Stability Of Functional Differential Systems Research Articles

Event-triggered delayed impulsive control for functional differential systems on networks

This article investigates the stability of functional differential systems on networks adopting event-triggered multi-delayed impulsive control. We construct a new event-triggered mechanism (ETM) type using the Lyapunov function and the network topology. Combined with the ETM and multi-delayed impulsive control, in which the impulsive jumps are related to the current and past states, we propose a multi-delayed event-triggered impulsive control strategy. This controller will execute the control tasks only when the systems violate the preset ETM. In the view of the Razumikhin method and the graph theory, some criteria are given to avoiding Zeno behavior under this ETM and achieving exponential stability, which is related to the event-triggered parameters, network topology, and impulsive intensity. As an application of our theoretical results, a class of coupled oscillation systems is considered and numerical simulations are presented to verify the practicability and effectiveness.

Stability of functional differential systems applied to the model of testosterone regulation

In this paper we propose a method for stability studies of functional differential systems. The idea of our method is to reduce the analysis of an n-dimensional system to one for an (n+m)-dimensional system, where m is a natural number, to obtain stability and then to come back and make conclusions on the stability of the given n-dimensional system. As an example, a model describing testosterone by distributed inputs feedback control is considered. The aim of the regulation is to hold testosterone concentration above an appropriate level. The feedback control with integral term is proposed. We have to increase the testosterone level to the normal one. The control we proposed could destroy the stability of the model. That is why we have to choose the parameters of our distributed control, namely a dosage or intensity of assimilation of a medicine in a human body in such a form that the stability of our system is preserved. Thus the problem of regulation of testosterone level leads us to the stability analysis of the functional differential system describing a connection between the concentrations of hormones (GnRH), (LH), and testosterone (Te). Constructing the system, we discard the connections which seem nonessential. To estimate the effect of these connections is an important problem. We construct the Cauchy matrix of integro-differential system to estimate this influence.

On Problem of Partial Stability for Functional Differential Systems with Holdover

The theory of systems of functional differential equations is a significant and rapidly developing sphere of modern mathematics which finds extensive application in complex systems of automatic control and also in economic, modern technical, ecological, and biological models. Naturally, the problems arises of stability and partial stability of the processes described by the class of the equation. The article studies the problem of partial stability which arise in applications either from the requirement of proper performance of a system or in assessing system capability. Also very effective is the approach to the problem of stability with respect to all variables based on preliminary analysis of partial stability. We suppose that the system have the zero equilibrium position. A conditions are obtained under which the uniform stability (uniform asymptotic stability) of the zero equilibrium position with respect to the part of the variables implies the uniform stability (uniform asymptotic stability) of this equilibrium position with respect to the other, larger part of the variables, which include an additional group of coordinates of the phase vector. These conditions include: 1) the condition for uniform asymptotic stability of the zero equilibrium position of the "reduced" subsystem of the original system with respect to the additional group of variables; 2) the restriction on the coupling between the "reduced" subsystem and the rest parts of the system. Application of the obtained results to a problem of stabilization with respect to a part of the variables for nonlinear controlled systems is discussed.

Exponential Stability of Functional Differential Systems

We present a novel approach to exponential stability of functional differential systems. Our approach is relied upon the theory of positive linear functional differential systems and a comparison principle. Consequently, we get some comparison tests and explicit criteria for exponential stability of functional differential systems. Two examples are given to illustrate the obtained results.

Hyers-Ulam-Rassias Stability of Functional Differential Systems with Point and Distributed Delays

This paper investigates stability and asymptotic properties of the error with respect to its nominal version of a nonlinear time-varying perturbed functional differential system subject to point, finite-distributed, and Volterra-type distributed delays associated with linear dynamics together with a class of nonlinear delayed dynamics. The boundedness of the error and its asymptotic convergence to zero are investigated with the results being obtained based on the Hyers-Ulam-Rassias analysis.

Exponential stability of functional differential systems with impulsive effect on random moments

The model of nonlinear functional differential systems with impulsive effect on random moments is brought forward by authors in this paper. Then, by means of Liapunov's direct method coupled with Razumikhin technique and comparison principle, sufficient conditions for p-moment exponential stability of the trivial solution to the systems are present, where the parameter λ( t) in ( dV( t, x( t)))/( dt) are not required to be negative. Finally, an example is discussed to show the applications of the obtained results.

Time lag versus stability

By employing Lyapunov functions and the theory of differential inequalities in the context of a suitable minimal class of functions, sufficient conditions are given for stability of functional differential systems. It is shown that the dominant diagonal property of a matrix provides a suitable mechanism to resolve "time-lag vs. stability" problem.