Tracking almost periodic signals under white noise perturbations

A result of the "law of large numbers" type is obtained for the cost of tracking almost periodic signals if the deterministic optimal feedback law is used in the presence of white noise perturbations     

Stability of generalized equations under nonlinear perturbations

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.     

Stability of plane Couette flows with respect to small periodic perturbations

We consider the plane Couette flow v₀=(x_n,0,…,0) in the infinite layer domain , where n≥2 is an integer. The exponential stability of v₀ in Lⁿ is shown under the condition that the initial perturbation is periodic in (x₁,…,x_n−1) and sufficiently small in the Lⁿ-norm.     

Asymptotic decay of solutions of the liouville equation under perturbations

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale. Translated frommatematicheskie Zametki, Vol. 68, No. 2, pp. 195–209, August, 2000.     

Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy

In this paper, we consider the dynamic behavior of an impulsive delay differential equation model with the effects of interspecific allelopathic interaction. A good understanding of the permanence, extinction and the existence of positive periodic solutions is gained. It turns out that the impulsive controls play a crucial role in shaping the above dynamics of the system. Numerical simulations are presented to substantiate the analytical results.     

Stability of linear stochastic delay under steady perturbations

Kiev. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 107–114, September–October, 1992.     

Stability of singular spectral types under decaying perturbations

We look at invariance of a.e. boundary condition spectral behavior under perturbations, W, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported W's to suitable short-range W. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.     

Stability of Martin boundary under non-local Feynman-Kac perturbations

Recently the authors showed that the Martin boundary and the minimal Martin boundary for a censored (or resurrected) -stable process Y in a bounded C^1,1-open set D with (1,2) can all be identified with the Euclidean boundary D of D. Under the gaugeability assumption, we show that the Martin boundary and the minimal Martin boundary for the Schrödinger operator obtained from Y through a non-local Feynman-Kac transform can all be identified with D. In other words, the Martin boundary and the minimal Martin boundary are stable under non-local Feynman-Kac perturbations. Moreover, an integral representation of nonnegative excessive functions for the Schrödinger operator is explicitly given. These results in fact hold for a large class of strong Markov processes, as are illustrated in the last section of this paper. As an application, the Martin boundary for censored relativistic stable processes in bounded C^1,1-smooth open sets is studied in detail.This research is supported in part by NSF Grant DMS-0071486 and a RRF Grant from University of WashingtonMathematics Subject Classification (2000):Primary 31C35, 60J45, 35J10; Secondary 60J50, 60J57     

Stability of large-scale discrete systems under structural perturbations

By using the matrix Lyapunov function, we establish conditions of (uniform) stability and (uniform) asymptotic stability of a large-scale discrete system under structural perturbations.     

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.     

The generalized Zakharov-Shabat system and the soliton perturbations

The nonlinear evolution equations and their inhomogeneous versions related through the inverse scattering method to the generalized Zakharov-Shabat systemL=id/dx+q(x)–J are studied. Here we assume that the potentialq(x)=[J,Q(x)] takes values in the simple Lie algebra g and thatJ is a nonregular element of the Cartan subalgebra . The corresponding systems of equations for the scattering data ofL are derived. These can be applied to the study of soliton perturbations of such equations as the matrix nonlinear Schrödinger equation, the matrixn-wave equations, etc.Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 292–299, May, 1994.     

Global well-posedness for the KP-II equation on the background of a non-localized solution

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in R×T and perturbations that are square integrable in R². In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.     

Stability of derivations under weak-2-local continuous perturbations

Let $${Omega}$$     

Stability of semi-infinite vector optimization problems under functional perturbations

This paper is devoted to the study of continuity properties of Pareto solution maps for parametric semi-infinite vector optimization problems (PSVO). We establish new necessary conditions for lower and upper semicontinuity of Pareto solution maps under functional perturbations of both objective functions and constraint sets. We also show that the necessary condition becomes sufficient for the lower and upper semicontinuous properties in the special case where the constraint set mapping is lower semicontinuous at the reference point. Examples are given to illustrate the obtained results.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.