Global Stability Results Research Articles

A note on the wave equation controlled with a dynamic saturating boundary control

This paper studies the nonlinear systems obtained by considering a wave equation in closed loop with a nonlinear dynamical boundary controller. The controller is subject to a magnitude limitation and modeled by a linear ordinary differential equation with a saturation map in the input. The well-posedness of the obtained infinite-dimensional system is first studied and then two stability results are given. These two stability results apply for two cascade cases and give sufficient conditions for the asymptotic stability of the equilibrium. The well-posedness is proven by using nonlinear semigroups techniques, whereas the global asymptotic stability results are obtained by Lyapunov-based arguments in infinite-dimensional state space.

Multi-time scale control and optimization via averaging and singular perturbation theory: From ODEs to hybrid dynamical systems

Multi-time scale techniques based on singular perturbations and averaging theory are among the most powerful tools developed for the synthesis and analysis of feedback control algorithms. This paper introduces some of the recent advances in singular perturbation theory and averaging theory for continuous-time dynamical systems modeled as ordinary differential equations (ODEs), as well as for hybrid dynamical systems that combine continuous-time dynamics and discrete-time dynamics. Novel multi-time scale analytical tools based on higher-order averaging and singular perturbation theory are also discussed and illustrated via different examples. In the context of hybrid dynamical systems, a class of sufficient Lyapunov-based conditions for global stability results are also presented. The analytical tools are illustrated through various new architectures and algorithms within the context of adaptive and extremum-seeking systems. These tools are suitable for the study of model-free optimization and stabilization problems that require the synergistic use of continuous-time and discrete-time feedback. The paper aims to acquaint the reader with a range of modern tools for studying multi-time scale phenomena in optimization and control systems, providing some guidelines for future research in this field.

Qualitative analysis on the diffusive Holling–Tanner predator–prey model

We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steady-state. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel'kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1 .

The effects of evolution on the stability of competing species

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabilization occurs when , when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against complex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when . In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation.

Global stability in a diffusive predator–prey model of Leslie–Gower type

We consider a diffusive predator–prey model of Leslie–Gower type, and obtain a new global stability result by combining the Lyapunov function method and the transformation technique used in Qi and Zhu, (2016). Our result partially answers the question proposed in [Y. H. Du and S. B. Hsu, J. Differential Equations 203(2004) 331–364]. In addition, we extend the result to a class of diffusive systems with a more general type of reaction-terms.

Propagation and blocking in a two-patch reaction-diffusion model

This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.

On the stability of the Lp-curvature

It is known that the Lp-curvature of a smooth, strictly convex body in Rn is constant only for origin-centered balls when 1≠p>−n, and only for balls when p=1. If p=−n, then the L−n-curvature is constant only for origin-symmetric ellipsoids. We prove ‘local’ and ‘global’ stability versions of these results. For p≥1, we prove a global stability result: if the Lp-curvature is almost a constant, then the volume symmetric difference of K˜ and a translate of the unit ball B is almost zero. Here K˜ is the dilation of K with the same volume as the unit ball. For 0≤p<1, we prove a similar result in the class of origin-symmetric bodies in the L2-distance. In addition, for −n<p<0, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with ‘almost’ constant Lp-curvature is ‘almost’ the unit ball. For p=−n, we prove a global stability result in R2 and a local stability result for n>2 in the Banach-Mazur distance.

Dynamical Behavior of a Fractional Order Model for Within-Host SARS-CoV-2

The prime objective of the current study is to propose a novel mathematical framework under the fractional-order derivative, which describes the complex within-host behavior of SARS-CoV-2 by taking into account the effects of memory and carrier. To do this, we formulate a mathematical model of SARS-CoV-2 under the Caputo fractional-order derivative. We derived the conditions for the existence of equilibria of the model and computed the basic reproduction number R0. We used mathematical analysis to establish the proposed model’s local and global stability results. Some numerical resolutions of our theoretical results are presented. The main result of this study is that as the fractional derivative order increases, the approach of the solution to the equilibrium points becomes faster. It is also observed that the value of R0 increases as the value of β and πv increases.

Stability analysis of within-host SARS-CoV-2/HIV coinfection model.

