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This article focuses on the controllability for a class of semilinear evolution equations and applications to some specific differential equations. Without assuming compactness or equicontinuity of the related semigroups, the existence of mild solutions of semilinear equations in Hilbert space is demonstrated via the noncompact measure tool and the fixed point trick. In order to study the controllability of semilinear equations, new concepts are proposed, namely, exact controllability along any A-bounded Lipschitz continuous curve, and approximate controllability along any continuous curve. Furthermore, two new approximation methods, "bisection method"" and "equisection method,"" are introduced. The exact controllability along any A-bounded Lipschitz continuous curve and the approximate controllability along any continuous curve in the sense of the graph norm of semilinear evolution equations are obtained under the asymptotic condition on the norm of the controllability Gramian inverse operator near the zero point. In fact, our conclusions show that this is not only a result control but also a process control. Finally, the results presented in this paper are employed in resistance, inductance, voltage source type electrical circuit systems, one-dimensional nonhomogeneous transport systems, as well as differential equations with delay which have important effects on economic systems. [ABSTRACT FROM AUTHOR]

Compartmental ordinary differential equation (ODE) models are used extensively in mathematical biology. When transit between compartments occurs at a constant rate, the wellknown linear chain trick can be used to show that the ODE model is equivalent to an Erlang distributed delay differential equation (DDE). Here, we demonstrate that compartmental models with nonlinear transit rates and possibly delayed arguments are also equivalent to a scalar distributed DDE. To illustrate the utility of these equivalences, we calculate the equilibria of the scalar DDE, and compute the characteristic function--without calculating a determinant. We derive the equivalent scalar DDE for two examples of models in mathematical biology and use the DDE formulation to identify physiological processes that were otherwise hidden by the compartmental structure of the ODE model. [ABSTRACT FROM AUTHOR]

We consider a differential equation with a state-dependent delay motivated by a queueing process. The time delay is determined by an algebraic equation involving the length of the queue for which a discontinuous differential equation holds. The new type of state-dependent delay raises some problems that are studied in this paper. We formulate an appropriate framework to handle the system, and show that the solutions define a Lipschitz continuous semiflow in the phase space. The second main result guarantees the existence of slowly oscillating periodic solutions. [ABSTRACT FROM AUTHOR]

We study a scalar delay differential equation (DDE) with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state-dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three contain only constant delays. We implemented this expansion and the computation of the normal form coeffcients in Matlab using symbolic differentiation and the resulting code HHnfDDEsd is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena. [ABSTRACT FROM AUTHOR]

The observation that time-delayed feedback can stabilize an inverted pendulum motivates the formulation of models of human balance control in terms of delay differential equations (DDEs). Recently the intermittent, digital-like nature of the neural feedback control of balance has become evident. Here, semidiscretization methods for DDEs are used to investigate an unstable dynamic system subjected to a digital controller in the context of a switching model for postural control. In addition to limit cycle and chaotic ("microchaos") oscillations, transiently stabilized balance states are possible even though both the open-loop and the closed-loop systems are globally unstable. The possibility that falls can be an intrinsic component of neural control of balance may provide new insights into how the risk of falling in the elderly can be minimized. [ABSTRACT FROM AUTHOR]

National boundaries have never prevented infectious diseases from reaching distant territories; however, the speed at which an infectious agent can spread around the world via the global airline transportation network has significantly increased during recent decades. We introduce an SEAIR-based, antigravity model to investigate the spread of an infectious disease in two regions which are connected by transportation. As a submodel, an age-structured system is constructed to incorporate the possibility of disease transmission during travel, where age is the time elapsed since the start of the travel. The model is equivalent to a large system of differential equations with dynamically defined delayed feedback. After describing fundamental but biologically relevant properties of the system, we detail the calculation of the basic reproduction number and obtain disease transmission dynamics results in terms of R0. We parametrize our model for influenza and use real demographic and air travel data for the numerical simulations. To understand the role of the different characteristics of the regions in the propagation of the disease, three distinct origin-destination pairs are considered. The model is also fitted to the first wave of the influenza A(H1N1) 2009 pandemic in Mexico and Canada. Our results highlight the importance of including travel time and disease dynamics during travel in the model: the invasion of disease-free regions is highly expedited by elevated transmission potential during transportation. [ABSTRACT FROM AUTHOR]

