Almost-Sure Stability of Some Linear Stochastic Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 175-178 ◽  
Author(s):  
S. T. Ariaratnam ◽  
B. L. Ly
Keyword(s):  
Stochastic Systems ◽  
Stability Boundary ◽  
Quadratic Function ◽  
Random Processes ◽  
Almost Sure Stability ◽  
Special Cases ◽  
Linear Stochastic Systems ◽  
The Matrix ◽  
The Stability ◽  
Second Order Systems

The almost-sure stability of linear second-order systems which are parametrically excited by ergodic, “nonwhite,” random processes is studied by an extension of the method of Infante. In this approach, a positive-definite quadratic function of the form V = x′Px is assumed and a family of stability boundaries depending on the elements of the matrix P is obtained. An envelope of these boundaries is then solved for by optimizing the stability boundary with respect to the elements of P. It is found that the optimum matrix P in general depends not only on the system constants but also on the excitation intensities. This approach is, in principle, applicable to study systems involving two or more random processes. The results reported in previous investigations are obtained as special cases of the present study.

On the stability of large-amplitude geostrophic flows in a two-layer fluid: the case of ‘strong’ beta-effect

1995 ◽  
Vol 284 ◽  
pp. 137-158 ◽  
Author(s):  
E. S. Benilov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Stability Boundary ◽  
Phase Speed ◽  
Stability Theorems ◽  
Beta Effect ◽  
Zonal Velocity ◽  
Special Cases ◽  
Geostrophic Flows ◽  
The Stability

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coefficient of the highest derivative in the corresponding boundary-value problem vanishes at the point where the depth profile has an extremum. Although this singularity is similar to a critical level, it cannot be regularized by the simplistic introduction of infinitesimal viscosity through the assumption that the phase speed of the disturbance is complex. In order to regularize the singularity properly, one should consider the problem within the framework of the original ageostrophic viscous equations and, having obtained the boundary-value problem for harmonic disturbance, take the limit Rossby number → 0, viscosity → 0.The results obtained analytically and (for special cases) numerically indicate that the stability of flows with non-monotonic profiles strongly depends on the depth of the upper layer. If the upper layer is ‘thick’ (i.e. if the average depth H1 of the upper layer is of the order of the total depth of the fluid H0), the stability boundary-value problem does not have any solutions at all, which means stability (however, this stability is structurally unstable, and the flow, generally speaking, can be made weakly unstable by any small effect such as external forcing, viscosity, or ageostrophic corrections). In the case of ‘thin’ upper layer (H1/H0 [lsim ] Ro), the order of the singularity changes and all non-monotonic flows are unstable regardless of their profiles. It is also demonstrated that thin-upper-layer flows do not have to be non-monotonic to be unstable: if u–βR20 (where u is the zonal velocity, β is the β-parameter, and R0 is the deformation radius) changes sign somewhere in the flow, the stability boundary-value problem has another singular point which leads to instability.

Statistics of an Appealing Class of Random Processes

2021 ◽  
pp. 260-276
Author(s):  
Shaival Hemant Nagarsheth ◽  
Shambhu Nath Sharma
Keyword(s):  
White Noise ◽  
Stochastic Systems ◽  
Correction Term ◽  
Random Processes ◽  
Random Perturbations ◽  
White Noise Process ◽  
Coloured Noise ◽  
Diffusion Operator ◽  
Noise Process ◽  
Special Cases

The white noise process, the Ornstein-Uhlenbeck process, and coloured noise process are salient noise processes to model the effect of random perturbations. In this chapter, the statistical properties, the master's equations for the Brownian noise process, coloured noise process, and the OU process are summarized. The results associated with the white noise process would be derived as the special cases of the Brownian and the OU noise processes. This chapter also formalizes stochastic differential rules for the Brownian motion and the OU process-driven vector stochastic differential systems in detail. Moreover, the master equations, especially for the coloured noise-driven stochastic differential system as well as the OU noise process-driven, are recast in the operator form involving the drift and modified diffusion operators involving an additional correction term to the standard diffusion operator. The results summarized in this chapter will be useful for modelling a random walk in stochastic systems.

On the generalized pantograph functional-differential equation

1993 ◽  
Vol 4 (1) ◽  
pp. 1-38 ◽  
Author(s):  
A. Iserles
Keyword(s):  
Differential Equation ◽  
Stability Boundary ◽  
Functional Differential Equations ◽  
Well Posedness ◽  
Almost Periodic ◽  
Pantograph Equation ◽  
Special Cases ◽  
Rotationally Symmetric ◽  
Functional Differential ◽  
The Stability

The generalized pantograph equation y′(t) = Ay(t) + By(qt) + Cy′(qt), y(0) = y0, where q ∈ (0, 1), has numerous applications, as well as being a useful paradigm for more general functional-differential equations with monotone delay. Although many special cases have been already investigated extensively, a general theory for this equation is lacking–its development and exposition is the purpose of the present paper. After deducing conditions on A, B, C ∈ ℂd×d that are equivalent to well-posedness, we investigate the expansion of y in Dirichlet series. This provides a very fruitful form for the investigation of asymptotic behaviour, and we duly derive conditions for limt⋅→∞y(t) = 0. The behaviour on the stability boundary possesses no comprehensive explanation, but we are able to prove that, along an important portion of that boundary, y is almost periodic and, provided that q is rational, it is almost rotationally symmetric. The paper also addresses itself to a detailed analysis of the scalar equation y′(t) = by(qt), y(0) = 1, to high-order pantograph equations, to a phenomenon, similar to resonance, that occurs for specific configurations of eigenvalues of A, and to the equation Y′(t) = AY(t) + Y(qt) B, Y(0) = Y0.

On the Stability of Linear Stochastic Systems with Additive Noise

1986 ◽  
Vol 19 (5) ◽  
pp. 157-160
Author(s):  
M.M. Konstantinov ◽  
N.D. Christov ◽  
P.Hr. Petkov
Keyword(s):  
Additive Noise ◽  
Stochastic Systems ◽  
Linear Stochastic Systems ◽  
The Stability
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