scholarly journals RESEARCH OF STABILIZATION CONDITIONS AND ROBUST STABILITY OF DISCRETE ALMOST CONSERVATIVE SYSTEMS

2017 ◽  
Vol 3 ◽  
pp. 22-30
Author(s):  
Olena Teteriatnyk
Keyword(s):  
Small Parameter ◽  
Solvability Condition ◽  
Lyapunov Equation ◽  
Coefficient Matrix ◽  
Positive Definiteness ◽  
Asymptotically Stable ◽  
Controllable System ◽  
Controlled System ◽  
Multiple Eigenvalues ◽  
Conservative Systems

Conditions for the stabilizability of discrete almost conservative systems in which the coefficient matrix of a conservative part has no multiple eigenvalues are investigated. It is known that a controllable system will be stabilized if its coefficient matrix is asymptotically stable. The system stabilization algorithm is constructed on the basis of the solvability condition for the Lyapunov equation and the positive definiteness of P0 and Q1. This theorem shows how to find the parameters of a controlled system under which it will be asymptotically stable for sufficiently small values of the parameter e (P > 0, Q > 0). In addition, for a small parameter e that determines the almost conservatism of the system, an interval is found in which the conditions for its stabilizability are satisfied (Theorem 2).

On the dynamics of linear non-conservative systems

1976 ◽  
Vol 352 (1668) ◽  
pp. 25-40 ◽  
Keyword(s):  
Arbitrary Function ◽  
Matrix Equation ◽  
Positive Definiteness ◽  
Free Motion ◽  
Transient Vibration ◽  
Harmonic Motion ◽  
Conservative Systems ◽  
The Matrix

A number of results are presented, relating to the matrix equation Aq̣̈ + Bq̇ + Cq = Q(t) . It is not assumed that the system matrices A, B and C possess any of the familiar properties (of symmetry, skew symmetry or positive definiteness). These results relate to free motion in which Q(t) = 0, to forced harmonic motion in which Q(t) = ϕ e iωt and to transient vibration in which Q(t) is an arbitrary function of time.

On the reducibility of two-dimensional linear quasi-periodic systems with small parameter

2014 ◽  
Vol 35 (7) ◽  
pp. 2334-2352 ◽  
Author(s):  
JUNXIANG XU ◽  
XUEZHU LU
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Lebesgue Measure ◽  
Periodic System ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Two Dimensional ◽  
Constant Matrix ◽  
Small Parameters ◽  
Real Analytic

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.

scholarly journals Necessary condition for solvability of the control problem of an asynchronous spectrum of linear almost periodic systems with trivial averaging of the coefficient matrix

2019 ◽  
Vol 55 (2) ◽  
pp. 176-181 ◽  
Author(s):  
A. K. Demenchuk
Keyword(s):  
Control Problem ◽  
Solvability Condition ◽  
Additive Group ◽  
Coefficient Matrix ◽  
Periodic Systems ◽  
Necessary Condition ◽  
Almost Periodic ◽  
Real Numbers ◽  
Frequency Module ◽  
Target Set

We consider a linear control system with an almost periodic matrix of coefficients. The control has a form of feedback and is linear in phase variables. It is assumed that the feedback coefficient is almost periodic and its frequency modulus, i.e. the smallest additive group of real numbers, including all Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix.The following problem is formulated: choose such a control from an admissible set so that the closed system has almost periodic solutions, the frequency spectrum (a set of Fourier exponents) of which contains a predetermined subset, and the intersection of the solution frequency modules and the coefficient matrix is trivial. The problem is called the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies.The aim of the work aws to obtain a necessary solvability condition for the control problem of the asynchronous spectrum of linear almost periodic systems with trivial averaging of coefficient matrix The estimate of the power of the asynchronous spectrum was found in the case of trivial averaging of the coefficient matrix.

scholarly journals On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Schrödinger Equation ◽  
Small Parameter ◽  
Hamiltonian Systems ◽  
Schrodinger Equation ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system ẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.

CHAOTIFICATION OF DISCRETE-TIME DYNAMICAL SYSTEMS: AN EXTENSION OF THE CHEN–LAI ALGORITHM

2005 ◽  
Vol 15 (01) ◽  
pp. 109-117 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Design Method ◽  
Control Design ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Discrete Time Dynamical Systems ◽  
Uniformly Bounded

In this article, we propose and study an extension of the Chen–Lai algorithm for chaotification of discrete-time dynamical systems. The proposed method is a simple but mathematically rigorous feedback control design method that can gradually make all the Lyapunov exponents of the controlled system strictly positive for any given n-dimensional dynamical system that has a uniformly bounded Jacobian but otherwise could be originally nonchaotic or even asymptotically stable.

FEEDBACK CONTROL OF LYAPUNOV EXPONENTS FOR DISCRETE-TIME DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (07) ◽  
pp. 1341-1349 ◽  
Author(s):  
GUANRONG CHEN ◽  
DEJIAN LAI
Keyword(s):  
Dynamical Systems ◽  
Feedback Control ◽  
Lyapunov Exponents ◽  
Design Method ◽  
Design Procedure ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Dimensional Dynamical System ◽  
Convenient Technique ◽  
Discrete Time Dynamical Systems

A simple, yet mathematically rigorous feedback control design method is proposed in this paper, which can make all the Lyapunov exponents of the controlled system strictly positive, for any given n-dimensional dynamical system that could be originally nonchaotic or even asymptotically stable. The argument used is purely algebraic and the design procedure is completely schematic, with no approximations used throughout the derivation. This is a rigorous and convenient technique suggested as an attempt for anticontrol of chaotic dynamical systems, with explicit computational formulas derived for applications.

scholarly journals Stabilization of the T system by an integrable deformation

2020 ◽  
Vol 34 ◽  
pp. 03009
Author(s):  
Cristian Lăzureanu ◽  
Cristiana Căplescu
Keyword(s):  
Chaotic Behavior ◽  
Linear Control ◽  
Asymptotically Stable ◽  
Controlled System ◽  
Integrable Deformation ◽  
T System

In this paper, some deformations of the T system are constructed. In order to stabilize the chaotic behavior of the T system, we particularize these deformations obtaining some external linear control inputs. In each case, we prove that the controlled system is asymptotically stable.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

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