Infinite frequency solutions to discontinuous control systems with variable delay

In this article, we study the frequency of zeros in the solution tou′(t)=f(t, u(t−τ(t))), whereτ is a delay that depends on time, andf is a discontinuous function. First we show examples of solutions that have infinite frequency for autonomous systems with variable delay, and for non-autonomous systems with constant delay. Then we prove that infinite frequency solutions cannot come from finite frequency data. Also we prove that under certain conditions on the delay, the zero solution is the only solution that has infinitely many zeros in each interval of a fixed length.     

Modulus Semigroups and Perturbation Classes for Linear Delay Equations in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper we study C₀-semigroups on X × L_p( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞). Dedicated to the memory of H. H. Schaefer     

Approximations for the delay probability in the M/G/s queue

This paper develops approximations for the delay probability in an M/G/s queue. For M/G/s queues, it has been well known that the delay probability in the M/M/s queue, i.e., the Erlang delay formula, is usually a good approximation for other service-time distributions. By using an excellent approximation for the mean waiting time in the M/G/s queue, we provide more accurate approximations of the delay probability for small values of s. To test the quality of our approximations, we compare them with the exact value and the Erlang delay formula for some particular cases.     

A Feynman-Kac type formula for a fixed delay CIR model

Stochastic delay differential equations (SDDE’s) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fixed delay CIR process. Moreover, for the fixed delay CIR process, we derive a Feynman-Kac type formula, leading to a generalized exponential-affine formula, which is used to determine a bond pricing formula when the interest rate follows the delay’s equation. It turns out that, for each maturity time T, the instantaneous forward rate is an affine function (with time dependent coefficients) of the rate process and of an auxiliary process (also depending on T). The coefficients satisfy a system of deterministic differential equations.     

Hopf bifurcation in a environmental defensive expenditures model with time delay

In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, τ₀. It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

Approximate controllability and regularity for semilinear retarded control systems

We deal with the approximate controllability for semilinear systems with time delay in a Hilbert space. First, we show the existence and uniqueness of solutions of the given systems with the more general Lipschitz continuity of nonlinear operator f fromR ×V toH. Thereafter, it is shown that the equivalence between the reachable set of the semilinear system and that of its corresponding linear system. Finally, we make a practical application of the conditions to the system with only discrete delay.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

Sharp and simple bounds for the Erlang delay and loss formulae

We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.     

Robust stabilization of uncertain linear dynamical systems with time-varying delay

In this paper, the problem of the robust stabilization for a class of uncertain linear dynamical systems with time-varying delay is considered. By making use of an algebraic Riccati equation, we derive some sufficient conditions for robust stability of time-varying delay dynamical systems with unstructured or structured uncertainties. In our approach, the only restriction on the delay functionh(t) is the knowledge of its upper boundh ^–. Some analytical methods are employed to investigate these stability conditions. Since these conditions are independent of the delay, our results are also applicable to systems with perturbed time delay. Finally, a numerical example is given to illustrate the use of the sufficient conditions developed in this paper.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

Delay Perturbed Sweeping Process

This paper is devoted to the study of a nonconvex perturbed sweeping process with time delay in the infinite dimensional setting. On the one hand, the moving subset involved is assumed to be prox-regular and to move in an absolutely continuous way. On the other hand, the perturbation which contains the delay is single-valued, separately measurable, and separately Lipschitz. We prove, without any compactness assumption, that the problem has one and only one solution.     

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.     

The delay effect on reaction–diffusion equations

In this work we study a reaction–diffusion problem with delay and we make an analysis of the stability of solutions by means of bifurcation theory. We take the delay constant as a parameter. Special conditions on the vector field assure existence of a spatially nonconstant positive equilibrium U_k , which is stable for small values of the delay. An increase of the delay destabilizes the equilibrium of U_k and leads to super or subcritical Hopf bifurcation.     

Delay analysis of extended rtPS for VoIP service in IEEE 802.16e by matrix analytic method

Extended real time polling service (ErtPS) is added to IEEE 802.16e-2005 standards in order for VoIP service to use uplink resources efficiently by considering on/off characteristic of voice source. Recently average queueing delay of ErtPS algorithm for VoIP service was investigated, and it was shown that ErtPS allows to admit more users than UGS algorithm. But we need the probability distribution of queueing delay rather than average queueing delay in order to provide a necessary information for QoS. In this paper we obtain the probability distribution of queueing delay of ErtPS for VoIP service by using the matrix analytic method for the GI/M/1 type and the M/G/1 type matrices in cases of the service time being exponential and deterministic respectively. By applying the results on deterministic service time we find the maximum allowable number of VoIP users with the required constraint on queueing delay. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

On the Instability of Linear Nonautonomous Delay Systems

The unstable properties of the linear nonautonomous delay system x(t) = A(t)x(t) + B(t)x(t – r(t)), with nonconstant delay r(t), are studied. It is assumed that the linear system y(t) = (A(t) + B(t))y(t) is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay r(t) is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function r(t) and the results depending on the asymptotic properties of the delay function.     

p-Moment Stability of Stochastic Nonlinear Delay Systems with Impulsive Jump and Markovian Switching

Abstract

This article is concerned with the problem of p-moment stability of stochastic differential delay equations with impulsive jump and Markovian switching. In this model, the features of stochastic systems, delay systems, impulsive systems, and Markovian switching are all taken into account, which is scarce in the literature. Based on Lyapunov–Krasovskii functional method and stochastic analysis theory, we obtain new criteria ensuring p-moment stability of trivial solution of a class of impulsive stochastic differential delay equations with Markovian switching.     

User equilibria for a parallel queueing system with state dependent routing

Consider a system of two queues in parallel, one of which is a ⋅|M|1 single-server infinite capacity queue, and the other a ⋅|G ^(N)|∞ batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs–Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the ⋅|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the Downs–Thomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably.     

Delay differential systems: Problem of control with target in function space

This paper deals with the control of linear delay differential systems with target in a function space. It is shown that the bang-bang property does not hold in the strict sense but in some approximate sense. Existence of a time-optimal control which steers to the null function is proved. Using degeneracy of linear autonomous delay differential systems, we describe a new behavior of some controlled delayed systems which we called theloss of memory phenomenon.     

Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems

This work presents two simple and robust techniques based on time delay estimation for the respective control and synchronization of chaos systems. First, one of these techniques is applied to the control of a chaotic Lorenz system with both matched and mismatched uncertainties. The nonlinearities in the Lorenz system is cancelled by time delay estimation and desired error dynamics is inserted. Second, the other technique is applied to the synchronization of the Lü system and the Lorenz system with uncertainties. The synchronization input consists of three elements that have transparent and clear meanings.Since time delay estimation enables a very effective and efficient cancellation of disturbances and nonlinearities, the techniques turn out to be simple and robust. Numerical simulation results show fast, accurate and robust performance of the proposed techniques, thereby demonstrating their effectiveness for the control and synchronization of Lorenz systems.