×

Theory and application of stability for stochastic reaction diffusion systems. (English) Zbl 1148.35106

This paper uses Lyapunov direct method for stochastic partial differential equations. For stochastic reaction-diffusion systems the authors establish Lyapunov stability theory, including stability in probability, asymptotic stability in probability, and exponential stability in mean square.
As the noise is finite dimensional the authors always use Itô’s formula to establish their results.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D20 Asymptotic stability in control theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bachelier L. Théorie de la speculation. Ann Sci Ecole Norm, 1900, 17: 21–86 · JFM 31.0241.02
[2] Itô K. On stochastic differential equations. Mem Amer Math Soc, 1951, 4 · Zbl 0045.07603
[3] Kac I J, Krasovskii N N. On stability of systems with random parameters. J Appl Mach Mech, 1960, 24: 1225–1246
[4] Has’minskii R Z. On the stability of trajectory of Markov processes. J Appl Mach Mech, 1962, 26: 1554–1565
[5] Has’minskii R Z. Stochastic Stability of Differential Equations. Sijthoff and Noordoff, Alphen aan den Rijn, 1981
[6] Friedman A. Stochastic Differential Equations and Applications. Academic Press, Vol.1, 1975 · Zbl 0323.60056
[7] Mao X. Exponential Stability of Stochastic Differential Equations. New York: Marcel Dekker, 1994 · Zbl 0806.60044
[8] Liao X X. Theory and Application of Stability for Dynamical Systems. Beijing: Defense Industry Press, 2000 · Zbl 0966.34001
[9] Holden H. Stochastic Partial Differential Equations – A Modeling White Noise Functional Analysis Approach. Boston: Birkhauser, 1996 · Zbl 0860.60045
[10] István Gyöngy, Carles Rovira. On L p-solutions of semilinear stochastic partial differential equations. Stoch Proc Their Appl, 2000, 90: 83–108 · Zbl 1046.60059
[11] Alòs E, Bonaccorsi S. Stochastic partial differential equations with Dirichlet white-noise boundary conditions. Ann I H Poincaré-PR38, 2002, 2: 125–154 · Zbl 0998.60065
[12] Dozzi M, Maslowski B. Non-explosion of solutions to stochastic reaction-diffusion equations, ZAMM _ Z. Angew Math Mech, 2002, 82: 11–12, 745–751 · Zbl 1009.60054
[13] Zhang X C. Quasi-sure limit theorem of parabolic stochastic partial differential equations. Acta Math Sin, 2004, 20(4): 719–730 · Zbl 1057.60064
[14] Pardouxa E, Piatnitskib A L. Homogenization of a nonlinear random parabolic partial differential equation. Stoch Proc Their Appl, 2003, 104: 1–27 · Zbl 1075.35003
[15] Debbi L, Dozzi M. On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stoch Proc Their Appl, 2005, 15: 1764–1781 · Zbl 1078.60048
[16] Nualart D, Vuillermot P-A. Variational solutions for a class of fractional stochastic partial Dofferential equations. C R Acad Sci Paris, 2005, 340: 281–286 · Zbl 1063.60091
[17] Nualart D, Vuillermo P-A. Variational solutions for partial differential equations driven by a fractional noise. J Funct Anal, 2005, 232: 390–454 · Zbl 1089.35097
[18] Duncan T E, Maslowski B, Pasik-Duncan B. Some properties of linear stochastic distributed parameter systems with fractional Brownian motion. In: Proceedings of the 40th IEEE Conference on Decision and Control. Orlando, Florida USA, December 2001, 808–812 · Zbl 1157.93034
[19] Duncan T E, Pasik-duncan B. Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch Dyn, 2002, 2(2): 225–250 · Zbl 1040.60054
[20] Dalang R C, Sanz-Solé M. Regularity of the samples paths of a class of second-order SPDE’s. J Funct Anal, 2005, 227: 304–337 · Zbl 1085.60042
[21] Dong Y, Han Q L. Delay-dependent exponential stability of stochastic systems with time-varying delays, nonlinearities and Markovian jump parameters. IEEE Trans Automat Contr, 2005, 50(2): 217–222 · Zbl 1365.93377
[22] Dong Y, Won S. Delay-dependent robust stability of stochastic systems with time delay and nonlinear uncertainties. Elect Lett (IEE), 2001, 37(15): 992–993 · Zbl 1190.93095
[23] Sigurd Assinga, Ralf Mantheyb. Invariant Measures for Stochastic Heat Equations with Unbounded Coefficients. Stochastic Processes and their Applications, 2003, 103: 237–256 · Zbl 1075.60542
