ALMOST SURE EXPONENTIAL STABILITY OF DELAY EQUATIONS WITH DAMPED STOCHASTIC PERTURBATION

2001 ◽  
Vol 19 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Xuerong Mao
Keyword(s):  
Exponential Stability ◽  
Stochastic Perturbation ◽  
Delay Equations ◽  
Almost Sure Exponential Stability

CONVERGENCE DYNAMICS OF STOCHASTIC REACTION–DIFFUSION RECURRENT NEURAL NETWORKS WITH DELAYS

2005 ◽  
Vol 15 (07) ◽  
pp. 2131-2144 ◽  
Author(s):  
JIANHUA SUN ◽  
LI WAN
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Reaction Diffusion ◽  
Mean Value ◽  
Almost Sure Exponential Stability ◽  
Transmission Delays ◽  
Mean Square Exponential Stability

Stochastic effects on convergence dynamics of reaction–diffusion recurrent neural networks (RNNs) with constant transmission delays are studied. Without assuming the boundedness, monotonicity and differentiability of the activation functions, nor symmetry of synaptic interconnection weights, by skillfully constructing suitable Lyapunov functionals and employing the method of variational parameters, M-matrix properties, inequality technique, stochastic analysis and non-negative semimartingale convergence theorem, delay independent and easily verifiable sufficient conditions to guarantee the almost sure exponential stability, mean value exponential stability and mean square exponential stability of an equilibrium solution associated with temporally uniform external inputs to the networks are obtained, respectively. The results are compared with the previous results derived in the literature for discrete delayed RNNs without diffusion or stochastic perturbation. Two examples are also given to demonstrate our results.

scholarly journals On stabilization of partial differential equations by noise

2001 ◽  
Vol 161 ◽  
pp. 155-170 ◽  
Author(s):  
Tomás Caraballo ◽  
Kai Liu ◽  
Xuerong Mao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Partial Differential Equations ◽  
Finite Dimension ◽  
Infinite Dimension ◽  
Stability Criteria ◽  
Partial Differential ◽  
Almost Sure Exponential Stability

Some results on stabilization of (deterministic and stochastic) partial differential equations are established. In particular, some stability criteria from Chow [4] and Haussmann [6] are improved and subsequently applied to certain situations, on which the original criteria commonly do not work, to ensure almost sure exponential stability. This paper also extends to infinite dimension some results due to Mao [9] on stabilization of differential equations in finite dimension.

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