A large number of human genes have been found to contain more than one polyadenylation site (poly(A) site). These lead to multiple messenger ribonucleic acid (mRNA) transcripts, although some of them do not yield protein products. Like alternative initiation and alternative splicing, alternative polyadenylation (APA) contributes to the complexity of the overall pool of mRNA transcripts in human cells. However, little is known about how the cell utilizes different poly(A) sites in response to different cell conditions and developmental stages. Several protein factors have been identified as being necessary for in vitro cleavage and polyadenylation. If the gene of a polyadenylation factor has multiple poly(A) sites, producing variable protein products with different functions, then their APA can control the polyadenylation activity under different cell conditions. The dynamics of the gene expression levels among APA products so far has been studied by experimental approaches only. Here we propose a novel mathematical model consisting of differential equations with a time delay to study the dynamical behavior of the gene expression levels. Numerical simulations have been provided for some of the theoretical analyses of the equations. Our model suggests an oscillatory pattern of APA of CSTF3, indicating APA is employed to maintain homeostasis of its function, perhaps general polyadenylation activity as well, in the cell. [ABSTRACT FROM AUTHOR]

In this paper, we derive an equation governing the dynamics of first-order forward sensitivities for a general system of parametric neutral delay differential equations. We also derive a formula which identifies the size of jumps that appear at discontinuity points when the sensitivity equations are integrated. The formula leads to an algorithm which can compute sensitivities for various types of parameters very accurately and efficiently. [ABSTRACT FROM AUTHOR]

We obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state-dependent delays. In the process, we transform the models with state-dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters. We find that the spindle speed control strategy that we investigated in [SIAM J. Appl. Math., 72 (2012), pp. 1-24] can provide essential improvement on the stability of turning processes with state-dependent delay, and furthermore we show the existence of a proper subset of the stability region which is independent of system damping. Numerical simulations are presented to illustrate the general results. [ABSTRACT FROM AUTHOR]

The article presents a research paper that examines an Itô stochastic differential equation with delay (SDDEs) on a finite interval of the form. It also examines a backward stochastic differential equation depending on X, with unknown processes (Y,Z). The authors' investigates properties of the resulting system, specially the identification of the process Z as a deterministic functional of X.

We study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions. Compared with the typical virus infection dynamical model, this model has two important and novel features. To give a more complex and general infection process, a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered, and two constant delays are incorporated into the model, which describe (i) the time needed for a newly infected cell to start producing viruses and (ii) the time needed for a newly produced virus to become infectious (mature), respectively. By the Lyapunov direct method and using the technology of constructing Lyapunov functionals, we establish global asymptotic stability of the infection-free equilibrium and the infected equilibrium. We also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays. Further, we generalize this type of Lyapunov functional to the model described by n-dimensional delay differential equations. [ABSTRACT FROM AUTHOR]

Continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations (ODEs), delay differential equations, neutral differential equations, integro-differential equations, etc. This paper deals with a continuous extension for the discrete approximate solution of ODEs generated by a class of block θ-methods. Existence and uniqueness for the continuous extension are discussed. Convergence and absolute stability of the continuous block θ-methods for ODEs are studied. As an application, we adopt the continuous block θ-methods to solve delay differential equations and prove that the continuous block θ-methods are GP-stable if and only if they are Aω-stable for ODEs. Several numerical experiments are given to illustrate the performance of the continuous block θ-methods. [ABSTRACT FROM AUTHOR]

In this paper we consider the convergence of the infinite dimensional version of the Kleinman–Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper the numerical solutions of nonlinear neutral differential equations with constant delay and with proportional delay, two typical examples of neutral functional differential equations, are investigated. The focus is on the stability of numerical solutions produced by θ- methods. It is proved that if ½ ≤ θ? ≤ 1 the numerical solution to neutral differential equations with constant delay is globally stable and asymptotically stable under some conditions which are sufficient for stability and asymptotic stability of the exact solution. For neutral differential equations with proportional delay, two discretization approaches, the approach to transforming it into neutral equations with constant delay and the approach to directly discretizing it based on full-geometric mesh, are discussed. The integration on the first mesh interval [0, T0], which needs to be considered for both approaches, is also investigated. By taking into account the order of the methods, we construct two algorithms for the integration on [0, T0]. It is also proved that under some assumptions the numerical solutions to the neutral equation with proportional delay produced by the two approaches are globally stable and asymptotically stable if ½ ≤ θ ≤ 1. [ABSTRACT FROM AUTHOR]

We describe a new Chebyshev spectral collocation method for systems of variable-coefficient linear delay differential equations with a single fixed delay. Computable uniform a posteriori bounds are given for this method. When the coefficients are periodic, the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system. The spectral method approximates this operator by a dense matrix of modest size. In cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator. Our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic. [ABSTRACT FROM AUTHOR]

