Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.     

Spectrum of the monodromy operator of the schrödinger operator with a potential which is periodic with respect to time

One considers the spectrum of the monodromy operator of the Schrödinger operator ? (t) =? Δ +x (x, t) with a potential which is periodic with respect to time. It is shown that under certain conditions on the potential there is no singular continuous spectrum and one investigates the point spectrum of the monodromy operator. Under certain conditions on the potentialq(x, t), periodic with respect to time, one shows the absence of the singular continuous spectrum for the monodromy operator corresponding to the Schrödinger operator ? Δ+q,(x,t).     

Controllability and stabilizability of linear time‐varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time‐varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.     

Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator

The Floquet theory provides a decomposition of a periodic
Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given. As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.

     

On the dichotomy of a system of linear differential equations with conditionally periodic coefficients

We show that a system of linear differential equations with conditionally periodic coefficients is exponentially dichotomous if and only if the spectrum of the monodromy operator does not meet the unit circle.     

Spectral gaps in the dirichlet and neumann problems on the plane perforated by a doubleperiodic family of circular holes

We show that the spectrum of the Dirichlet and Neumann problems for the Laplace operator in the plane perforated by a double–periodic family of circular holes contains gaps (even any a priori given number of gaps) of certain radii of holes. The result is obtained by asymptotic analysis of the cell spectral problem, interpreted as a problem in a domain with thin bridges. Some open questions are stated.     

On the range of use of the tight-binding approximation method for a complex-valued potential

We consider a one-dimensional Schrödinger operator with a periodic potential constructed as the sum of the shifts of a given complex-valued potential q that decreases at infinity. Mathematical justification of the tight-binding approximation method is presented. Let λ₀ be an isolated eigenvalue of the Schrödinger operator with a potential q. Then, for the corresponding operator with a periodic potential, a continuous spectrum exists in the neighborhood of λ₀. The asymptotic behavior of this part of the spectrum as the period increases infinitely is studied for the cases of one- and two-dimensional invariant subspaces corresponding to λ₀.     

Spectral properties of a thin layer with a doubly periodic family of thinning regions

We show that the spectrum of the Dirichlet problem for the Laplace operator in a layer with a doubly periodic structure has gaps and determine several characteristics of their location. The result is obtained by asymptotic analysis of a model spectral problem on the periodicity cell.     

Lower part of the spectrum for the two-dimensional Schrödinger operator periodic in one variable and application to quantum dimers

We study the semiclassical asymptotic approximation of the spectrum of the two-dimensional Schrödinger operator with a potential periodic in x and increasing at infinity in y. We show that the lower part of the spectrum has a band structure (where bands can overlap) and calculate their widths and dispersion relations between energy and quasimomenta. The key role in the obtained asymptotic approximation is played by librations, i.e., unstable periodic trajectories of the Hamiltonian system with an inverted potential. We also present an effective numerical algorithm for computing the widths of bands and discuss applications to quantum dimers.     

Asymptotic form of the spectrum of operators associated with p-adic fields

We consider a locally compact nonconnected nondiscrete field and study a linear operator given by the sum of the operator of multiplication by a function and the operator of convolution with a generalized function. We derive the asymptotic form of the spectrum of that linear operator. In this problem, we use the generalized p-adic Feynman-Kac formula.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

Operator Hamiltonians and anticanonic equations with periodic coefficients

In a Hilbert space we study Hamiltonians and anticanonic equations with periodic coefficients. We prove existence theorems for the solutions of ill-posed Cauchy problems for the given equations. Following Krein we define the notion of the genus of the spectrum points of the monodromy operator of an equation of the class being studied. We formulate existence and uniqueness theorems for the solutions when determining the reflected and the transmitted waves for a specified incident wave. The theory developed is applied to the study of cylindrical waveguides with a periodic filling.Translated from Problemy Matematicheskogo Analiza. No. 4: Integralnye i Differentsial'nye Operatory. Differentsial'nye Uraveniya, pp. 9–36, 1973.     

Schrödinger operator with a perturbed small steplike potential

We consider the Schrödinger operator with a potential that is periodic with respect to two variables and has the shape of a small step perturbed by a function decreasing with respect to a third variable. We show that under certain conditions on the magnitudes of the step and the perturbation, a unique level that can be an eigenvalue or a resonance exists near the essential spectrum. We find the asymptotic value of this level.     

On the second term of the spectral asymptotics of the transmission problem

This article is devoted to the investigation of spectral asymptotics for the second-order differential operator with transmission boundary conditions. A two-term asymptotic formula is obtained for the distribution function of eigenvalues under some assumptions not excluding the reach set of periodic billiard trajectories.     

Opening of a gap in the continuous spectrum of a periodically perturbed waveguide

It is established that a small periodic singular or regular perturbation of the boundary of a cylindrical three-dimensional waveguide can open up a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator in the resulting periodic waveguide. A singular perturbation results in the formation of a periodic family of small cavities while a regular one leads to a gentle periodic bending of the boundary. If the period is short, there is no gap, while if it is long, a gap appears immediately after the first segment of the continuous spectrum. The result is obtained by asymptotic analysis of the eigenvalues of an auxiliary problem on the perturbed cell of periodicity.     

用于周期有限带Dirac谱问题非线性化的迹公式

本文研究具有周期有限带位势的Dirac算子,利用Dirac算子与单值算子的交换性,定义Bloch函数和乘子曲线,获得Dubrovin-Novikov型公式;进而通过复球面上的留数计算及规范变换,分别得到相应于谱带左端点、右端点以及双侧端点的特征函数的迹公式.作为应用,将Dirac谱问题非线性化得到在Liouville意义下完全可积的Hamilton系统.     

On the density of states for the periodic Schrödinger operator

An asymptotic formula for the density of states of the polyharmonic periodic operator (?δ) ^l +V inR ⁿ ,n≥2,l>1/2 is obtained. Special consideration is given to the case of the Schrödinger equationn=3,l=1,V being a periodic potential, where the second term of the asymptotic is found.