Lyapunov Stability of a Class of Distributed Parameter Systems

1971 ◽  
Vol 93 (2) ◽  
pp. 79-85
Author(s):  
A. Frank D’Souza
Keyword(s):  
Differential Operator ◽  
Green’S Function ◽  
Green's Function ◽  
Linear Differential Operator ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Partial Differential ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Function Matrix

The mathematical model of dynamical systems is represented as an initial-boundary value problem described by nonlinear vector-matrix valued partial differential equations. The linear partial differential operator associated with the nonlinear system is restricted to a time-invariant operator with domain dense in Hilbert space. New stability results reported in this paper show the existence of quadratic Lyapunov functions that yield both necessary and sufficient conditions for asymptotic stability of linear systems satisfying certain restrictions and the use of these forms for the stability investigation of a class of nonlinear systems. The proofs of the stability theorems employ the spectral representation of the Green’s function matrix of the associated linear differential operator. Therefore, in the earlier part of the paper well-known properties of linear operators are stated in order to express the Green’s function matrix in the form of spectral expansion.

scholarly journals Analysis of the interaction of thermoacoustic modes with a Green’s function approach

2018 ◽  
Vol 10 (4) ◽  
pp. 326-336 ◽  
Author(s):  
Alessandra Bigongiari ◽  
Maria Heckl
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Perturbation ◽  
Time History ◽  
Function Approach ◽  
Stability Maps ◽  
Flame Describing Function ◽  
The Stability ◽  
Dynamics Simulations ◽  
Fast Prediction

In this paper, we will present a fast prediction tool based on a one-dimensional Green's function approach that can be used to bypass numerically expensive computational fluid dynamics simulations. The Green’s function approach has the advantage of providing a clear picture of the physics behind the generation and evolution of combustion instabilities. In addition, the method allows us to perform a modal analysis; single acoustic modes can be treated in isolation or in combination with other modes. In this article, we will investigate the role of higher-order modes in determining the stability of the system. We will initially produce the stability maps for the first and second mode separately. Then the time history of the perturbation will be computed, where both the modes are present. The flame will be modelled by a generic Flame Describing Function, i.e. by an amplitude-dependent Flame Transfer Function. The time-history calculations show the evolution of the two modes resulting from an initial perturbation; both transient and limit-cycle oscillations are revealed. Our study represents a first step towards the modelling of nonlinearity and non-normality in combustion processes.

Eigenvalue problems for the wave equation with strong damping

1997 ◽  
Vol 127 (4) ◽  
pp. 755-771 ◽  
Author(s):  
Pedro Freitas
Keyword(s):  
Stationary Solution ◽  
Wave Equations ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Linear Operators ◽  
Stability And Instability ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Strongly Damped

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

Dynamical analysis of cellular networks based on the Green's function matrix

2009 ◽  
Vol 261 (2) ◽  
pp. 248-259 ◽  
Author(s):  
Thanneer M. Perumal ◽  
Yan Wu ◽  
Rudiyanto Gunawan
Keyword(s):  
Green’S Function ◽  
Cellular Networks ◽  
Green's Function ◽  
Dynamical Analysis ◽  
Function Matrix
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