Fast Convergence using difference Equations for the Uncertained Framework

 

Dr B. Venkateswarlu, Dr. Madhusudhana Rao

Department of  Mathematics, School of Advanced Sciences (SAS), VIT University, Vellore, Tamilnadu, India

​Department of Information Technology, Higher College of Technology, Muscat, Oman

 

ABSTRACT:

In this paper, we establish the almost sure asymptotic stability and decay solutions of a scalar stochastic difference equation with a non-hyperbolic equilibrium at the origin which is perturbed by a random term with a fading state-independent intensity. Also we arrived the convergence speed because it determines the maximum rate of change of the input non –stationaries in the uncertained domain that was tracked by the stochastic network.

 

 

 

INTRODUCTION:

In this paper, we establish the almost sure asymptotic stability and decay solutions of a scalar stochastic difference equation with a non-hyperbolic equilibrium at the origin [5].Also we study the almost sure convergence to the equilibrium point. This equilibrium can be taken to be zero, without loss of generality [10].In this difference equation, linearization of the equation close to the equilibrium does not determine the asymptotic behaviour [1],because the terms which depend on the state of the system are  as.

 

The equations studied may be viewed as stochastically perturbed versions of  stochastic perturbed difference equations, where the random perturbation is independent of the state [7].Such kind of adaptive implementation is based on solving the following stochastic equation in the uncertained domain [4],

 

CONCLUSIONS:

In this paper, the necessity to achieve a decay rate of solutions of a stochastic difference equation with non-hyperbolic equilibrium at the origin is discussed with suitable theorem. The computationalComplexity refers to the number of operations required to update and optimal the network from one time instant to the next. We deals with the results on the asymptotic stability of general scalar stochastic difference equations with state- independent perturbations .we studied the exact polynomialrate of convergence of solutions of stochastic scalar difference equations. The results are applied in thealmost sure polynomial rate of decay of solutions of stochastic difference equations.

 

REFERENCES:

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[3]     J.A.D.Apple by, X.Mao and A. Rodkina, On the asymptotic, stability of polynomial stochastic delay differential equations, Funct.Differ.Equ, 12(1-2), 35-66, 2005

[4]     J.A.D.Appleby, Stabilisation of functional differential equations by noise, system controlLett., 2005.

[5]     C.Brezinski, History of Continued Fractions and Pade Approximants, Springer-Verlag, Newyork, 1991.

[6]     B.Carlson,Algorithms involving arithmetic and geometric means,496-505,1971.

[7]     P. Hartman, Difference equations disconjugacy, Principal solutions, Green’s function, Complete monotonicity, Trans.AMS Math 246, 1-30,1998.

[8]     R. Joothilakshmi, Effectiveness of the Extended Kalman Filter through Difference Equations, Nonlinear Dynamics and Systems Theory,15 (13) 290927,2015.

[9]     T. Karatzas and S.E Shreve,Brownian Motion and Stochastic Calculus, Springer, Newyork, 1991.

[10]   V. Lakshmikanthan and DonatoTrrigiante, Marcel Dekker, Theory of Difference equations-Numerical method and the applications, 2^nd Edition New York,2002

[11]   A. Rodkina and H. Schurz,On global asymptotic stability of solutions to cubic stochastic difference equations, Advances in difference equations, 3:249-260,2004

 

 

Accepted on 03.11.2016        © RJPT All right reserved

Research J. Pharm. and Tech 2016; 9(12):2372-2376.