Functional Differential Systems Research Articles

Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h(⋅)↦∫0⋅U(⋅,s)h(s)ds:L1([0,T];Y)→C([0,T];Y), where Y is a Banach space and U(⋅,⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.

A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.

О верхних и нижних разрешающих функциях в игровых задачах динамики

Изучаются квазилинейные конфликтно-управляемые процессы общего вида на предмет сближения траекторий с заданным цилиндрическим множеством. В основу исследований положен метод верхних и нижних решающих функций. Основное внимание уделено ситуации, когда нет места условию Понтрягина, к тому же телесная часть терминального множества не является выпуклой. Предложена схема метода, которая позволяет в случае невыпуклости телесной части зафиксировать некоторую точку в ней, точку прицеливания, и реализовать процесс сближения. Получены достаточные условия для решения задачи сближения для разных классов стратегий. При этом использованы стробоскопические стратегии Хайека, определяющие управление М.М. Красовским. Процесс сближения состоит из двух этапов: активного и пассивного. На активном этапе накапливается верхняя разрешающая функция первого типа, а после момента переключения используется нижняя разрешающая функция второго типа. Эти функции дают возможность построить измерительное управление первого игрока на основе теорем об измеримом выборе, в частности теоремы Филиппова-Кастена. Полученные результаты для обобщенных квазилинейных процессов позволяют охватить широкий круг функционально-дифференциальных систем, систем с дробными и частными производными. Указаны возможности для развития предложенной методики.

Fundamental Results on Determining Matrices for a Certain Class of Hereditary Systems

Three major tools are required to investigate the controllability of control systems, namely, determining matrices, index of control systems and controllability Grammian. Determining matrices are the preferred choice for autonomous control systems due to the fact that they are devoid of integral operators in their computations. This article developed the structure of certain parameter-ordered determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for Euclidean controllability of the system. Expressions for the relevant determining matrices were formulated and it was established that the determining matrices for double time-delay linear autonomous functional differential control systems do not exist if one of the time-delays is not an integer multiple of the other paving the way for the investigation of the Euclidean controllability of generic double time-delay control systems.

Global Structure of Determining Matrices for a Class of Differential Control Systems

This paper developed and established unprecedented global results on the structure of determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for the Euclidean controllability of the system. The computational process and implementation of the controllability matrix were demonstrated on the MATLAB platform to determine the controllability disposition of a small-problem instance. Finally, the work examined the computing complexity of the determining matrices.

Corrigendum : Stabilization of stochastic functional differential systems by steepest descent feedback controls

<b>Corrigendum</b> : Stabilization of stochastic functional differential systems by steepest descent feedback controls

Asymptotic behavior analysis of Markovian switching neutral-type stochastic time-delay systems

A new integral inequality method is put forward to analyze the general decay stability for Markovian switching neutral stochastic functional differential systems. At first, in order to get around the dynamic analyses difficulty induced by the coinstantaneous presence of neutral term, Markovian switching and Brownian motion noise, an new integral inequality as a powerful tool is gained. Then, based on the integral inequality, general decay stability in the sense of pth(p>0) moment and the almost sure can be taken out by utilizing the nonnegative semimartingale convergence theorem and Lyapunov stability theory. The obtained results can be especially applied to two special types of neutral stochastic differential systems that have been studied in the literature. Finally, an example has been performed to verify the obtained analytical results.

Time optimal control to a partial stochastic differential system with pseudo almost periodic coefficients

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the α-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses.

Stabilization of stochastic functional differential systems by steepest descent feedback controls

Stabilization of stochastic functional differential systems by steepest descent feedback controls

非线性分数阶泛函微分方程组边值问题的可解性

非线性分数阶泛函微分方程组边值问题的可解性

Stability of stochastic functional differential equations with random switching and applications

This paper develops a new stability theory for stochastic functional differential systems with random switching. It examines general systems that are time inhomogeneous, past-dependent, and perturbed by a Brownian motion and modulated by a switching process. In contrast to the advances in the literature, this paper provides weaker and more verifiable conditions. Examples and discussion are also given to demonstrate the applicability of our results.

