Exponential Mean‐Square Stability of Stochastic String Hybrid Systems Under Continuous Non‐Gaussian Excitation

2018 ◽  
Vol 20 (6) ◽  
pp. 2116-2129 ◽  
Author(s):  
Lesław Socha
Keyword(s):  
Hybrid Systems ◽  
Mean Square ◽  
Mean Square Stability ◽  
Non Gaussian ◽  
Exponential Mean Square Stability

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

EXPONENTIAL MEAN SQUARE STABILITY OF STOCHASTICALLY FORCED INVARIANT MANIFOLDS FOR NONLINEAR SDEs

2007 ◽  
Vol 07 (03) ◽  
pp. 389-401 ◽  
Author(s):  
L. B. RYASHKO
Keyword(s):  
Linear Extension ◽  
Invariant Manifolds ◽  
Function Technique ◽  
Nonlinear Stochastic System ◽  
General Notion ◽  
Mean Square ◽  
Extension System ◽  
Mean Square Stability ◽  
The Stability ◽  
Exponential Mean Square Stability

An exponential mean square stability for the invariant manifold [Formula: see text] of a nonlinear stochastic system is considered. The stability analysis is based on the [Formula: see text]-quadratic Lyapunov function technique. The local dynamics of the nonlinear system near manifold is described by the stochastic linear extension system. We propose a general notion of the projective stability (P-stability) and prove the following theorem. The smooth compact manifold [Formula: see text] is exponentially mean square stable if and only if the corresponding stochastic linear extension system is P-stable.

Stability of switched Markovian jump systems with time-varying generally bounded transition rates

2021 ◽  
pp. 014233122098753
Author(s):  
Dunke Lu ◽  
Xiaohang Li
Keyword(s):  
Sufficient Conditions ◽  
Markovian Jump Systems ◽  
Time Varying ◽  
Transition Rates ◽  
Mean Square ◽  
Markovian Jump ◽  
Mean Square Stability ◽  
Special Cases ◽  
Jump Systems ◽  
Exponential Mean Square Stability

This paper addresses the exponential mean-square stability for a kind of switched Markovian jump systems, which have time-varying generally bounded transition rates and mode-dependent time delay. Since these transition rates are time-varying and generally bounded, they turn out to be more practical. In fact, those existing transition rates can be treated as special cases of the proposed ones in this paper. By constructing a new Lyapunov-Krasovskii function, sufficient conditions in a tractable form are derived for the exponential mean-square stability of the considered systems. For good measure, a numerical example is given to show the efficiency and potential of the proposed method.

The exponential behavior and stabilizability of quasilinear parabolic stochastic partial differential equation

2021 ◽  
pp. 1-13
Author(s):  
Xiuwei Yin ◽  
Guangjun Shen ◽  
Jiang-Lun Wu
Keyword(s):  
Exponential Stability ◽  
Mean Square ◽  
Exponential Behavior ◽  
Partial Differential ◽  
Stochastic Perturbations ◽  
Mean Square Stability ◽  
The Mean ◽  
The Stability ◽  
Exponential Mean Square Stability ◽  
Quasilinear Parabolic

In this paper, we study the stability of quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The exponential mean square stability and pathwise exponential stability of the solutions are established. Moreover, under certain hypothesis on the stochastic perturbations, pathwise exponential stability can be derived, without utilizing the mean square stability.

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