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Delay differential systems with time-varying delay: new directions for stability theory

James Louisell (2001)

Kybernetika

In this paper we give an example of Markus–Yamabe instability in a constant coefficient delay differential equation with time-varying delay. For all values of the range of the delay function, the characteristic function of the associated autonomous delay equation is exponentially stable. Still, the fundamental solution of the time-varying system is unbounded. We also present a modified example having absolutely continuous delay function, easily calculating the average variation of the delay function,...

Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch

F. Crauste (2009)

Mathematical Modelling of Natural Phenomena

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the...

Delay perturbed evolution problems involving time dependent subdifferential operators

Soumia Saïdi, Mustapha Fateh Yarou (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We investigate in the present paper, the existence and uniqueness of solutions for functional differential inclusions involving a subdifferential operator in the infinite dimensional setting. The perturbation which contains the delay is single-valued, separately measurable, and separately Lipschitz. We prove, without any compactness condition, that the problem has one and only one solution.

Delay-dependent robust stability conditions and decay estimates for systems with input delays

Kostas Hrissagis, Olga I. Kosmidou (1998)

Kybernetika

The robust stabilization of uncertain systems with delays in the manipulated variables is considered in this paper. Sufficient conditions are derived that guarantee closed-loop stability under state-feedback control in the presence of nonlinear and/or time-varying perturbations. The stability conditions are given in terms of scalar inequalities and do not require the solution of Lyapunov or Riccati equations. Instead, induced norms and matrix measures are used to yield some easy to test robust stability...

Delay-dependent stability conditions for fundamental characteristic functions

Hideaki Matsunaga (2023)

Archivum Mathematicum

This paper is devoted to the investigation on the stability for two characteristic functions f 1 ( z ) = z 2 + p e - z τ + q and f 2 ( z ) = z 2 + p z e - z τ + q , where p and q are real numbers and τ > 0 . The obtained theorems describe the explicit stability dependence on the changing delay τ . Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

Delay-dependent stability of high-order neutral systems

Yanbin Zhao, Guang-Da Hu (2021)

Kybernetika

In this note, we are concerned with delay-dependent stability of high-order delay systems of neutral type. A bound of unstable eigenvalues of the systems is derived by the spectral radius of a nonnegative matrix. The nonnegative matrix is related to the coefficient matrices. A stability criterion is presented which is a necessary and sufficient condition for the delay-dependent stability of the systems. Based on the criterion, a numerical algorithm is provided which avoids the computation of the...

Delays induced in population dynamics

Eva Sánchez (2003)

Banach Center Publications

This paper provides an introduction to delay differential equations together with a short survey on state-dependent delay differential equations arising in population dynamics. Our main goal is to examine how the delays emerge from inner mechanisms in the model, how they induce oscillations and stability switches in the system and how the qualitative behaviour of a biological model depends on the form of the delay.

Density of chaotic dynamics in periodically forced pendulum-type equations

Elena Bosetto, Enrico Serra, Susanna Terracini (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We announce that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics for a dense set of forcing terms in a space of periodic functions with zero mean value. The approach is based on global variational methods.

Derivative of the norm of a linear mapping and its application to differential equations

František Tumajer (1992)

Applications of Mathematics

In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.

Derived cones to reachable sets of a nonlinear differential inclusion

Aurelian Cernea (2014)

Mathematica Bohemica

We consider a nonlinear differential inclusion defined by a set-valued map with nonconvex values and we prove that the reachable set of a certain variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. In order to obtain the continuity property in the definition of a derived cone we use a continuous version of Filippov's theorem for solutions of our differential inclusion. As an application, in finite dimensional spaces, we obtain...

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