New results on stability for non‐linear Markov switched stochastic functional differential systems

For Markov switched stochastic functional differential systems (SFDSs), asymptotic property is one of the most desired issues. Recently, a new class of delay-dependent asymptotic stability for non-linear Markov switched SFDSs was investigated. However, the existing references do not take the convergent speed and non-autonomous factor into consideration. Therefore, by means of multiple Lyapunov–Krasovskii functionals, this study is devoted to examine the exponential stability for highly non-linear autonomous Markov switched SFDSs and the exponential stability, polynomial stability and polynomial growth at most for highly non-linear non-autonomous systems, where all the criteria rely on the intervals lengths of continuous delays. In addition, the properties of boundedness and asymptotic stability are as well explored.

Stability analysis for nonlinear Markov jump neutral stochastic functional differential systems

Recently, the asymptotic stability for Markov jump stochastic functional differential systems (SFDSs) was studied, whose stability criteria relied on the intervals lengths of continuous delays. Whereas, so far all the existing references require the rigorous global Lipschitz condition for the delay parts of the drift coefficients and do not consider the challenging factors of exponential decay and neutral issue. Motivated by the aforementioned considerations and the advantages of the degenerate functionals, this paper aims to weaken the global Lipschitz condition for the delay parts of the drift coefficients and investigate the delay-dependent exponential stability and asymptotic boundedness for highly nonlinear Markov jump neutral SFDSs with the method of multiple degenerate functionals. Of course, the delay-independent assertions are as well derived here.

Razumikhin-type theorem for pth exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function

This paper is concerned with the pth moment exponential stability of stochastic functional differential equations with impulses. Based on average dwell-time method, Razumikhin-type technique and vector Lyapunov function, some novel stability criteria are obtained for impulsive stochastic functional differential systems. Two examples are given to demonstrate the validity of the proposed results.

Some Economic Dynamics Problems for Hybrid Models with Aftereffect

In this paper, we consider a class of economic dynamics models in the form of linear functional differential systems with continuous and discrete times (hybrid models) that covers many kinds of dynamic models with aftereffect. The focus of attention is periodic boundary value problems with deviating argument, control problems with respect to general on-target vector-functional and questions of stability to solutions. For boundary value problems, some sharp sufficient conditions of the unique solvability are obtained. The attainability of on-target values is under study as applied to control problems with polyhedral constraints with respect to control, some estimates of the attainability set as well as estimates to a number of switch-points of programming control are presented. For a class of hybrid systems, a description of asymptotic properties of solutions is given.

On a class of linear continuous-discrete systems with discrete memory

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

Flocking behaviours of a delayed collective model with local rule and critical neighbourhood situation

It is of particular significance in both theories and applications to understand how self-organized particles use limited environment and simple rules to organize into ordered emergence. In this paper, we study a modified Cucker–Smale-type system with a simple and local cut-off weight. Also, a communication delay is introduced into both the velocity adjoint terms and the cut-off weight. We extend the previous criteria for flocking to the delayed model, using an approach based on the invariant subspace decomposition in the infinite-dimensional space. For the noncritical neighbourhood situation, a criterion of flocking emergence with an exponential convergent rate is established by the standard arguments on the functional differential system. For the general neighbourhood situation, when all the switching intervals have a uniform minimum gap, another criterion of flocking emergence is also found.

Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

Abstract In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

On stability functional differential equation with delay variable by using fixed point-theory

I will explain how to use the Banach fixed point theory in the asymptotic stability of nonlinear differential equations; I will obtain appropriate generalizations and strong forms of some of the results in [11]. Specifically, in the above-mentioned paper, asymptotic stability is achieved, while I will discuss how to achieve a asymptotic stability as well as stability by making a simple observation, also circulate the previous asymptotic stability results to the Functional Differential Equations systems, not only on the scalar Functional Differential Equations as is the case in the mentioned paper. This raises the question of how much this particular method can afford us, and what are the limitations of this technique. I will refer to the important limitation of the fixed point theory on the uniqueness of solutions only within the complete metric space area where they are not specified. If the metric space onto which the contraction mapping principle is applied is very small, i do not get a satisfactory result. I will discuss this in detail below.

Basic reproduction ratios for almost periodic compartmental models with time delay

The theory of basic reproduction ratio R0 is established for a large class of almost periodic and time-delayed compartmental population models. We first present some dynamical properties for linear almost periodic functional differential systems. By using the product space and evolution semigroup approach, we then prove that R0−1 has the same sign as the exponential growth bound of an associated linear system. As an application, we apply the developed theory to an almost periodic SEIR model with an incubation period and obtain a threshold result on its global dynamics in terms of R0. Finally, we present some numerical simulations. Numerical simulations indicate that prolonging the length of incubation period is beneficial for the control of the disease. In addition, a simple model shows that the basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one.