[24] Cerrai S, Röckner M. Large deviations for invariant measures of stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann I H Poincaré-PR, 2005, 41: 69–105 · Zbl 1066.60029
[25] Buckdahn R, Ma J. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I. Stoch Proc Their Appl, 2001, 93: 181–204 · Zbl 1053.60065
[26] Bally V, Saussereau B. A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs. J Funct Anal, 2004, 210: 465–515 · Zbl 1055.60051
[27] Zhang Q M, Zhang W G, Nie Z K. Convergence of the Euler scheme for stochastic functional partial differential equations. Appl Math Comp, 2004, 155: 479–492 · Zbl 1059.65008
[28] Goldys B, Maslowski B. Exponential ergodicity for stochastic burgers and 2D Navier-Stokes equations. J Funct Anal, 2005, 226: 230–255 · Zbl 1078.60049
[29] Gradinaru M, Nourdin I, Tindel S. Itô’s and Tanaka’s-type formulate for the stochastic heat equation: The linear case. J Funct Anal, 2005, 228: 114–143 · Zbl 1084.60039
[30] Badri Narayanan V A, Zabaras N. Variational multiscale stabilized FEM formulations for transport equations: Stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations. J Comp Phys, 2005, 202: 94–133 · Zbl 1061.76029
[31] Dauer J P, Mahmudov N I, Matar M M. Approximate controllability of backward stochastic evolution equations in Hilbert spaces. J Math Anal Appl, 2006, 323: 42–56 · Zbl 1103.60058
[32] Donati-Martin C, Pardoux E. White noise driven SPDEs with reaction. IEEE Trans Autom Contr, 1993, 95: 1–24 · Zbl 0794.60059
[33] Kotelenez P. Comparison methods for a class of function valued stochastic partial differential equations. Probab Theory Related Fields, 1992, 93: 1–19 · Zbl 0767.60053
[34] Mueller C. On the support of solutions to the heat equation with noise. Stochastics, 1991, 37: 225–245 · Zbl 0749.60057
[35] Shiga T. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can J Math, 1994, 46(2): 415–437 · Zbl 0801.60050
[36] Assing S. Comparison of systems of stochastic partial differential equations. Stoch Proc Their Appl, 1999, 82: 259–282 · Zbl 0997.60064
[37] Curtain R F. Stochastic differential equation in Hilbert space. J Diff Eq, 1971, 10: 412–430 · Zbl 0225.60028
[38] Haussmann U G. Asymptotic stability of the linear Itô equation in infinite dimension. J Math Anal Appl, 1978, 65: 219–235 · Zbl 0385.93051
[39] Chow P L. Stability of nonlinear stochastic evolution equations. J Math Anal Appl, 1982, 89: 400–419 · Zbl 0496.60059
[40] Itô K. Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS Notes. Vol. 47. Baton Rouge: SIAM, 1984 · Zbl 0547.60064
[41] Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications. Vol. 44. Cambridge: Cambridge University Press, 1992 · Zbl 0761.60052
[42] Luo Q, Deng F Q, Bao J D, et al. Stabilization of stochastic Hopfield neural network with distributed parameters. Sci China Ser F-Inf Sci, 2004, 47(6): 752–762 · Zbl 1187.35128
[43] Hopfield J J. Neural networks and physical systems with emergent collect computational abilities. Proc Natl Acad Sci USA, 1982, 79: 2554–2558 · Zbl 1369.92007
[44] Hopfield J J, Tank D W. Computing with neural circuits: a model. Science, 1986, 233: 625–633 · Zbl 1356.92005
[45] Liao X X. Mathematical theory of cell neural network (II). Sci China Ser A, 1995, 38(5): 542–551 · Zbl 0836.68094
[46] Liang X B, Wu L D. Globally exponential stability of Hopfield neural network and its application. Sci China Ser A, 1995, 38(6): 757–768
[47] Liao X X, Mao X R. Stability of stochastic neural networks. Neur Paral Sci Comp, 1996, 4: 205–224 · Zbl 1060.92502
[48] Liao X X, Mao X R. Exponential stability and instability of stochastic neural networks. Stoch Anal Appl, 1996, 14(2): 165–185 · Zbl 0848.60058
[49] He Q M, Kang L. Existence and stability of global solution for generalized Hopfield neural network system. Neural, Parallel & Sci Comp, 1994, 2: 165–176 · Zbl 0815.92002
[50] Liao X X, Yang S Z, Cheng S J, et al. Stability of general neural networks with reaction-diffusion. Sci China Ser F-Inf Sci, 2001, 44(5): 389–395 · Zbl 1238.93071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.