The world has been suffering from the coronavirus disease 2019 (COVID‐19) since late 2019. COVID‐19 is caused by a virus called the severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2). The human immunodeficiency virus (HIV) coinfection with SARS‐CoV‐2 has been reported in many patients around the world. This has raised the alarm for the importance of understanding the dynamics of coinfection and its impact on the lives of patients. As in other pandemics, mathematical modeling is one of the important tools that can help medical and experimental studies of COVID‐19. In this paper, we develop a within‐host SARS‐CoV‐2/HIV coinfection model. The model consists of six ordinary differential equations. It depicts the interactions between uninfected epithelial cells, infected epithelial cells, free SARS‐CoV‐2 particles, uninfected CD4+ T cells, infected CD4+ T cells, and free HIV particles. We confirm that the solutions of the developed model are biologically acceptable by proving their nonnegativity and boundedness. We compute all possible steady states and derive their positivity conditions. We choose suitable Lyapunov functions to prove the global asymptotic stability of all steady states. We run some numerical simulations to enhance the global stability results. Based on our model, weak CD4+ T cell immune response or low CD4+ T cell counts in SARS‐CoV‐2/HIV coinfected patient increase the concentrations of infected epithelial cells and SARS‐CoV‐2 viral load. This causes the coinfected patient to suffer from severe SARS‐CoV‐2 infection. This result agrees with many studies which showed that HIV patients are at greater risk of suffering from severe COVID‐19 when infected. More studies are needed to understand the nature of SARS‐CoV‐2/HIV coinfection and the role of different immune responses during infection.

Global Stability of Fluid Flows Despite Transient Growth of Energy.

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct nonquadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the the energy stability threshold found by Orr in 1907 [The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid, Proc. R. Ir. Acad. Sect. A 27, 69 (1907)]. This is the first global stability result for any flow that surpasses the energy method.

Modelling the impact of health care providers in transmission dynamics of COVID-19

In this paper, a mathematical model is proposed and analysed to assess the impacts of health care providers in transmission dynamics of COVID-19. The stability theory of differential equations is used to examine a mathematical model. The results of both local and global stability of disease-free equilibrium points were determined by using Routh–Hurwitz criteria and Metzler matrix method which verified that was locally and globally asymptotically stable. Also, the endemic equilibrium point was determined by the Lyapunov function which showed that E∗ was globally asymptotically stable under strict conditions. The findings revealed that non-diagnosed and undetected health care providers seems to contribute to high spread of COVID-19 in a community. Also, it illustrates that an increase in the number of non-diagnostic testing rates of health care providers may result in high infection rates in the community and contaminations of hospitals’ equipment. Therefore, the particular study recommend that there is a necessity of applying early diagnostic testing to curtail the COVID-19 transmission in the health care providers’ community and reduce contaminations of hospital’s equipment.

Analysis of a continuously stirred two tank reactor cascade with Haldane kinetics

Biological reactors are employed in industrial applications to break down organic waste. We view the cascade of two open loop continuously stirred tank reactors with Haldane growth function as chemostats with bacterial inputs. A function of some of the reactor parameters is derived, the sign of which determines the maximum number of critical points a reactor can have. This allows us to determine the parameter combinations which ensure a reactor has only a single critical point for all bacterial removal rates (dilution rate plus death rate). Where a simple condition on the above function is confirmed to hold, if the first reactor in a cascade only has a single critical point for all bacterial removal rates then, the next reactor will also only have a single critical point for all bacterial removal rates. A global stability result is also given for some of these cases. A simple proof is given for the local stability of critical points of a reactor with a general class of bacterial growth functions, bacteria and substrate input, and a death rate. For the special case where the first reactor has zero bacteria input, we compare a two reactor cascade with a single reactor under various conditions, long and short residence times, and different death rates. This follows the pattern of similar papers that considered cascades using the Monod and Contois growth functions.

Global stability dynamics and sensitivity assessment of COVID-19 with timely-delayed diagnosis in Ghana

Abstract In this paper, we study the dynamical effects of timely and delayed diagnosis on the spread of COVID-19 in Ghana during its initial phase by using reported data from March 12 to June 19, 2020. The estimated basic reproduction number, ℛ0, for the proposed model is 1.04. One of the main focus of this study is global stability results. Theoretically and numerically, we show that the disease persistence depends on ℛ0. We carry out a local and global sensitivity analysis. The local sensitivity analysis shows that the most positive sensitive parameter is the recruitment rate, followed by the relative transmissibility rate from the infectious with delayed diagnosis to the susceptible individuals. And that the most negative sensitive parameters are: self-quarantined, waiting time of the infectious for delayed diagnosis and the proportion of the infectious with timely diagnosis. The global sensitivity analysis using the partial rank correlation coefficient confirms the directional flow of the local sensitivity analysis. For public health benefit, our analysis suggests that, a reduction in the inflow of new individuals into the country or a reduction in the inter community inflow of individuals will reduce the basic reproduction number and thereby reduce the number of secondary infections (multiple peaks of the infection). Other recommendations for controlling the disease from the proposed model are provided in Section 7.

Global Stability of a Three-Species System with Attractive Prey-Taxis

This paper reports the global asymptotic stability of a three-species predator-prey system involving the prey-taxis. With the assumptions, we establish the global asymptotic stability results of its equilibria, respectively. Our results illustrate that 1) the global asymptotic stability of the semi-trivial equilibrium does not involve the prey-taxis coefficients χ, ξ; 2) the global asymptotic stability of two boundary equilibria relies on a single prey-taxis coefficient χ and ξ, respectively; 3) the global asymptotic stability of the unique positive equilibrium depends on two prey-taxis coefficients χ and ξ.

