scholarly journals Dynamics of Unilateral and Bilateral Control Systems with State Feedback for Renewable Resource Management

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Mingzhan Huang ◽  
Shouzong Liu ◽  
Xinyu Song ◽  
Lansun Chen
Keyword(s):  
Periodic Solution ◽  
Control System ◽  
Periodic Solutions ◽  
State Feedback ◽  
Control Strategies ◽  
Management Strategies ◽  
Sufficient Conditions ◽  
Renewable Resource ◽  
Bilateral Control ◽  
Unilateral Control

In this paper, mathematical models for the management of biological resources based on a given predator-prey relationship are proposed, and two types of control strategies, unilateral and bilateral control with impulsive state feedback, are studied. The existence of the order-1 homoclinic orbit, order-1 periodic solution, and bifurcation of homoclinic of the unilateral control system are obtained, and the attraction region of this system is also discussed. Besides, sufficient conditions for the existence and stability of the order-1 and order-2 periodic solutions of the bilateral control system are also gained. A series of numerical simulations including bifurcation diagrams of periodic solution are performed, which not only verify the theoretical results we get but also reveal some peculiar dynamical phenomena, such as the appearance of high-order periodic solutions and existence of parameter intervals with drastic order change of periodic solution. By comparing the two management strategies, our study encourages bilateral control rather than unilateral control for the risk of predator extinction.

scholarly journals Dynamics of a Guanaco-sheep Competitive System with Unilateral and Bilateral Control

2021 ◽  
Author(s):  
Jing Xu ◽  
Mingzhan Huang ◽  
Xinyu Song
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Control System ◽  
Control Strategy ◽  
Impulsive Control ◽  
Bilateral Control ◽  
Competitive System ◽  
Existence And Stability ◽  
Uncontrolled System ◽  
Unilateral Control

Abstract In this paper, based on a guanaco-sheep competitive system, we develop and analyze mathematical models with unilateral and bilateral control for the management of overgrazing. We first analyze the dynamics of the uncontrolled system. Then, we investigate the system with impulsive control by differential equation geometry theory. And we mainly prove the existence and stability of order-1 periodic solution for unilateral control system and order-2 periodic solution for bilateral control system. Comparing the unilateral and bilateral control strategy, we encourage bilateral control rather than unilateral control for the management of sheep species.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

scholarly journals On periodic solutions to autonomous retarded functional differential equations

1990 ◽  
Vol 41 (3) ◽  
pp. 347-354
Author(s):  
Zhanyuan Hou
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Retarded Functional Differential Equations ◽  
Retarded Functional Differential Equation ◽  
Functional Differential ◽  
Necessary And Sufficient

Under the assumption that Ca = C([−r, 0], Sn−1(a)) is positively invariant for a > 0, two necessary and sufficient conditions are obtained for an autonomous retarded functional differential equation to have a non-trivial periodic solution in Ca. Moreover, a feasible sufficient condition is given, which is better for n = 2 than that given by Dos Reis and Baroni.

scholarly journals PERIODIC SOLUTIONS OF A MODEL OF LOTKA-VOLTERRA WITH VARIABLE STRUCTURE AND IMPULSES

2015 ◽  
Vol 11 (6) ◽  
pp. 5317-5325
Author(s):  
Katya Dishlieva ◽  
Katya Dishlieva
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Variable Structure ◽  
Impulsive Effects ◽  
Volterra Model ◽  
Right Hand

We consider a generalized version of the classical Lotka Volterra model with differential equations. The version has a variable structure (discontinuous right hand side) and the solutions are subjected to the discrete impulsive effects. The moments of right hand side discontinuity and the moments of impulsive effects coincide and they are specific for each solution. Using the Brouwer fixed point theorem, sufficient conditions for the existence of periodic solution are found.

scholarly journals Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hongyun Yan ◽  
Yuanhua Qiao ◽  
Lijuan Duan ◽  
Ling Zhang
Keyword(s):  
Neural Networks ◽  
Lyapunov Function ◽  
Fractional Order ◽  
State Feedback ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
State Feedback Control ◽  
Bam Neural Networks ◽  
Linear State ◽  
The Stability

