · Zbl 1123.34323
[4] Kolmanovskii, V.B.; Myshkis, A.D., Introduction to the theory and applications of functional differential equations, (1999), Kluwer Academic Dordrecht · Zbl 0907.39012
[5] Shaikhet, L.E.; Shaikhet, L.E., Asymptotic stability of linear stochastic differential equations of neutral type, Matematicheskiye zametki, Matematicheskiye zametki, 52, 2, 159-147, (1992), (in Russian)
[6] Kolmanovskii, V.B., On stability of some hereditary systems, Avtomatika i telemekhanika, 11, 45-59, (1993) · Zbl 0279.93049
[7] Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E., A method for constructing Lyapunov functionals for stochastic systems with aftereffect, Differentialniye uravneniya, Differentialniye uravneniya, Differential equations, 29, 11, 1657-1666, (1994), (Translation) · Zbl 1123.34323
[8] Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E., Stability of stochastic systems with aftereffect, Avtomatika i telemekhanika, Automat. remote control, 54, 7, 1087-1107, (1993), Part 1, (Translation)
[9] 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
[10] Crisci, M.R.; Kolmanovskii, V.B.; Russo, E.; Vecchio, A., Stability of continuous and discrete Volterra integro-differential equations by Liapunov approach, Journal of integral equations, 7, 4, 393-411, (1995) · Zbl 0858.45008
[11] Crisci, M.R.; Kolmanovskii, V.B.; Russo, E.; Vecchio, A., Stability of difference Volterra equations: direct Liapunov method and numerical procedures, IAM rap tec 114/95, (1995), Napoli, Italy · Zbl 0858.45008
[12] Kolmanovskii, V.B., About application of Lyapunov’s second method to Volterra difference equations, Avtomatika i telemekhanika, 11, 50-64, (1995), (in Russian)
[13] Kolmanovskii, V.B.; Rodionov, A.M., The stability of certain discrete Volterra processes, Avtomatika i telemekhanika, 2, 3-13, (1995), (in Russian)
[14] Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E., A method for constructing Lyapunov functionals for stochastic differential equations of neutral type, Differentialniye uravneniya, Differentialniye uravneniya, Differential equations, Differential equations, 31, 11, 1996-1825, (1995), (Translation) · Zbl 0869.93052
[15] Kolmanovskii, V.B.; Shaikhet, L.E., General method of Lyapunov functionals construction for stability investigations of stochastic difference equations, (), 397-439 · Zbl 0846.93083
[16] Shaikhet, L.E.; Shaikhet, L.E.; Shaikhet, L.E., On the stability of solutions of stochastic Volterra equations, Avtomatika i telemekhanika, Automat. remote control, Automat. remote control, 56, 8, 1996-1137, (1995), Part 2, (Translation) · Zbl 0843.93086
[17] Shaikhet, L.E., Stability in probability of nonlinear stochastic hereditary systems, Dynamic systems and applications, 4, 2, 199-204, (1995) · Zbl 0831.60075
[18] Shaikhet, L.E.; Shaikhet, L.E., Stability in probability of nonlinear stochastic systems with delay, Matematicheskiye zametki, Math. notes, 57, 1-2, 103-106, (1995), (Translation) · Zbl 0843.93086
[19] Kolmanovskii, V.B.; Shaikhet, L.E.; Kolmanovskii, V.B.; Shaikhet, L.E., Asymptotic behaviour of some discrete-time systems, Avtomatika i telemekhanika, Automat. remote control, 57, 12, 1735-1742, (1997), Part 1, (Translation) · Zbl 1148.93346
[20] Kuchkina, N.V.; Shaikhet, L.E.; Kuchkina, N.V.; Shaikhet, L.E., Stability of the stochastic difference Volterra equations, Theory of stochastic processes, Theory of stochastic processes, 2, N.3-4, 79-86, (1996) · Zbl 0939.60071
[21] Shaikhet, L.E.; Shaikhet, L.E., Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems, Theory of stochastic processes, Theory of stochastic processes, 2, N.1-2, 248-259, (1996) · Zbl 0893.60029
[22] Shaikhet, L.E.; Shaikhet, L.E., Stability of stochastic hereditary systems with Markov switching, Theory of stochastic processes, Theory of stochastic processes, 2, N.3-4, 180-184, (1996) · Zbl 0939.60049
[23] Ford, N.J.; Edwards, J.T.; Roberts, J.A.; Shaikhet, L.E., Stability of a difference analogue for a nonlinear integro differential equation of convolution type, ()
[24] Kolmanovskii, V.; Shaikhet, L., Matrix Riccati equations and stability of stochastic linear systems with nonincreasing delays, Functional differential equations, 4, 3-4, 279-293, (1997) · Zbl 1148.93346
