China
Africa
Explore chapters and articles related to this topic
Introduction
Published in Magdi S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, 2018
In what follows we collect information and mathematical definitions related to functional differential equations (FDEs). Unless stated otherwise, all quantities and variables under consideration are real. It is well-known from mathematical sciences that, an ordinary differential equation (ODE) is an equation connecting the values of an unknown function and some of its derivatives for one and the same argument value, for example H (t, a;, x, i ) = 0. Following [4,6], a functional equation (FE) is an equation involving an unknown function of different argument values. For example, x(t) + 3x(41) = 2, x (t) = sin (t)x(t + 2) + cos(t + l)a;2( --3) = 2 are FEs. By combining the notions of differential and functional equations, we obtain the notion of a functional differential equation (FDE). Thus, FDE is an equation connecting the unknown function and some of its derivatives for, generally speaking, different argument values. Looked at in this light, the notion of FDE generalizes all equations of mathematical analysis for functions of a continuous argument. This assertion is greatly justified by examining models of several applications [6-13]. We take note that all fundamental
*
Published in A.A. Martynyuk, Advances in Stability Theory at the End of the 20th Century, 2002
A.A. Martynyuk, J.H. Shen, I.P. Stavroulakis
It is now being recognized that the theory of impulsive differential equations is not only richer than the theory of differential equations without impulses but also represents a more natural framework for mathematical modelling of many real world phenomena (cf. [18, 25]). The stability theory of impulsive differential equations goes back to the work of Mil'man and Myshkis [22]. In the last few decades the stability theory of impulsive differential equations marked a rapid development, and most research focuses on impulsive ordinary differential equations. See, for example, [5,18,25] and the references cited therein. Now there also exists a well-developed stability theory of functional differential equations (cf. [6-78910,12-17,26,32,385,37,38]). However, not so much has been developed in the direction of the stability theory of impulsive functional differential equations. In the few publications dedicated to this subject, earlier works were done by Anokhin [1] and Gopalsamy and Zhang [11]. Recently, stability problems on some linear impulsive delay differential equations are systematically investigated in several papers. See, for example, [2-4,34,36]. However, so far stability problems on impulsive functional differential equations in more general form attracted little attention, and the well-known Lyapunov's second method applied to such equations remain neglected unlike in functional differential equations and impulsive ordinary differential equations.
Pseudo-almost periodic C 0 solutions to the evolution equations with nonlocal initial conditions
Published in Applicable Analysis, 2023
Bassem Meknani, Jun Zhang, Talaat Abdelhamid
Functional differential equations and inclusions arise in a variety of areas of biology, physics, and engineering. Such equations have received much attention in recent years. A good guide to the literature for functional differential equations is given by the books Hale [26], Hale and Verduyn Lunel [27], Kolmanovskii and Myshkis [28], and the references therein. During the last decades, existence and uniqueness of mild, strong, classical, almost periodic, almost automorphic solutions of semi-linear functional differential equations and inclusions have been studied extensively by many authors. These use the semigroup theory, fixed point argument, degree theory, and measures of noncompactness. We mentioned, for instance, the books by Ahmed [29], Diagana [30], Engel and Nagel [31], Kamenskii et al. [32], Pazy [33], Wu [34], Zheng [35], and the references therein. In recent years, there has been a significant development in evolution equations and inclusions; see the monograph of Perestyuk et al. [36], the papers of Baliki and Benchohra [37], Benchohra and Medjedj [38] and the references therein.
Almost sure and pth moment stability of uncertain differential equations with time-varying delay
Published in Engineering Optimization, 2022
Delay differential equations or functional differential equations express the fact that the velocity of a physical system depends not only on the state of the system at a given instant but also depends upon on its past states (Deng, Wu, and Li 2006; Llibre and Tarta 2007; Nesterov 2018). For example, cell division is not instantaneous once activated and patients show symptoms of illnesses until some time after the infection. The delay differential equation with plays an important role in an increasing number of system models in biology, engineering, economics, physics and other sciences (Awwad, Gyori, and Hartung 2012; Nouri, Nazari, and Keramati 2017). There exists an extensive literature dealing with delay differential equations and their applications.
Feedback stabilisation and polynomial decay estimate for distributed bilinear parabolic systems with time delay
Published in International Journal of Control, 2021
A. El Houch, A. Tsouli, Y. Benslimane, A. Attioui
The study of functional differential equations comes from the fact that when we want to model some evolution phenomena arising in biology, physics and engineering sciences, etc., a time delay can appear in the variables. Some of the sources inducing delays are the reaction times of the sensors or actuators, the transmission times of the information, the transfer times of the molecules or the measurement. One of the most important and interesting problem in the analysis of functional differential equations is the stability notion. This problem has been studied by many researchers (see, e.g. Arino & Sanchez, 1995; Kunisch & Schappacher, 1983; Wu, 1996).