scholarly journals Addendum to ‘On an index policy for restless bandits'

1991 ◽  
Vol 23 (2) ◽  
pp. 429-430 ◽  
Author(s):  
Richard R. Weber ◽  
Gideon Weiss
Keyword(s):  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Index Policy ◽  
Fluid Approximation ◽  
Globally Asymptotically Stable ◽  
Restless Bandits

We show that the fluid approximation to Whittle's index policy for restless bandits has a globally asymptotically stable equilibrium point when the bandits move on just three states. It follows that in this case the index policy is asymptotic optimal.

scholarly journals Addendum to ‘On an index policy for restless bandits'

1991 ◽  
Vol 23 (02) ◽  
pp. 429-430 ◽  
Author(s):  
Richard R. Weber ◽  
Gideon Weiss
Keyword(s):  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Index Policy ◽  
Fluid Approximation ◽  
Globally Asymptotically Stable ◽  
Restless Bandits

We show that the fluid approximation to Whittle's index policy for restless bandits has a globally asymptotically stable equilibrium point when the bandits move on just three states. It follows that in this case the index policy is asymptotic optimal.

scholarly journals Dynamics of transposable elements under the selection model

1999 ◽  
Vol 74 (2) ◽  
pp. 159-164 ◽  
Author(s):  
A. TSITRONE ◽  
S. CHARLES ◽  
C. BIÉMONT
Keyword(s):  
Equilibrium Point ◽  
Dynamic Characteristics ◽  
Time Scale ◽  
Copy Number ◽  
Stable Equilibrium ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Weak Selection ◽  
Strong Selection ◽  
Long Time

We examine an analytical model of selection against the deleterious effects of transposable element (TE) insertions in Drosophila, focusing attention on the asymptotic and dynamic characteristics. With strong selection the only asymptotically stable equilibrium point corresponds to extinction of the TEs. With very weak selection a stable and realistic equilibrium point can be obtained. The dynamics of the system is fast for strong selection and slow, on the human time scale, for weak selection. Hence weak selection acts as a force that contributes to the stabilization of mean TE copy number. The consequence is that under weak selection, and ‘out-of-equilibrium’ situation can be maintained for a long time in populations, with mean TE copy number appearing stabilized.

scholarly journals Markus–Yamabe Conjecture for Asymptotic Stability of Planar Piecewise Linear Systems

2021 ◽  
Author(s):  
Yuhong Zhang ◽  
Xiao-Song Yang
Keyword(s):  
Linear Systems ◽  
Stable Equilibrium ◽  
Piecewise Linear ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Piecewise Linear Systems ◽  
Straight Line ◽  
Straight Lines ◽  
Discontinuous Piecewise Linear Systems

Abstract We present in this paper a detailed study on the Markus–Yamabe conjecture in planar piecewise linear systems. We consider discontinuous piecewise linear systems with two zones separated by a straight line, in which every subsystem is asymptotically stable. We prove the existence of limit cycles under explicit parameter conditions and give more different counterexamples to the Markus-Yamabe conjecture in addition to the counterexamples given by Llibre and Menezes. In particular, we consider continuous planar piecewise linear systems. For such a system with n + 1 zones separated by n parallel straight lines in phase space, we prove that if each of subsystems is asymptotically stable, then this system has a globally asymptotically stable equilibrium point, therefore the Markus–Yamabe conjecture still holds. Some examples are given to illustrate the main results.Mathematics Subject Classification (2020) 34C05 · 34C07 · 37G15

scholarly journals Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 336
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko
Keyword(s):  
Optimal Control ◽  
Equilibrium Point ◽  
Trajectory Optimization ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Control Object ◽  
Symbolic Regression ◽  
Optimal Location ◽  
Stable Equilibrium Point ◽  
Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

scholarly journals Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Qilin Sun ◽  
Lequan Min
Keyword(s):  
Drug Resistance ◽  
Antiretroviral Therapy ◽  
Hiv Infection ◽  
Equilibrium Point ◽  
Infection Rate ◽  
Reproductive Number ◽  
Infection Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Hiv Rna

This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive numberR0of the model is less than one, then the infection-free equilibrium point of the model is globally asymptotically stable; ifR0of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ anti-HIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive numberR0<1then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’sR0<1, this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.

The Qualitative Analysis of Two Populations Commensalisms Model

2012 ◽  
Vol 524-527 ◽  
pp. 3705-3708
Author(s):  
Guang Cai Sun
Keyword(s):  
Qualitative Analysis ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Mathematics Model ◽  
Stable Equilibrium Point ◽  
The Stability ◽  
Two Populations

This paper deals with the mathematics model of two populations Commensalisms symbiosis and the stability of all equilibrium points the system. It has given the conclusion that there is only one stable equilibrium point the system. This paper also elucidates the biology meaning of the model and its equilibrium points.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

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