Oscillation of second-order nonlinear delay dynamic equations on time scales

By using the generalized Riccati transformation and the inequality technique, we establish one new oscillation criterion for the second-order nonlinear delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales.     

Stability for first order delay dynamic equations on time scales

In this paper, we establish some sufficient conditions for the uniform stability and the uniformly asymptotical stability of the first order delay dynamic equation

where is a time scale, p(.) is rd-continuous and positive, the delay function . Our results unify the corresponding ones for differential and difference equations. To the best of our knowledge, this is the first time to discuss the asymptotical behavior of delay dynamic equations on time scales.     

Oscillation theorems for second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criteria for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

Some integral inequalities useful in the theory of certain partial dynamic equations on time scales

Our aim in this paper is to investigate some integral inequalities in two independent variables on time scales, which unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.     

Some delay integral inequalities on time scales

In this paper, using Gronwall’s inequality, we investigate some delay integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some delay integral inequalities and their corresponding discrete analogues. The inequalities given here can be used as tools in the qualitative theory of certain classes of delay dynamic equations on time scales.     

Sensitivity of certain dynamic systems with respect to a small delay

A class of non-linear differential equations of order one with pure delay is examined. It is found that the ‘effective’ order of an equation of this class need not be constant for all values of the delay parameter, even if the latter is small. In the autonomous case periodic solutions are determined and an estimate of the domain of influence of a constant steady state is made. In the non-autonomous case it is found that the periodic and almost periodic solutions are of more varied types than for analoguous differential equations without pure delay.     

Stability of moving invariant sets and uncertain dynamic systems on time scales

In this paper, we study the stability of moving invariant sets and uncertain dynamic systems on time scale. A control application is considered.     

Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems

This paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain two-dimensional fractional differential systems established in a recent paper of the authors. The main generalizations come from an elementary inequality and by means of the modification of Medve?’s de-singular approach.     

On some qualitative properties for solutions of a certain two-dimensional fractional differential systems

By means of the modification of Medve?’s de-singular method and a result of two-dimensional linear integral inequalities, components-wise (not on some norm) upper bounds are obtained for solutions of a class of nonlinear two-dimensional systems of fractional differential equations. The uniqueness and continuous dependence of the solutions are also discussed here.     

Robust adaptive fault-tolerant control for time delay uncertain nonlinear systems with time-varying performance bounds

In this paper, a robust adaptive fault tolerant controller guaranteeing with time-varying performance bounds is designed for a class of time delay uncertain nonlinear systems subject to actuator failures and external disturbance. The influence of time delay on the system is mitigated and the system performance can be guaranteed by introducing a positive nonlinear control gain function and the generalised restricted potential function. A new method with more design degrees of freedom is developed to ensure the norm of the system state within a-priori, user-defined time varying performance bounds. Using the online estimation information provided by adaptive mechanism, a robust adaptive fault-tolerant control method guaranteeing time varying performance bounds is proposed. It is shown that all the signals of the resulting closed-loop system are bounded and the system state less than a-priori, user-defined performance bounds. Finally, simulation results are given to demonstrate the efficacy of the proposed fault-tolerant control method.     

Maximum allowable delay bounds of networked control systems

This paper proposes a new method to obtain a maximum allowable delay bound for a scheduling of networked control systems. The proposed method is formulated in terms of linear matrix inequalities and can give a much less conservative delay bound than the existing methods. A network scheduling method is presented based on the delay obtained through the proposed method, the bandwidth of a network is allocated to each node and the sampling period of each sensor and controller is determined. The presented method can handle three types of data (periodic data, sporadic data, and message) and guarantees real-time transmission of periodic and sporadic data, and minimum network utilization for non-real time message.     

Backward linear control systems on time scales

We show how a linear control systems theory for the backward nabla differential operator on an arbitrary time scale can be obtained via Caputo's duality. More precisely, we consider linear control systems with outputs defined with respect to the backward jump operator. Kalman criteria of controllability and observability, as well as realisability conditions, are proved.     

Oscillation of certain two-dimensional nonlinear difference systems

Several new oscillation criteria for two-dimensional nonlinear difference systems are established. Examples which dwell upon the importance of our results are also included.     

Observability of nonlinear control systems on time scales

The problem of observability of the family of nonlinear control systems defined on any time scale and parameterised by time instant t ₀ is developed. Necessary and sufficient conditions for observability are given. Because time scale is a rich model of time that includes continuous- and discrete-time models as standard cases, the obtained results extend classical observability rank conditions given for continuous-time systems and discrete-time systems.