Absolute exponential stability of nonlinear systems of differential equations with lag

Constructive sufficient conditions of absolute exponential stability for a class of nonlinear systems of differential equations with lag are found.     

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4⁺ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12 : 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4⁺ T cells. Copyright © 2014 John Wiley & Sons, Ltd.     

Robust exponential stability of switched delay interconnected systems under arbitrary switching

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.     

Eventual stability properties in a non-autonomous model of population dynamics

We prove that (λ^∗,C/λ^∗) is an eventually uniform-asymptotically stable point in the large of the system

     

A nonsmooth Lyapunov function and stability for ODE’s of Caratheodory type

The problem of Lyapunov stability for systems of ODE’s of Caratheodory type is considered. It is proved that without the necessity of calculating Dini or Clarke generalized gradients, the locally Lipschitz Lyapunov function can follow different kinds of stability.     

Three-dimensional time-varying nonlinear systems containing a Hamilton system

In this paper the following three-dimensional nonlinear system is considered:

     

Input-to-state stability of Runge–Kutta methods for nonlinear control systems

In the paper we introduce input-to-state stability (ISS) of Runge–Kutta methods for control systems. The ISS properties of Runge–Kutta methods are studied for linear control systems and nonlinear control systems, respectively. The previously reported results in literature are special cases of ISS of Runge–Kutta methods.     

Global exponential stability in DCNNs with distributed delays and unbounded activations

We study delayed cellular neural networks (DCNNs) whose state variables are governed by nonlinear integrodifferential differential equations with delays distributed continuously over unbounded intervals. The networks are designed in such a way that the connection weight matrices are not necessarily symmetric, and the activation functions are globally Lipschitzian and they are not necessarily bounded, differentiable and monotonically increasing. By applying the inequality pa^p-1b?(p-1)a^p+b^p${\mathit{pa}}^{p-1}b?(p-1)a^p+b^p$, where p   denotes a positive integer and a,b$a,b$ denote nonnegative real numbers, and constructing an appropriate form of Lyapunov functionals we obtain a set of delay independent and easily verifiable sufficient conditions under which the network has a unique equilibrium which is globally exponentially stable. A few examples added with computer simulations are given to support our results.     

Exponential stability for linear skew-product flows

We give general characterizations for uniform exponential stability of linear skew-product flows. We present a unified treatment for discrete and integral conditions for uniform exponential stability. As applications, for the particular case of evolution families, we generalize some results due to Przyluski, Rolewicz and Zabczyk.     

Novel delay-dependent asymptotical stability of neutral systems with nonlinear perturbations

This paper address the asymptotical stability of neutral systems with nonlinear perturbations. Some novel delay-dependent asymptotical stability criteria are formulated in terms of linear matrix inequalities (LMIs). The resulting delay-dependent stability criteria are less conservative than the previous ones, owing to the introduction of free-weighting matrices, based on a class of novel augment Lyapunov functionals. Numerical examples are given to demonstrate that the derived conditions are much less conservative than those given in the literature.     

Aggregation and stability analysis of nonlinear complex systems

The problem of stability of large-scale systems in critical cases is investigated. New form of aggregation for essentially nonlinear complex systems is suggested. With the help of this form the sufficient conditions of asymptotic stability are determined. The results obtained are used for the stability analysis of complex systems by the nonlinear approximation and for the investigation of absolute stability conditions for a certain class of nonlinear systems.     

Stability analysis of neutral type systems in Hilbert space

The asymtoptic stability properties of neutral type systems are studied mainly in the critical case when the exponential stability is not possible. We consider an operator model of the system in Hilbert space and use recent results on the existence of a Riesz basis of invariant finite-dimensional subspaces in order to verify its dissipativity. The main results concern the conditions of asymptotic non-exponential stability. We show that the property of asymptotic stability is not determinated only by the spectrum of the system but essentially depends on the geometric spectral characteristic of its main neutral term. Moreover, we present an example of two systems of neutral type which have both the same spectrum in the open left-half plane and the main neutral term but one of them is asymptotically stable while the other is unstable.     

Positivity and stability for some partial functional differential equations

The aim of this work is to study the stability for some linear partial functional differential equations. We assume that the linear part is non-densely defined and satisfies the Hille-Yosida condition. Using the positiveness, we give nessecary and sufficient conditions independently of the delay to ensure the uniform exponential stability of the solution semigroup. An application is given for a reaction diffusion equation with several delays. RID="h1" ID="h1"This work is supported by the Moroccan Grant PARS MI 36 and TWAS Grant under contract: No. 00-412 RG/MATHS/AF/AC.     

Stability analysis for Lotka–Volterra systems based on an algorithm of real root isolation

Based on the theory of monotone flows of solutions of systems of differential equations, the Routh–Hurwitz theorem and a real root isolation algorithm of multivariate polynomials are applied to a class of Lotka–Volterra diffusion systems. An algorithm to determine the location of equilibria and the stability of the nonlinear dynamics systems is implemented in Maple.     

The converse Lyapunov’s theorem for ODEs of Carathéodory type

The converse problem of Lyapunov stability for systems of ODEs of Caratheodory type is considered. It is proved that if the right hand side of an ODE satisfies only the Osgood condition, the uniform stability of the origin is sufficient to the existence of a locally Lipschitz continuous Lyapunov function. Actually, the uniform stability is equivalent to the robust stability in this case. Moreover, as an auxiliary result, the generalization of the famous Gronwall-Bellman-Bihari inequality is also proved.     

Poisson stability for impulsive semidynamical systems

In this paper, the concept of Poisson stability is investigated for impulsive semidynamical systems. Recursive properties are also investigated.     

Robust stability and stabilization methods for a class of nonlinear discrete-time delay systems

This paper establishes new robust delay-dependent stability and stabilization methods for a class of nonlinear discrete-time systems with time-varying delays. The parameter uncertainties are convex-bounded and the unknown nonlinearities are time-varying perturbations satisfying Lipschitz conditions in the state and delayed-state. An appropriate Lyapunov functional is constructed to exhibit the delay-dependent dynamics and compensate for the enlarged time-span. The developed methods for stability and stabilization eliminate the need for over bounding and utilize smaller number of LMI decision variables. New and less conservative solutions to the stability and stabilization problems of nonlinear discrete-time system are provided in terms of feasibility-testing of new parametrized linear matrix inequalities (LMIs). Robust feedback stabilization methods are provided based on state-measurements and by using observer-based output feedback so as to guarantee that the corresponding closed-loop system enjoys the delay-dependent robust stability with an L₂ gain smaller that a prescribed constant level. All the developed results are expressed in terms of convex optimization over LMIs and tested on representative examples.     

General theorems for stability and boundedness for nonlinear functional discrete systems

We consider the nonlinear functional discrete system

     

Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.