Mathematical modeling of the dynamics of vector-borne diseases transmitted by mosquitoes : taking into account aquatic stages and gonotrophic cycle

Abstract In this paper, we formulate a mathematical model of vector-borne disease dynamics. The model is constructed by considering two models : a baseline model of vector population dynamics due to Lutambi et al. that takes into account the development of the aquatic stages and the female mosquitoes gonotrophic cycle and an SI-SIR model describing the interaction between mosquitoes and human hosts. We briefly study the baseline model of vectors dynamics and, for the transmission model, we explicitly compute the equilibrium points, and by using the method of Van den Driesshe and J. Watmough, we derive the basic reproduction number ℛ0. Otherwise, thanks to Lyapunov’s principle, Routh-Hurwitz criteria and a favorable result due to Vidyasagar, we establish the local and global stability results of the equilibrium points. Furthermore, we establish an interesting relationship between the mosquito reproduction number ℛ v and the basic reproduction number ℛ0. It then follows that aquatic stages and behavior of adult mosquitoes have a significant impact on disease transmission dynamics. Finally, some numerical simulations are carried out to support the theoretical findings of the study.

Rapid exponential stabilization by boundary state feedback for a class of coupled nonlinear ODE and $ 1-d $ heat diffusion equation

&lt;p style='text-indent:20px;'&gt;In this paper, we solve the problem of rapid exponential stabilization for coupled nonlinear ordinary differential equation (ODE) and &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 1-d $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; unstable linear heat diffusion. The control acts at a boundary of the heat domain and the heat equation enters in the ODE by Dirichlet connection. We show that the infinite dimensional backstepping transformation introduced recently for stabilization of coupled linear ODE-PDE can deal with a nonlinear ODE and obtain a global stabilization result. Our result is innovative and no similar result can be found in the literature as it combines the three following factors, i) nonlinear term in the ODE subsystem, ii) unstable PDE subsystem and iii) mixed boundary condition. Not only this, the techniques used in this work can provide answers to fundamental questions, such as the stabilization of coupled systems where both subsystems may contain nonlinear terms.&lt;/p&gt;

Quantifying the Landscape of Decision Making From Spiking Neural Networks.

The decision making function is governed by the complex coupled neural circuit in the brain. The underlying energy landscape provides a global picture for the dynamics of the neural decision making system and has been described extensively in the literature, but often as illustrations. In this work, we explicitly quantified the landscape for perceptual decision making based on biophysically-realistic cortical network with spiking neurons to mimic a two-alternative visual motion discrimination task. Under certain parameter regions, the underlying landscape displays bistable or tristable attractor states, which quantify the transition dynamics between different decision states. We identified two intermediate states: the spontaneous state which increases the plasticity and robustness of changes of minds and the “double-up” state which facilitates the state transitions. The irreversibility of the bistable and tristable switches due to the probabilistic curl flux demonstrates the inherent non-equilibrium characteristics of the neural decision system. The results of global stability of decision-making quantified by barrier height inferred from landscape topography and mean first passage time are in line with experimental observations. These results advance our understanding of the stochastic and dynamical transition mechanism of decision-making function, and the landscape and kinetic path approach can be applied to other cognitive function related problems (such as working memory) in brain networks.

Boundary sliding mode control of a system of linear hyperbolic equations: A Lyapunov approach

This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. The main difference with the existing literature is the definition of the sliding variable, given here by the gradient of a Lyapunov functional. We state and prove an existence theorem and a global asymptotic stability result. The efficiency of our feedback-law is illustrated by some numerical simulations relying on implicit schemes.

Mathematical modeling of fruit trees’ growth under scarce watering

As a consequence of the Global Warming, the decrease in water resources is a problem that affects the agricultural systems worldwide. In particular, the Global Warming has produced a sustained decrease in precipitation in Mediterranean-type climatic zones and the more frequent presence of droughts. A dry-and hot environment could negatively affect the physiology of the fruit tree, compromising its growth and development rates, especially, the yield of a fruit tree results from the interactions between genotype, climate, soil, crop management, and irrigation. This work theoretically studies the growth dynamics of fruit trees with a limited water resource by a mathematical model. A system of non-linear ordinary differential equations was used as a method to describe the growing phenomenon. A qualitative analysis of the system was carried out, and local and global stability results were obtained. The analysis showed that the limitation of water resources compromises the growth dynamics of the fruit tree, causing a total loss of the fruit tree.

Further improved global exponential stability result for neural networks with time-varying delay

Further improved global exponential stability result for neural networks with time-varying delay