×

zbMATH — the first resource for mathematics

The exponential stability for stochastic delay partial differential equations. (English) Zbl 1125.60063
This paper deals with the exponential stability problem for nonlinear stochastic functional differential equations with finite delays. The author investigates the existence of a solution satisfying an energy equality and obtains the conditions for exponential stability of the zero solution using the evaluations of the coefficient functions.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B35 Stability in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caraballo, T.; Liu, K., On exponential stability criteria of stochastic partial differential equations, Stochastic process. appl., 83, 289-301, (1999) · Zbl 0997.60065
[2] Caraballo, T.; Liu, K.; Truman, A., Stochastic functional partial differential equations; existence, uniqueness and asymptotic decay property, Proc. R. soc. lond. ser. A math. phys. eng. sci., 456, 1775-1802, (2000) · Zbl 0972.60053
[3] Caraballo, T.; Garrido-Atienza, M.J.; Real, J., Existence and uniqueness of solutions for delay stochastic evolution equations, Stoch. anal. appl., 20, 1225-1256, (2002) · Zbl 1028.60056
[4] Caraballo, T.; Lang, J.; Taniguchi, T., The exponential behaviour and stabilizability of stochastic 2D-navier – stokes equations, J. differential equations, 714-737, (2002) · Zbl 0990.35138
[5] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. anal. appl., 90, 12-44, (1982) · Zbl 0497.93055
[6] Ichikawa, A., Absolute stability of a stochastic evolution equation, Stochastics, 11, 143-158, (1983) · Zbl 0531.93065
[7] Kwiecinska, A., Stabilization of evolution equation by noise, Proc. amer. math. soc., 130, 3067-3074, (2001) · Zbl 1003.35074
[8] Leha, G.; Maslowski, B.; Ritter, G., Stability of solutions to semilinear stochastic evolution equations, Stoch. anal. appl., 17, 1009-1051, (1999) · Zbl 0943.60049
[9] Liu, K., Lyapunov functionals and asymptotic stability of stochastic delay evolution equations, Stoch. stoch. rep., 63, 1-26, (1998) · Zbl 0947.93037
[10] Liu, R.; Mandrekar, V., Stochastic semilinear evolution equations: Lyapunov functions, stability, and ultimate boundedness, J. math. anal. appl., 212, 537-553, (1997) · Zbl 0885.60044
[11] Liu, K.; Mao, X., Exponential stability of non-linear stochastic evolution equations, Stochastic process. appl., 78, 173-193, (1998) · Zbl 0933.60072
[12] Maslowski, B., Stability of semilinear equations with boundary and pointwise noise, Ann. sc. norm. super. Pisa cl. sci., IV, 55-93, (1995) · Zbl 0830.60056
[13] Taniguchi, T., Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stoch. stoch. rep., 53, 41-52, (1995) · Zbl 0854.60051
[14] Taniguchi, T., Almost sure exponential stability for stochastic partial functional differential equations, Stoch. anal. appl., 16, 965-975, (1998) · Zbl 0911.60054
[15] Taniguchi, T.; Liu, K.; Truman, A., Existence, uniqueness, and asymptotic behavior of mild solution to stochastic functional differential equations in Hilbert spaces, J. differential equations, 181, 72-91, (2002) · Zbl 1009.34074
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.