Stability in systems with aftereffect when there are singularities in the integral kernels

Systems with aftereffect are considered. The state of these systems is described by integrodifferential equations of the Volterra type, which depend on functionals in integral form and, in particular, on analytic functionals which are represented by Frechet series. The integral kernels can allow of singularities of Abel kernel singularities. The total stability (i.e. stability under persistent disturbances) is investigated, and the structure of the general solution is investigated in the neighbourhood of zero for an equation with a holomorphic non-linearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. The conditions for instability are given in the critical case of a single zero root, which generalise results obtained previously.     

Stability analysis of the zero solution for two class evolution equations

By constructing some suitable Lyapunov-type functionals and applying the theory of the definite-quadratic form, we obtain the stability of the zero solution of two classes of evolution equations with delays, including reaction–diffusion equations and damped wave equations. Our criteria depend on the derivatives of delays. Consequently, when the delays are constants, these criteria are independent of the magnitudes of the delays, so the delays are harmless for the stability of the zero solution.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

Nonmonotone functionals in the Lyapunov direct method for equations of the neutral type

We present new tests for the stability and asymptotic stability of trivial solutions of equations with deviating argument of the neutral type. Unlike well-known results, here we use nonmonotone indefinite Lyapunov functionals. Our class of functionals contains both Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions as natural special cases. This class of functionals is broad enough that, in a number of stability tests, we have been able to omit the a priori requirement of stability of the corresponding difference operator. In addition, we present tests for the asymptotic stability of solutions of equations of the neutral type with unbounded right-hand side and new estimates for the magnitude of perturbations that do not violate the asymptotic stability if it holds for the unperturbed equation. The obtained estimates single out domains of the phase space in which perturbations should be small and domains in which essentially no constraints are imposed on the perturbation magnitude.     

On the expected discounted penalty function at ruin of a surplus process with interest

In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

We study global and local stabilities of the stationary zero solution to certain infinite-dimensional stochastic differential equations. The stabilities are in terms of fractional powers of the linear part of the drift. The abstract results are applied to semilinear stochastic partial differential equations with non-Lipschitzian drift terms and, in particular, to some specific models of population dynamics. We also expose the stabilizing effect of noise on the otherwise unstable zero solution

As a basic tool we use the Forward Inequality, a generalization of Kolmogorov's forward equation; it is an application of Lyapunov's second method with a sequence of Lyapunov functionals     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger's equation

A Banach algebra A of functionals on C[a, b] is introduced and it is proved that the operator-valued Feynman integral recently defined by Cameron and Storvick exists for functionals in A. Two existence theorems of Cameron and Storvick are seen to be special cases of this result; in fact, even in these cases, the present theorem gives improved results.Cameron and Storvick have used their function space integral to give a solution to an integral equation formally equivalent to Schroedinger's equation; using our existence theorem, we give a relatively brief and transparent proof of this result.     

Solution procedures for three-dimensional eddy current problems

The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwell's equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.     

Dynamics of a hysteretic neuron model

A mathematical model of a single isolated artificial neuron with hysterisis is formulated by means of a neutral delay differential equation. The asymptotic and exponential stability of such a model are investigated. Sufficient conditions for the exponential stability of a linear integral difference inequality are obtained. In the absence of hysterisis effect, our model reduces to a known model of a single neuron. Usually asymptotic stability of neutral delay differential equations is studied by means of degenerate Lyapunov–Kravsovskii functionals. In this article, perhaps for the first time exponential stability of a class of neutral differential equations are studied by means of the exponential stability of an affiliated difference inequality. While generalization to Hopfield type hysteretic neural networks is possible, such a generalization is not considered in this article.     

Averaging of electrical properties of anisotropic fiber-metal composites

The electrical properties of fiber-metal composites with anisotropic components were averaged within the framework of a regular structure model. The corresponding current problem was reduced to a system of two integral equations with elliptical kernels. The effective electrical conductivity tensor of such materials was defined as several functionals based on integral equation solutions for the boundary problem. The calculation results are given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 263–268, March–April, 1997.     

On the stability of the solutions of linear stochastic integro-differential equations encountered in elasticity and viscoelasticity problems

The asymptotic stability, almost sure also in the mean square, of a viscoelastic system subjected to a load in the form of a random stationary broadband ergodic process is investigated. The behaviour of this system is described by integro-differential equations with stochastic parameters. The stability is considered with respect to the perturbation of the initial conditions. The governing relation is taken in an integral form with a creep (or relaxation) kernel of convolution type, which satisfies the condition of limited creep of the material. Using the fundamental solution of the corresponding deterministic integro-differential equation and its maximum Lyapunov exponent, the sufficient condition for stability of the zero solution of the initial equation or, which is the same thing, the equilibrium position of the viscoelastic system, is obtained.     

On boundary-volume element discretization of inhomogenous elastodynamic problems

This paper presents an integral equation formulation and its discretization scheme for the elastodynamic problem in which the material properties are prescribed as arbitrary, continuous and differentiable functions of the spatial coordinates. The formulation is made by using the Green's function for the corresponding problem in homogenous elasticity. From a weighted residual statement of the problem, the governing differential equation is transformed into a set of the integral equations in the inner domain as well as on the boundary. These integral equations are discretized by introducing a finite number of the boundary-volume-time elements, and the solution for the system of linear equations thus obtained is discussed.     

Numerical methods for solving the eigenvalue problem for a positive branch confocal unstable resonator

The resonator problem for a positive branch confocal unstable resonator reduces to a Fredholm homogeneous integral equation of the second kind, whose numerical solution here is based on a sequence of algebraic eigenvalue problems. We compare two algorithms for the solution of an optical resonator problem. These are obtained by (i) successive degenerate kernel approximation by Taylor polynomials of the Fredholm kernel and (ii) Nyström’s method with Simpson’s rule as the subordinate numerical integration method. The numerical results arising from these routines compare well with other published results, and have the added advantage of simplicity and easy adaptability to other resonator problems.     

一类带扰动的随机脉冲泛函微分方程解的渐近性北大核心CSCD

该文讨论了一类带扰动的随机脉冲泛函微分方程解的渐近性.通过比较扰动方程的解和原方程的解,得到了两者逼近的充分条件.首先,两者在有限的时间区间上相互逼近;其次,当扰动趋于零时,区间长度趋于无穷大,在这个区间上两个解仍然是相互逼近的.最后,举例说明了结果的有效性.     

Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel

It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method.In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation.     

Limit-periodic solutions of integro-differential equations in a critical case

We consider equations with nonlinear terms representable by power series in the variable and functionals in integral form. The equation depends on a small exponentially limitperiodic perturbation, i.e., on a function that exponentially tends to a periodic function as the independent variable increases. In the Lyapunov critical case of one zero root, we prove the existence of a family of exponentially limit-periodic solutions of the equation in the form of power series in the small parameter and arbitrary initial values of the noncritical variables.     

Explicit stability conditions for integro-differential equations with periodic coefficients

New explicit stability conditions are derived for a linear integro-differential equation with periodic operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations. The other approach is based on transforming the equation under consideration in such a way that an appropriate ‘differential’ part of the new equation would possess some reserve of stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the stability problem for a viscoelastic bar compressed by periodic forces. The effect of material and structural parameters on the critical load is studied numerically. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.