[25] Kolmanovskii, V.; Shaikhet, L.; Kolmanovskii, V.; Shaikhet, L., On the stability of some stochastic equations of Volterra type, Minsk, differentialniye uravneniya, Minsk, differentialniye uravneniya, 33, 11, 1581-1501, (1997), (in Russian) · Zbl 1148.93346
[26] Shaikhet, L.E., Necessary and sufficient conditions of asymptotic Mean square stability for stochastic linear difference equations, Appl. math. lett., 10, 3, 111-115, (1997) · Zbl 0883.39005
[27] Shaikhet, L.E.; Shaikhet, L.E.; Shaikhet, L.E., Problems of the stability for stochastic difference equations, (), N.3-4, 403-411, (1997) · Zbl 0691.93053
[28] Shaikhet, L.E., (), 257-262
[29] Beretta, E.; Kolmanovskii, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Delay systems, Mathematics and computers in simulation, 45, 3-4, 269-277, (1998), Special Issue of · Zbl 1017.92504
[30] Crisci, M.R.; Kolmanovskii, V.B.; Russo, E.; Vecchio, A., Stability of difference Volterra equations: direct Liapunov method and numerical procedure, Advances in difference equations II, Computers math. applic., 36, 10-12, 77-97, (1998), Special Issue of · Zbl 0933.39007
[31] Kolmanovskii, V.B.; Myshkis, A.D., Stability in the first approximation of some Volterra difference equations, Journal of difference equations and applications, 3, 563-569, (1998) · Zbl 0907.39012
[32] Kolmanovskii, V.; Shaikhet, L., On the stability of difference equations, (), 143-145 · Zbl 1107.39301
[33] Kolmanovskii, V.; Shaikhet, L., (), 97-100, Durban, South Africa
[34] Kolmanovskii, V.B.; Shaikhet, L.E., Riccati equations in stability of stochastic linear systems with delay, Avtomatika i telemekhanika, 10, 35-54, (1998), (in Russian) · Zbl 1123.34323
[35] Shaikhet, G.; Shaikhet, L., (), 101-104, Durban, South Africa
[36] Shaikhet, L., Stability of predator-prey model with aftereffect by stochastic perturbations, Stability and control: theory and application, 1, 1, 3-13, (1998)
[37] Shaikhet, L.; Shaikhet, L.; Shaikhet, L., Stability of systems of stochastic linear difference equations with varying delays, (), 4, N.1-2, 258-273, (1998) · Zbl 0938.93059
[38] Kolmanovskii, V.B., The stability of certain discrete-time Volterra equations, Journal of applied mathematics and mechanics, 63, 4, 537-543, (1999)
[39] Kolmanovskii, V.; Kosareva, N.; Shaikhet, L., About one method of Lyapunov functionals construction, Differentsialniye uravneniya, 35, 11, 1553-1565, (1999), (in Russian)
[40] Paternoster, B.; Shaikhet, L., Stability in probability of nonlinear stochastic difference equations, Stability and control: theory and application, 2, 1-2, 25-39, (1999)
[41] Borne, P.; Kolmanovskii, V.; Shaikhet, L.; Borne, P.; Kolmanovskii, V.; Shaikhet, L., Steady-state solutions of nonlinear model of inverted pendulum, (), 5, 21, 203-209, (2000), Kyiv, Ukraine.
[42] Borne, P.; Kolmanovskii, V.; Shaikhet, L., ()
[43] In Proceedings, CD-ROM, Session Papers, 154-2, 5 p. · Zbl 0951.65148
[44] Ford, N.J.; Edwards, J.T.; Roberts, J.A.; Shaikhet, L.E., Stability of a difference analogue for a nonlinear integro differential equation of convolution type, Stability and control: theory and application, 3, 1, 24-37, (2000)
[45] Ford, N.J.; Edwards, J.T.; Roberts, J.A.; Shaikhet, L.E.; Ford, N.J.; Edwards, J.T.; Roberts, J.A.; Shaikhet, L.E., (), 6-163, Proceedings, CD-ROM, Session Papers
[46] Mao, X.; Shaikhet, L.; Mao, X.; Shaikhet, L., (), 6-155, Proceedings, CD-ROM, Session Papers
[47] Mao, X.; Shaikhet, L., Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching, Stability and control: theory and application, 3, 2, 88-101, (2000)
[48] Paternoster, B.; Shaikhet, L.; Paternoster, B.; Shaikhet, L., (), 5-156, Proceedings, CD-ROM, Session Papers
[49] Paternoster, B.; Shaikhet, L., About stability of nonlinear stochastic difference equations, Appl. math. lett., 13, 5, 27-32, (2000) · Zbl 0959.60056
[50] Paternoster, B.; Shaikhet, L., Integrability of solutions of stochastic difference second kind Volterra equations, Stability and control: theory and application, 3, 1, 78-87, (2000) · Zbl 0959.60056
[51] Borne, P.; Kolmanovskii, V.; Shaikhet, L., Stabilization of inverted pendulum by control with delay, Dynamic systems and applications, 9, 4, 501-515, (2000) · Zbl 1045.70015
[52] Kolmanovskii, V.; Shaikhet, L., Some peculiarities of the general method of Lyapunov functionals construction, Appl. math. lett., 15, 3, 355-360, (2002) · Zbl 1015.39001
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