Global Stability Results Research Articles

ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

AbstractWe study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function

In this article, we consider a predator–prey system with constant rate of harvesting, which exhibits Hopf and Bogdanov–Takens bifurcations under certain parametric conditions. The parametric space under which the system enters into Hopf bifurcation is investigated. By constructing suitable Lyapunov function, global stability results are obtained. Here, death rate and harvesting rate are taken as the Bogdanov–Takens bifurcation parameters. The canonical form of Bogdanov–Takens bifurcation is derived with the use of repeated nonlinear analytic transformation of coordinates. Later, we include the spatiotemporal effect on the same system and observed some relevant outcomes like Turing pattern, Turing–Bogdanov–Takens bifurcation, Turing–Hopf bifurcation and asynchrony of predator and prey in the space. The present study renders important tools for investigations of the dynamics of biotic organisms (predator and prey) for the management and control of overharvesting. Extensive numerical examples are given in support of the physical existence of the model system under consideration.

Expectational stability in aggregative games

In aggregative games individuals’ payoffs depend on players’ own contributions (or strategies) and on the aggregate contribution of other players. The replacement function, which defines the optimal contribution of a player in an aggregate contribution, is a convenient tool to analyze aggregative games. In this paper, the replacement function is used to define an adjustment process of expectations with respect to the aggregate strategy of the game. The Nash equilibrium of the game is interpreted as the rational expectations equilibrium (REE) of the dynamical system defined by that adjustment process. The expectational stability of the REE is analyzed and its local stability is characterized in terms of the fundamentals and the REE itself. Stronger results of global stability can be obtained when the model is applied to specific aggregative games with explicit payoff functions. Examples of Cournot oligopoly and public goods provision games are presented to illustrate global stability.

The Landau Equation with the Specular Reflection Boundary Condition

The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi–Nash–Moser theory for the kinetic Fokker–Planck equations (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem.

On the role of the epithelium in a model of sodium exchange in renal tubules

In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle’s loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding interstitium. In a specific range of parameters, we show that explicitly accounting for transport across the apical and basolateral membranes of epithelial cells, instead of assuming a single barrier, affects the axial concentration gradient, an essential determinant of the urinary concentrating capacity. We present the solution related to the stationary system, and we perform numerical simulations to understand the physiological behaviour of the system. We prove that when time grows large, our dynamic model converges towards the stationary system at an exponential rate. In order to prove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.

A Novel Tuning Method of PD With Gravity Compensation Controller for Robot Manipulators

Proportional-Derivative (PD) control is one of the most widely used controllers, especially for robot manipulators. When the robot presents gravitational terms, PD control cannot guarantee position convergence, therefore compensation is required such as PD with gravity compensation, PD+G. PD+G control requires knowledge of the gravitational term and there exist several results that prove global asymptotic stability. However, there is no method to tune the PD gains. In this work, a novel method to tune the PD+G controller is proposed. The tuning method is obtained using the global asymptotic stability result of the La Salle's theorem and robot dynamics properties. A comparison between previous works is realized via simulations and experiments to verify our approach. The results show fast and smooth convergence to the desired reference without overshoots.

Sliding mode strategies for monitoring and compensation of cyber-attacks to bCyber-Physical Systems

In this paper, the problem of detection and reconstruction of cyber-attacks in linear cyber-physical systems is considered. The class of cyber-attacks described in this paper can corrupt the states or the outputs of a cyber-physical system. An attack monitor based on High-Order sliding mode is proposed to reconstruct the cyber-attack. A First Order Approximation Filter is proposed to ensure global stability and convergence results. Using sliding mode techniques, an attack compensation is developed for square plants, guarateeing finite time convergence to the output tracking while rejecting the effects of the cyber-attack.

Study of the global asymptotic stability in nonlinear neutral integro-dynamic

The main purpose of this paper is to establish the global asymptotic stability of the zero solution of a class of nonlinear neutral integro-dynamic equations in C1 rd. Global asymptotic stability results are based on the Banach fixed point theorem. The results obtained here extend the work of Huang, Zhao and Liu [19].

On Global Asymptotic Stability for the LSW Model with Subcritical Initial Data

The main result of the paper is a global asymptotic stability result for solutions to the Lifschitz-Slyozov-Wagner (LSW) system of equations. This extends some local asymptotic stability results of Niethammer-Vel\'{a}zquez (2006). The method of proof is along similar lines to the one used in a previous paper of the authors. This previous paper proves global asymptotic stability for a class of infinite dimensional dynamical systems for which no Lyapounov function is (apparently) available.

On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

Global stability for an age-structured multistrain virus dynamics model with humoral immunity

In this paper, we present an age-structured multistrain mathematical model for infectious disease in vivo with infection age of cells. The model considers strain specific immune variables and the effect of absorption of pathogens into uninfected cells. By the construction of a Lyapunov functional, we prove the global stability results of the model, and show that several strains can survive at an equilibrium. We present full mathematical details of the proof of the global stability. The proof contains persistence arguments.

Global solutions to three-dimensional generalized MHD equations with large initial data

We investigate the initial value problem for the three-dimensional generalized incompressible MHD equations. Firstly, global stability result is established by energy method in the Fourier space. Then for a class of large initial data, global solutions are obtained in the critical function space $$\mathcal {X}^{1-2\alpha }$$ by global stability result.

