scholarly journals Asymptotic Stability for Some Types of Nonlinear Fractional Order Differential-Algebraic Control Systems

2021 ◽  
pp. 623-638
Author(s):  
Sameer Qasim Hasan
Keyword(s):  
Feedback Control ◽  
Asymptotic Stability ◽  
Control Systems ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Stable Solution ◽  
Asymptotically Stable ◽  
Fractional Differential ◽  
Necessary And Sufficient ◽  
Asymptotically Stable Solution

The aim of this paper is to study the asymptotically stable solution of nonlinear single and multi fractional differential-algebraic control systems, involving feedback control inputs, by an effective approach that depends on necessary and sufficient conditions.

scholarly journals Asymptotic stability of a full term linear difference equation with two parameters

2015 ◽  
Vol 63 (1) ◽  
pp. 283-290
Author(s):  
Petr Tomášek
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Full Term ◽  
Asymptotically Stable ◽  
Term Linear ◽  
Asymptotic Stability Region ◽  
Two Parameters ◽  
Necessary And Sufficient

Abstract The paper introduces an efficient form of necessary and sufficient conditions for a special full term linear difference equation with two real parameters to be asymptotically stable. The result is obtained utilizing the Schur Cohn criterion. The asymptotic stability region in the parameters plane is also illustrated in the paper.

scholarly journals Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

2021 ◽  
Vol 24 (1) ◽  
pp. 225-253
Author(s):  
Oana Brandibur ◽  
Eva Kaslik
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Order ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Fractional Order Differential Equations ◽  
Stability And Instability ◽  
Instability Regions ◽  
Necessary And Sufficient

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

Global stability and optimal control of a two-patch tuberculosis epidemic model using fractional-order derivatives

2020 ◽  
Vol 13 (03) ◽  
pp. 2050008
Author(s):  
Hossein Kheiri ◽  
Mohsen Jafari
Keyword(s):  
Optimal Control ◽  
Effective Treatment ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Maximum Level ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
Fractional Differential ◽  
The Impact

In this paper, we propose a fractional-order and two-patch model of tuberculosis (TB) epidemic, in which susceptible, slow latent, fast latent and infectious individuals can travel freely between the patches, but not under treatment infected individuals, due to medical reasons. We obtain the basic reproduction number [Formula: see text] for the model and extend the classical LaSalle’s invariance principle for fractional differential equations. We show that if [Formula: see text], the disease-free equilibrium (DFE) is locally and globally asymptotically stable. If [Formula: see text] we obtain sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. We extend the model by inclusion the time-dependent controls (effective treatment controls in both patches and controls of screening on travel of infectious individuals between patches), and formulate a fractional optimal control problem to reduce the spread of the disease. The numerical results show that the use of all controls has the most impact on disease control, and decreases the size of all infected compartments, but increases the size of susceptible compartment in both patches. We, also, investigate the impact of the fractional derivative order [Formula: see text] on the values of the controls ([Formula: see text]). The results show that the maximum levels of effective treatment controls in both patches increase when [Formula: see text] is reduced from 1, while the maximum level of the travel screening control of infectious individuals from patch 2 to patch 1 increases when [Formula: see text] limits to 1.

scholarly journals Stability conditions for linear continuous-time fractional-order state-delayed systems

2016 ◽  
Vol 64 (1) ◽  
pp. 3-7
Author(s):  
M. Busłowicz
Keyword(s):  
Asymptotic Stability ◽  
Fractional Order ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
The State ◽  
Necessary And Sufficient Conditions ◽  
Order System ◽  
State Matrix ◽  
The Stability ◽  
Necessary And Sufficient

Abstract The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

Positive stable realizations of discrete-time linear systems

2012 ◽  
Vol 60 (3) ◽  
pp. 605-616
Author(s):  
T. Kaczorek
Keyword(s):  
Transfer Function ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Numerical Examples ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

Abstract The problem of existence and determination of the set of positive asymptotically stable realizations of a proper transfer function of linear discrete-time systems is formulated and solved. Necessary and sufficient conditions for existence of the set of the realizations are established. A procedure for computation of the set of realizations are proposed and illustrated by numerical examples.

Stabilizing Feedback Controls for Nonlinear Hamiltonian Systems and Nonconservative Bilinear Systems in Elasticity

1982 ◽  
Vol 104 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. N. Singh
Keyword(s):  
Asymptotic Stability ◽  
Hamiltonian Systems ◽  
Attitude Control ◽  
Sufficient Conditions ◽  
Bilinear Systems ◽  
Feedback Controls ◽  
Control Laws ◽  
Nonlinear Maps ◽  
The Stability ◽  
Necessary And Sufficient

Using the invariance principle of LaSalle [1], sufficient conditions for the existence of linear and nonlinear control laws for local and global asymptotic stability of nonlinear Hamiltonian systems are derived. An instability theorem is also presented which identifies the control laws from the given class which cannot achieve asymptotic stability. Some of the stability results are based on certain results for the univalence of nonlinear maps. A similar approach for the stabilization of bilinear systems which include nonconservative systems in elasticity is used and a necessary and sufficient condition for stabilization is obtained. An application to attitude control of a gyrostat Satellite is presented.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

scholarly journals Positive 2D Discrete-Time Linear Lyapunov Systems

2009 ◽  
Vol 19 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Przemysław Przyborowski ◽  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Necessary And Sufficient ◽  
Time Linear

Positive 2D Discrete-Time Linear Lyapunov SystemsTwo models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.

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