Nonstationary Coefficients Research Articles

Mean-square stability of stochastic differential time-lag systems

Two sets of sufficient conditions for the mean-square asymptotic stability of the equilibrium state of linear differential time-lag systems with non-stationary random coefficients are derived. Each of the sets of sufficient conditions is then shown to be equivalent to a corresponding unconstrained optimization problem. Hence, it can readily be solved by standard efficient optimization techniques. The usual trial-and-error method of finding the required parameters to satisfy the stability conditions can now be avoided.

Iterative k-step methods for computing possibly repulsive fixed points in Banach spaces

We discuss iterative methods of the form x n : = μ 0 Φ( x x − 1 ) + μ 1 x n − 1 + … + μ k x n − k ( n = k, k + 1,…) for computing a fixed point x ∗ of a Fréchet-differentiable self-mapping Φ of a sub-set in a Banach space. By suitably choosing the coefficients μ 0,…, μ k the local convergence is established under certain assumptions on the spectrum of Φ′ x ∗ . This spectrum need not be contained in the unit disk however; if it is, convergence can often be speeded up considerably compared to Picard iteration. The methods are generalizations of the methods of V. N. Kublanovskaya and W. Niethammer for linear systems of equations. A more general type of iteration with nonstationary coefficients is considered also. For the proof of their local convergence a generalization of the convergence theorems of Perron, Ostrowski and Kitchen is presented. The connection with other methods, in particular Mann's iterative processes, is also discussed.

Mean-square stability of delay-differential equations with non-stationary random coefficients

Abstract The mean-square asymptotic stability of the equilibrium stale of a system of linear stochastic delay-differential equations with general non-stationary random coefficients is studied. The Lyapunov method of stability analysis is employed and several criteria for the mean-square stability of the equilibrium state of the system are established. A few examples are solved and the application of the method is discussed.

Adaptive Relaxation Labeling

Current implementation of probabilistic relaxation labeling (PRL) is based on stationary compatibility coefficients (SCC's). Such labeling frequently diverges from an achieved minimum labeling error. In this correspondence it is shown that by having nonstationary compatibility coefficients (NSCC's) the PRL stabilizes about the minimum error which is obtained during the early iterations. Also, a noniterative labeling algorithm which uses NSCC and has a performance similar to that of the modified PRL is suggested.

On the mean square stability of a class of nonstationary coupled partial differential equations

The stability of the equilibrium solution of a class of coupled partial differential equations with nonstationary random coefficients is studied. Several theorems in regard to mean square stability of the equilibrium state of the system are established. Some examples of the applications of the results to engineering problems are presented and significant improvements on stability region are observed.

The stability of a class of continuous systems with non-stationary random coefficients

The stability of a class of continuouous systems with non-stationary random coefficients is studied. It is shown that the trivial equilibrium solution is moan-square asymptotic stable as long as the expected values of the absolute value of the random coefficients are integrable. Several examples are considered and discussed.

An application of analog computers to the statistical analysis of time-variable networks

IN RECENT years an extensive body of mathematical techniques has been developed for the analysis of the response of linear constant coefficient control systems to stationary random processes as inputs. In many cases it is possible to achieve direct synthesis of the optimum system for an assigned task. For nonstationary inputs or variable coefficient systems, no corresponding theory exists, even though problems of this nature arise quite often in practice. In the present paper an analog method is presented for the rms error analysis of a class of nonstationary problems. However, no attempt is made at the synthesis of an optimum system. Following a brief discussion of analog methods applicable to the general nonstationary case, our attention is concentrated on the special problem of a variable coefficient linear system with a stationary random input. Exploitation of the properties of the adjoint system in this case is shown to reduce considerably the labor in computing rms errors in comparison with the general method for nonstationary inputs. The simulation of the adjoint system is shown to be readily obtainable from the simulation of the original system.