Small noise asymptotics of nonlinear filters with nonobservable limiting deterministic system

2002 ◽  
Author(s):  
M. Joannides ◽  
F. LeGland
Keyword(s):  
Nonlinear Filters ◽  
Deterministic System ◽  
Small Noise ◽  
Small Noise Asymptotics

STABILITY OF CELLULAR NEURAL NETWORKS WITH DISTRIBUTIVE TIME DELAYS AND NOISE PERTURBATIONS

2007 ◽  
Vol 17 (02) ◽  
pp. 535-544
Author(s):  
CHENG-HSIUNG HSU ◽  
TZI-SHENG YANG
Keyword(s):  
Neural Networks ◽  
Time Delays ◽  
Internal Noise ◽  
External Noise ◽  
Cellular Neural Networks ◽  
The Other ◽  
Deterministic System ◽  
Noise Strength ◽  
Small Noise ◽  
Globally Exponential Stability

This work investigates the global stability of cellular neural networks with distributive time delays and noise perturbations. Two types of noise perturbations are considered in the system, one is internal noise and another is external noise. We show that the deterministic system preserves the globally exponential stability when it is perturbed by the internal noise with small noise strength. On the other hand, the deterministic system can only preserve weaker stability when it is perturbed by the external noise. We also provide some numerical simulations to support our theoretical analysis.

scholarly journals OPTIMAL FLUCTUATIONS AND THE CONTROL OF CHAOS

2002 ◽  
Vol 12 (03) ◽  
pp. 583-604 ◽  
Author(s):  
D. G. LUCHINSKY ◽  
S. BERI ◽  
R. MANNELLA ◽  
P. V. E. McCLINTOCK ◽  
I. A. KHOVANOV
Keyword(s):  
Control Problem ◽  
Chaotic Attractor ◽  
Initial Conditions ◽  
Control Function ◽  
Optimal Path ◽  
Hamiltonian Formulation ◽  
Chaotic Oscillator ◽  
Deterministic System ◽  
Small Noise ◽  
The Cost

The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel–Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identified with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

ON DELAY ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 05 (02) ◽  
pp. 333-342 ◽  
Author(s):  
YURY A. KUTOYANTS
Keyword(s):  
Maximum Likelihood ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Maximum Likelihood Estimator ◽  
Delay Estimation ◽  
Likelihood Estimator ◽  
Small Noise ◽  
Noise Limit ◽  
Small Noise Asymptotics ◽  
Regular Problems

We present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to large samples limit. In both cases we describe the properties of the maximum likelihood estimator and Bayesian estimators with especial attention to asymptotic efficiency of the estimators. We show that the first asymptotic corresponds to regular problems of mathematical statistics and the second is close to non regular problems. In small noise asymptotics we give the next after the Gaussian term of asymptotic expansion of the maximum likelihood estimator.

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