We study an approximation scheme for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of X(s) for s ϵ [t - τ, t], where τ is a constant. The approximating state is the piecewise linear Euler-Maruyama scheme, and the observation process is a noisy function of the approximating state. The rate of convergence of this scheme is computed. [ABSTRACT FROM AUTHOR]

We analyze the optimal (global and local) orders of superconvergence of collocation solutions uh on uniform meshes Ih for delay Volterra integro-differential equations with proportional delay functions given by θ(t) = qt (0 < q < 1, t ϵ [0, T]). In particular, we show that if u h is a continuous piecewise polynomial of degree m ≥ 2, and if collocation is at the Gauss (-Legendre) points, then the (optimal) order of local superconvergence on Ih is p* = m+2. It turns out that the same order p* holds for nonlinear (strictly increasing) delay functions vanishing at t = 0. However, on judiciously chosen geometric meshes, collocation at the Gauss points yields the order 2m - ϵN, where ϵN → 0 as the number N of mesh points tends to infinity. Optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis. [ABSTRACT FROM AUTHOR]

By applying M-matrix theory and some new analysis techniques, we have succeeded in establishing the asymptotic stability of the zero solution of the "pure-delay type" neutral system with variable delays. The criteria obtained in this paper extend and improve the existing ones. [ABSTRACT FROM AUTHOR]

In [D. Breda, S. Maset, and R. Vermiglio, IMA J. Numer. Anal., 24 (2004), pp. 1--19.] and [D. Breda, The Infinitesimal Generator Approach for the Computation of Characteristic Roots for Delay Differential Equations Using BDF Methods, Research report UDMI RR17/2002, Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Udine, Italy, 2002.] the authors proposed to compute the characteristic roots of delay differential equations (DDEs) with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by Runge--Kutta (RK) and linear multistep (LMS) methods, respectively. In this work the same approach is proposed in a new version based on pseudospectral differencing techniques. We prove the "spectral accuracy" convergence behavior typical of pseudospectral schemes, as also illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper analyzes the dynamics of a G0 cell cycle model of pluripotential stem cells so we can understand the origin of the long period oscillations of blood cell levels observed in periodic chronic myelogenous leukemia (PCML). The model dynamics are described by a system of two delayed differential equations. We give conditions for the local stability of the nontrivial steady state. We use these conditions to study when stability is lost and oscillations occur, and how various parameters modify the period of these oscillations. We consider a limiting case of the original model in order to compute an explicit solution and give an exact form of the period and the amplitude of oscillations. We illustrate these results numerically and show that the main parameters controlling the period are the cellular loss (the differentiation rate $\delta$ and the apoptosis rate $\gamma$), while the cell regulation parameters (proliferation rate $\beta$ and cell cycle duration $\tau$) mainly influence the amplitude. [ABSTRACT FROM AUTHOR]

This paper investigates the structure of two-dimensional Riemann problems for Hamilton-Jacobi equations. The solutions to such problems are fundamental building blocks for constructing solutions to more general problems, in particular, for numerical construction using methods such as front tracking. Here we prove the existence of a particular class of Riemann problem for which the viscosity solutions contain closed characteristic orbits, enclosing furthermore a periodic sonic structure, which n turn encloses a parabolic structure. The existence of such examples elucidates the difficulties encountered in designing construction methods for viscosity solutions to Riemann problems n dimension ≥ 2. This investigation was prompted by the discovery of numerical evidence of examples displaying an even richer internal structure. [ABSTRACT FROM AUTHOR]

In this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations (DDEs). Periodic solutions are found by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for DDEs, in contrast to the case of ordinary differential equations. We investigate three collocation methods based on piecewise polynomials. We discuss computational issues and show numerical orders of convergence using an extensive number of tests. We compare our numerical results with known theoretical convergence results for initial value problems for DDEs. In particular, we show how superconvergence at the mesh points can be lost or recovered depending on the DDE model under consideration and on the choice of collocation discretization. We end with a brief discussion of adaptive mesh selection. [ABSTRACT FROM AUTHOR]

We have recently developed a generic approach to solving retarded and neutral delay differential equations (DDEs). The approach is based on the use of an explicit continuous Runge­Kutta formula and employs defect control. The approach can be applied to problems with state-dependent delays and vanishing delays. In this paper we determine convergence properties for the numerical solution associated with methods that implement this approach. We first analyze such properties for retarded DDEs and then the analysis is extended to the case of neutral DDEs (NDEs). Such an extension is particularly important since NDEs have received little attention in the literature. The main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect. Our analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge­Kutta formulas for DDEs. [ABSTRACT FROM AUTHOR]