Some Contributions to Stochastic Asymptotic Stability and Boundedness via Multiple Lyapunov Functions

Most of the existing results on stochastic stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations. From them follow many useful results on stochastic asymptotic stability and boundedness, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on stochastic asymptotic stability is a special case of our more general results. These show clearly the power of our new results.

[1]  K. Elworthy.  Stochastic Differential Equations on Manifolds , 1982 .

[2]  V. Kolmanovskii,et al.  Applied Theory of Functional Differential Equations , 1992 .

[3]  L. Arnold,et al.  Large noise asymptotics of invariant measures, with applications to Lyapunov exponents , 1996 .

[4]  Xuerong Mao,et al.  Stochastic Versions of the LaSalle Theorem , 1999 .

[5]  Hans Crauel,et al.  Stabilization of Linear Systems by Noise , 1983 .

[6]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[7]  J. P. Lasalle.  Stability theory for ordinary differential equations. , 1968 .

[8]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[9]  A. Friedman,et al.  Asymptotic stability and spiraling properties for solutions of stochastic equations , 1973 .

[10]  X. Mao,et al.  Stability of Stochastic Differential Equations With Respect to Semimartingales , 1991 .

[11]  V. Lakshmikantham,et al.  Stability Analysis of Nonlinear Systems , 1988 .

[12]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[13]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[14]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[15]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[16]  V. Lakshmikantham,et al.  Random differential inequalities , 1980 .

[17]  S. Mohammed.  Stochastic functional differential equations , 1984 .

[18]  V. Lakshmikantham,et al.  Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems , 1991 .