Graininess Function Research Articles

Dissipativity of dynamical systems on time scales and the relationship between dissipative differential and dynamical systems

This work is devoted to study the dissipativity property of dynamical systems on time scales and relationship between the dissipativity of systems of dynamic equations on time scales and the corresponding systems of ordinary differential equations. It is established that the dissipativity property is preserved when transitioning from equations on time scales $\mathbb{T}_\lambda$ to the corresponding ordinary differential equations and vice versa, provided that the graininess function $\mu_\lambda$ converges to zero as $\lambda\to0$.

On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales

Abstract This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.

On the relation between oscillation of solutions of differential equations and corresponding equations on time scales

This paper studies oscillatory properties of solutions of a dynamic equation on the set of time scales $\mathbf{T}_\lambda$ provided that the graininess function $\mu_\lambda$ approaches zero as $\lambda\to 0$. We derived the conditions under which oscillation of solutions of differential equations implies that of solutions of the corresponding equations defined on time scales with the same initial data, and vice versa.

Higher order differences on arbitrary discrete time scales and related generating functions

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the n-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed n, the n-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

Solving Riccati Type q−Difference Equations via Difference Transform Method

In this paper, we deal on the time scale that its delta derivative of graininess function is a nonzero positive constant. Based on the Taylor formula for this time scale, we investigate the difference transform method (DTM). This method has been applied successfully to solve Riccati type difference equations in quantum calculus. To demonstrate the ability and efficacy of this method, some examples have been provided.

The weighted discrete dynamic inequalities for 4-convex functions, and its generalization on time scales with constant graininess function

Motivated by Pečarić et al. in (J. Math. Inequal. 11(2):543–5502017), we established here weighted discrete dynamic inequalities for the difference of second divided difference of 4-convex functions. Further we extend and unify the two inequalities, by establishing the theory of n-convexity on time scales having constant graininess function.

The Taylor series method and trapezoidal rule on time scales

The Taylor series method for initial value problems associated with dynamic equations of first order on time scales with delta differentiable graininess function is introduced. The trapezoidal rule for the same types of problems is derived and applied to specific examples. Numerical results are presented and discussed.

Observer-based output feedback consensus protocol for double-integrator multi-agent systems under intermittent sampled position measurements

This paper deals with the leader-follower consensus problem for double-integrator multi-agent systems using sampled position data information. An observer-based output feedback controller is designed to study this problem while taking into account intermittent sensor failures. The non-uniformity and randomness of the sampling times due to intermittent information lead to a µ-varying linear system on a discrete stochastic time domain for the closed-loop system dynamics (here µ is the graininess function). Some necessary and sufficient conditions for the observer and controller gains are derived, using positive perturbation and Lyapunov operators on the space of symmetric matrices, to guarantee mean-square exponential stability for the observation and tracking errors. Some simulation results illustrate the effectiveness of the proposed observer-based output feedback controller.

Lagrange stability of delayed switched inertial neural networks

As an extension of single/multiple stability notion, Lagrange stability is considered here. A class of switched inertial neural networks (SINNs) is studied on both continuous-time and discrete-time domain. Two kinds of activation functions are evolved for the network. Through characteristic function approach, matrix measure strategy, and theory of time scales, Lagrange stability of delayed continuous-/discrete-time SINNs with lurie-type and bounded-type activation functions are addressed, respectively. The results also show that the criteria corresponding to discrete-time network approach to those corresponding to continuous-time one as the graininess function tends to zero. Two examples are given to show the effectiveness of the main results.

Local pseudo almost automorphic functions with applications to semilinear dynamic equations on changing-periodic time scales

In this paper, based on the concepts of changing-periodic time scales, we introduce a notion of local pseudo almost automorphic functions on an arbitrary time scale with a bounded graininess function μ. Then, some properties of local pseudo almost automorphic functions are investigated. As applications, by adopting the concept of a Π-semigroup on time scales, we obtain some new sufficient conditions for the existence of local pseudo almost automorphic mild solutions for a class of semilinear dynamic equations on arbitrary time scales.

Stability analysis of switched systems on time scales with all modes unstable

In this paper, global uniform asymptotic stability problems of switched systems on time scales with all modes unstable are considered. First, a more general stability criterion is proposed for nonlinear switched systems on time scales, in which the value of Lyapunov function at some switching instants may not necessarily decrease. Second, based on the discretized Lyapunov function technique, a sufficient condition is derived for stability analysis of linear switched systems with all subsystems unstable on a special kind of time scale (the graininess function is constant). It is shown that this approach does not require the system matrix of one subsystem must commute with the one of the other subsystem, which is essential for stability analysis of linear switched systems on non-uniform time domains by the eigenvalue based approach in the existing results. Moreover, the idea of this paper not only provides a unified approach to study continuous switched systems and their discrete counterparts simultaneously, but also is effective to investigate stability problems of other systems on uniform or non-uniform time domains. Two examples are given for illustrating the effectiveness of the proposed results.