New stability results for Takagi–Sugeno fuzzy Cohen–Grossberg neural networks with multiple delays

This work focuses on global asymptotic stability of Takagi–Sugeno fuzzy Cohen–Grossberg neural networks with multiple time delays. By using the standard Lyapunov stability techniques and nonsingular M-matrix condition of matrices together with employing the nonlinear Lipschitz activation functions, a new easily verifiable sufficient criterion is obtained to guarantee global asymptotic stability of the Cohen–Grossberg neural network model which is represented by a Takagi–Sugeno fuzzy model. A constructive numerical example is studied to demonstrate the effectiveness of the proposed theoretical results. This numerical example is also used to make a comparison between the global stability condition obtained in this study and some of previously published global stability results. This comparison reveals that the condition we propose establishes a novel and alternative stability result for Takagi–Sugeno fuzzy Cohen–Grossberg neural networks of this class.

Impulsive Control of Nonlinear Systems With Time-Varying Delay and Applications.

Impulsive control of nonlinear delay systems is studied in this paper, where the time delays addressed may be the constant delay, bounded time-varying delay, or unbounded time-varying delay. Based on the impulsive control theory and some analysis techniques, a new theoretical result for global exponential stability is derived from the impulsive control point of view. The significance of the presented result is that the stability can be achieved via the impulsive control at certain impulse points despite the existence of impulsive perturbations which causes negative effect to the control. That is, the impulsive control provides a super performance to allow the existence of impulsive perturbations. In addition, we apply the theoretical result to the problem of impulsive control of delayed neural networks. Some results for global exponential stability and synchronization control of neural networks with time delays are derived via impulsive control. Three illustrated examples are given to show the effectiveness and distinctiveness of the proposed impulsive control schemes.

Mixing in Reaction-Diffusion Systems: Large Phase Offsets

We consider Reaction-Diffusion systems on $${\mathbb{R}}$$ , and prove diffusive mixing of asymptotic states $${u_0(kx - \phi_{\pm}, k)}$$ , where u0 is a spectrally stable periodic wave. Our analysis is the first to treat arbitrarily large phase-offsets $${\phi_d = \phi_{+}- \phi_{-}}$$ , so long as this offset proceeds in a sufficiently regular manner. The offset $${\phi_d}$$ completely determines the size of the asymptotic profiles in any topology, placing our analysis in the large data setting. In addition, the present result is a global stability result, in the sense that the class of initial data considered is not near the asymptotic profile in any sense. We prove the global existence, decay, and asymptotic self-similarity of the associated wavenumber equation. We develop a functional analytic framework to handle the linearized operator around large Burgers profiles via the exact integrability of the underlying Burgers flow. This framework enables us to prove a crucial, new mean-zero coercivity estimate, which we then combine with a nonlinear energy method.

Global Stabilization of a Class of Nonlinear Reaction-Diffusion Partial Differential Equations by Boundary Feedback

This paper provides global exponential stabilization results by means of boundary feedback control for one dimensional nonlinear unstable reaction-diffusion partial differential equations (PDEs) with nonlinearities of superlinear growth. The class of systems studied are parabolic PDEs with nonlinear reaction terms that provide “damping” when the norm of the state is large (the class includes reaction-diffusion PDEs with polynomial nonlinearities). The case of Dirichlet actuation at one end of the domain is considered and a Control-Lypunov Functional construction is applied in conjunction with Stampacchia's truncation method. The paper also provides several important auxiliary results, among which is an extension of Wirtinger's inequality, used here for the construction of the Control-Lyapunov functional.

Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations

This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\mathbb{R}^d}K(y)g(v(t-h,x-y))dy, x \in \mathbb{R}^d,\ t >0;$ where $h>0$ and $d\in\mathbb{Z}_+$. We give two general results for $d\geq1$: on the global stability of semi-wavefronts in $L^p$-spaces with unbounded weights and the local stability of planar wavefronts in $L^p$-spaces with bounded weights. We also give a global stability result for $d = 1$ which yields to the global stability in Sobolev spaces with bounded weights. Here $g$ is not assumed to be monotone and the kernel $K$ is not assumed to be symmetric, therefore non-monotone semi-wavefronts and backward semi-wavefronts appear for which we show their stability. In particular, the global stability of critical wavefronts is stated.

Stability Analysis of Periodic Solutions of Some Duffing’s Equations

In this paper, some stability results were reviewed. A suitable and complete Lyapunov function for the hard spring model was constructed using the Cartwright method. This approach was compared with the existing results which confirmed a superior global stability result. Our contribution relies on its application to high damping door constructions. (2010 Mathematics Subject Classification: 34B15, 34C15, 34C25, 34K13.)

Recursive dynamics in a class of systems on q-timescales with unbounded delays

This work deals with a class of retarded equations on q-time scales which involves unbounded delays. A global exponential stability result is obtained and a dynamic of the recursive steps due to the delay function is discussed. A formal representation of natural numbers with respect to an unbounded delay function is featured with an asymptotic behavior of the recursion in an extended class of retarded equations.

Stabilization of a class of nonlinear parametrized systems by a flexible switching adaptive backstepping

In this paper, we consider global adaptive feedback control of nonlinear systems with unknown parameters entering nonlinearly. Such unknown parameters are also not required to lie in a known compact set. Unlike previous results, our proposed adaptive controller is a new double dynamical switching-type controller in which the controller parameter is tuned in a flexible switching manner via a monotonically decreasing switching logic and the controller combines the traditional adaptive theorem with the switching scheme perfectly. Global stability results of the closed-loop system have been proved.