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

IMPULSIVE CONTROL STRATEGIES FOR PEST MANAGEMENT

2007 ◽  
Vol 15 (02) ◽  
pp. 235-260 ◽  
Author(s):  
HONG ZHANG ◽  
LANSUN CHEN ◽  
PAUL GEORGESCU
Keyword(s):  
Periodic Solution ◽  
Pest Management ◽  
Impulsive Control ◽  
Control Strategies ◽  
Integrated Management ◽  
Management Strategies ◽  
Critical Value ◽  
Globally Asymptotically Stable ◽  
Pest Eradication ◽  
Low Levels

In this paper, we propose two impulsive differential systems concerning biological and, respectively, integrated pest management strategies. In each case, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than a certain critical value. When the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a non-trivial periodic solution is also studied by means of numerical simulations. In the case in which a single control is used, one can only use the amount of infective pests which are periodically released in order to control pests at desirable low levels, while in the case in which integrated management is used, one can use the proportion of pests removed by means of spraying chemical pesticides together with the amount of infective pests which are periodically released to control pests at desirable low levels.

Periodic solutions of dissipative neutral differential systems with singular potential and P -Laplacian

2008 ◽  
Vol 45 (2) ◽  
pp. 251-271
Author(s):  
Yong Li ◽  
Bing Liu
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Topological Degree ◽  
Singular Potential ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Topological Degree Theory ◽  
Differential Systems

In this paper, by using topological degree theory and some analysis skill, we consider the periodic solutions for the dissipative neutral differential systems with singular potential and p -Laplacian: ( ϕp ( x ′( t ) − μx ′( t − τ1 )))′ + \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{d} {{dt}}$$ \end{document} grad G ( x ( t − τ2 )) = e ( t ). Sufficient conditions to guarantee the existence of periodic solution for the systems are obtained under having no restriction on the damping forces \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{d} {{dt}}$$ \end{document} grad F ( x ).

scholarly journals The Effect of Pulse Vaccination and Treatment on SIR Epidemic Model with Media Impact

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Zhi-Long He ◽  
Lin-Fei Nie
Keyword(s):  
Periodic Solution ◽  
Control Strategies ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Positive Order ◽  
Media Impact ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
State Dependent ◽  
Control Threshold

We propose a novel SIR epidemic dynamical control model with media impact, where the state dependent pulse vaccination and medication treatment control strategies are being introduced to prevent the spread of disease at different control threshold values. By using the geometry theory of differential equation and method of successor function, the existence of positive order-1 periodic solution is studied. Further, some sufficient conditions of the orbitally asymptotical stability for positive order-1 periodic solution are given by the analog Poincaré criterion. Furthermore, numerical simulations are carried to illustrate the feasibility of our main results presented here.

scholarly journals The Mathematical Study of Pest Management Strategy

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Jinbo Fu ◽  
Yanzhen Wang
Keyword(s):  
Mathematical Model ◽  
Periodic Solution ◽  
Pest Management ◽  
Management Strategy ◽  
State Feedback ◽  
Sufficient Conditions ◽  
State Feedback Control ◽  
Successor Function ◽  
Mathematical Study ◽  
The Mathematical Model

The theory of impulsive state feedback control is used to establish a mathematical model in the pest management strategy. Then, the qualitative analysis of the mathematical model was provided. Here, a successor function in the geometry theory of differential equations is used to prove the sufficient conditions for uniqueness of the 1-periodic solution. It proved the orbital asymptotic stability of the periodic solution. In addition, numerical analysis is used to discuss the application significance of the mathematical model in the pest management strategy.

PERIODIC SOLUTION FOR GENERALIZED HIGH-ORDER DELAY DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 03 (01) ◽  
pp. 31-43
Author(s):  
Zhibo Cheng ◽  
Jingli Ren ◽  
Stefan Siegmund
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Existence Of Periodic Solutions ◽  
Delay Differential ◽  
New Inequalities

In this paper we consider a generalized n-th order delay differential equation, by applying Mawhin's continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions. Moreover, an example is given to illustrate the results.

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