Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

Abstract

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.     

New Interior-Point Algorithm for Symmetric Optimization Based on a Positive-Asymptotic Barrier Function

Abstract

We define a new interior-point method (IPM), which is suitable for solving symmetric optimization (SO) problems. The proposed algorithm is based on a new search direction. In order to obtain this direction, we apply the method of algebraically equivalent transformation on the centering equation of the central path. We prove that the associated barrier cannot be derived from a usual kernel function. Therefore, we introduce a new notion, namely the concept of the positive-asymptotic kernel function. We conclude that this algorithm solves the problem in polynomial time and has the same complexity as the best known IPMs for SO.     

On measure solutions of the Boltzmann equation,part I: Moment production and stability estimates

The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.     

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

Abstract

We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.     

POSITIVITY METHODS FOR DICHOTOMY OF LINEAR SKEW-PRODUCT FLOWS ON HILBERT SPACES

Abstract

In [Na-Rh] we developed a method based on positivity in order to characterize the stability of the evolution family corresponding to the nonautonomous Cauchy problem in Hilbert spaces. This method is extended to the study of hyperbolicity of linear skew-products. We also show that exponential dichotomy of a linear skew-product flow is equivalent to the existence of a Hermitian valued solution of some linear Riccati equation.     

输入带有时滞的线性系统的镇定

考虑带有输入时滞的线性系统的镇定问题.通过把时滞写成一阶传播方程,带有输入时滞的镇定问题转化为常微分方程和一阶双曲方程组成的串联系统的镇定问题.与现有Backstepping方法不同,文章给出了新的变换,其核函数是一阶倒向向量值常微分方程,这使得控制的设计更加简单.文章给出了新的状态反馈控制器,并证明了闭环系统解的适定性和指数稳定性.数值模拟说明,给出的方法是非常有效的.     

An inverse spectral problem for integro-differential Dirac operators with general convolution kernels

ABSTRACT

An integro-differential Dirac system with a convolution kernel consisting of four independent functions is considered. We prove that the kernel is uniquely determined by specifying the spectra of two boundary value problems with one common boundary condition. The proof is based on the reduction of this nonlinear inverse problem to solving some nonlinear integral equation, which we solve globally. On this basis we also obtain a constructive procedure for solving the inverse problem along with necessary and sufficient conditions for its solvability in an appropriate class of kernels.     

Dynamic behavior of a heat equation with memory

This paper addresses the spectrum‐determined growth condition for a heat equation with exponential polynomial kernel memory. By introducing some new variables, the time‐variant system is transformed into a time‐invariant one. The detailed spectral analysis is presented. It is shown that the system demonstrates the property of hyperbolic equation that all eigenvalues approach a line that is parallel to the imaginary axis. The residual spectral set is shown to be empty and the set of continuous spectrum is exactly characterized. The main result is the spectrum‐determined growth condition that is one of the most difficult problems for infinite‐dimensional systems. Consequently, a strong exponential stability result is concluded. Copyright © 2008 John Wiley & Sons, Ltd.     

On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.     

Stabilization of the Korteweg-de Vries-Burgers equation with non-periodic boundary feedbacks

We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,1]. We derive a control law which guaranteesL ²-global exponential stability,H ³-global asymptotic stability, andH ³-semiglobal exponential stability.     

Third-order forms used to modify an L. T. Grujic theorem

The possibility of using third-order forms to modify the Grujic theorem of exponential stability is examined. The modification obtained is employed in a study of the absolute stability of an equation describing longitudinal movement of an aircraft.Translated from Dinamicheskie Sistemy, No. 5, pp. 127–131, 1986.     

High-order exponential spline method for pricing European options

ABSTRACT

The key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.     

Inequalities and exponential stability and instability in finite delay Volterra integro-differential equations

We use Liapunov functionals to obtain sufficient conditions that ensure exponential stability of the nonlinear Volterra integro-differential equation $$x^{\prime }(t)=p(t)x(t)-\int \limits _{t-\tau }^{t}q(t,s)x(s)ds,$$ where the constant $\tau $ is positive, the function $p$ does not need to obey any sign condition and the kernel $q$ is continuous. Our results improve the results obtained in literature even in the autonomous case. In addition, we give a new criteria for instability.     

Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity

ABSTRACT

Here we consider a class of non-linear heat equation with polynomial non-linearity. We prove a non-uniqueness result for mild solutions which take values in a critical Lebesgue space. To this end we extend to the entire space a counter-example of Ni and Sacks in the case where the underlying space is the ball of center 0 and of radius 1. We also propose a new criterion of uniqueness optimal with respect to the given counter-examples. The proof of our results lie on some estimates for the heat kernel in Lorentz spaces introduced by Meyer in the Navier–Stokes context.     

Exponential Stability in Mean Square for a General Class of Discrete-Time Linear Stochastic Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and Markovian switching is investigated. The case of the linear systems whose coefficients depend both to present state and the previous state of the Markov chain is considered. Three different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other two definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions.

The results developed in this article may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems in the presence of a delay in the transmission of the data.     

Alternative Solution to a Quorum Queueing System

Abstract

This paper considers the quorum queueing system with exponential/Erlangian service times and derives the system content steady-state probabilities explicitly in terms of the roots inside and outside the unit ball of the characteristic equation.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ_τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.     

Energy decay to Timoshenko system with indefinite damping

We consider the classical Timoshenko system for vibrations of thin rods. The system has an indefinite damping mechanism, ie, it has a damping function a=a(x) possibly changing sign, present only in the equation for the vertical displacement. We shall prove that exponential stability depends on conditions regarding of the indefinite damping function a and a nice relationship between the coefficient of the system. Finally, we give some numerical result to verify our analytical results.