Abstract
The purpose of this work is to present a review of the results related to some extensions of Lyapounov methods in the stability theory for the solutions of infinite-dimensional (stochastic and deterministic) evolution equations. As a consequence, one can derive results on the stability of solutions of stochastic partial and delay differential equations.
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1Supported in part by ONR N00014-91-J-1087
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References
Gellman R., (1960), Introduction to matrix analysis, McGraw Hill, New York.
Chow P.L., (1982), Stability of non-linear stochastic evolution equations, J. Math. Anal. and Appins., 89, pp. 400–409.
Curtain R.F., (1981), Stability of stochastic partial differential equation, J. Math. Anal. and Appins., 79, pp. 352–369.
Daletskii Ju.L., Krein M.G., (1974), Stability of solutions of differential equations in Banach space, Transi. of Amer. Math. Soc. 43, Providence, R.I.
Daprato G. and Zabezyk J., (1992), Stochastic equations in infinite-dimensions, Cambridge University Press, New York.
Datko, R., (1970), Extending a theorem of A.M. Lyapounov to Hilbert space, J. Math. Anal. and Appins, 39, pp. 610–616.
Dunford N. and Schwartz J.T., (1963), Linear operators II Interscience, New York.
Hausman U.G., (1978), Asymptotic stability of the linear Ito equation in infinite dimensions, J. Math. Anal. Appins., 65, pp. 219–235.
Ichikawa A., (1979), Dynamic programing approach to stochastic evolution equations, SIAM J. Control and Optim., 17, pp. 152–174.
Ichikawa A., (1982), Stability of semilinear stochastic evolution equations, J. Math. Anal. Appins., 90, pp. 12–44.
Ichikawa A., (1984), Semilinear stochastic evolution equations: Boundedness, stability and invariant measures, Stochastics, 12, pp. 1–39.
Khasminskii R.Z., (1980), Stochastic stability of differential equations, Sijthoff and Noordoff, Netherlands.
Khasminskii R.Z. and Mandrekar V., On stability of solutions of stochastic evolution equations. (To appear).
Krylov N.V. and Rozovskii B.L., (1981), Stochastic evolution equations, J. Soviet Math., 16, pp. 1233–1277.
Loéve, M., (1959), Probability Theory. Van Nostrand, Princeton..
Metivier M. and Pellaumail J., (1980), Stochastic Integration, Academic Press, New York.
Pardoux E., (1979), Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3, pp. 127–167.
Pazy A., (1983), Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York.
Pazy A., (1972), On the applicability of Lyapounov’s theorem in Hilbert space, SIAM J. Math. and Anal., 3, pp. 291–294.
Rozovskii B.L., (1990), Stochastic evolution systems, Kuwer Academic Publishers, Boston.
Slemrod, M., (1976), Asymptotic behaviour of Co-semigroups as determined by the spectrum of the generator, Indiana University, Math. J., 783–792.
Tanabe H., (1979), Equations of evolution, Pitman.
Yosida K., (1965), Functional analysis, Springer-Verlag, New York.
Zabezyk J., (1979), On the stability of infinite-dimensional linear stochastic systems, Banach Center Publ., 5, PWN-Polish Scientific Publishers.
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© 1994 Springer Science+Business Media Dordrecht
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Mandrekar, V. (1994). On Lyapounov Stability Theorems for Stochastic (Deterministic) Evolution Equations1 . In: Cardoso, A.I., de Faria, M., Potthoff, J., Sénéor, R., Streit, L. (eds) Stochastic Analysis and Applications in Physics. NATO ASI Series, vol 449. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0219-3_9
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DOI: https://doi.org/10.1007/978-94-011-0219-3_9
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