The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.

Exponential stability of nonlinear positive systems on time scales

Positive uniform exponential stability of positive nonlinear systems on arbitrary time scales with bounded graininess function is studied. It is proved that linearization preserves positivity of systems. Then it is shown that positive uniform exponential stability of the linear approximation of a nonlinear system implies positive uniform exponential stability of the nonlinear system.

Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales

Abstract In this paper, by using the concept of changing-periodic time scales and composition theorem of time scales introduced in 2015, we establish a local phase space for functional dynamic equations with infinite delay (FDEID) on an arbitrary time scale with a bounded graininess function μ. Through Krasnoseľskiĭ’s fixed point theorem, some sufficient conditions for the existence of local-periodic solutions for FDEID are established for the first time. This research indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function.

On consensus in the Cucker–Smale type model on isolated time scales

This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual influences in the group of agents do not change. A sufficient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.

Krause's model of opinion dynamics on isolated time scales

We analyze a bounded confidence model, introduced by Krause, on isolated time scales. In this model, each agent takes into account only the assessments of the agents whose opinions are not too far away from its own opinion. We show that the behavior of the model depends strongly on the graininess function μ: If μ takes values in the interval ]0,1], then our discrete time scale model behaves similarly to the classical one, but if μ takes values in ]1,+∞[, then the model has different properties. Simulations are performed to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

Changing-periodic time scales and decomposition theorems of time scales with applications to functions with local almost periodicity and automorphy

In this work, we address the periodic coverage phenomenon on arbitrary unbounded time scales and initiate a new idea, namely, we introduce the concept of changing-periodic time scales. We discuss some properties of this new concept and illustrate several examples. Specially, we establish a basic decomposition theorem of time scales which provides bridges between periodic time scales and an arbitrary time scale with a bounded graininess function μ. Based on this result, we introduce local-almost periodic and local-almost automorphic functions on changing-periodic time scales and study some related properties. The concept of changing-periodic time scales introduced in this paper will help in understanding and removing the serious deficiency which arises in the study of classical functions on time scales.

Region of exponential stability of switched linear systems on time scales

This paper deals with the stability analysis of a class of linear time-invariant switched systems on non-uniform time domains. The considered class consists of a set of linear continuous-time and linear discrete-time subsystems. Some sufficient and necessary conditions are derived to guarantee the exponential stability of this class of systems on time scales with bounded graininess function, introducing a region of exponential stability. Some examples illustrate these results.

Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions

This paper deals with the stability analysis of a class of uncertain switched systems on non-uniform time domains. The considered class consists of dynamical systems which commute between an uncertain continuous-time subsystem and an uncertain discrete-time subsystem during a certain period of time. The theory of dynamic equations on time scale is used to study the stability of these systems on non-uniform time domains formed by a union of disjoint intervals with variable length and variable gap. Using the concept of common Lyapunov function, sufficient conditions are derived to guarantee the asymptotic stability of this class of systems on time scale with bounded graininess function. The proposed scheme is used to study the leader–follower consensus problem under intermittent information transmissions.

ON HENSTROCK INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

In this paper we introduce the interval-valued Hen- stock integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was flrst deflned by Henstock (2) in 1963. The Henstock integral is more powerful and simpler than the Lebesgue, Wiener and Feynman integrals. The Henstock delta integral on time scales was introduced by Allan Peterson and Bevan Thompson (5) in 2006. In 2000, Congxin Wu and Zengtai Gong introduced the concept of the Henstock integral of interval-valued functions (6). In this paper we introduce the concept of the Henstock delta integral of interval-valued function on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real numberR with the subspace topology inherited from the standard topology ofR. For t 2 T we deflne the forward jump operator ae(t) = inffs 2 T : s > tg where inf ` = supfTg, while the backward jump operator ‰(t) = supfs 2 T : s t, we say that t is right-scattered, while if ‰(t) < t, we say that t is left-scattered. If ae(t) = t, we say that t is right-dense, while if ‰(t) = t, we say that t is left-dense. The forward graininess function (t) of t 2 T is deflned by (t) = ae(t) i t, whlie the backward graininess function (t) of t 2 T is deflned by (t) = t i ‰(t). For a;b 2 T we denote the closed interval (a;b)T = ft 2 T : atbg.