Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure

Abstract

Abstract. The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.Key words. neutral system, strong stability, spectral theory Notations.C set of complex numbers [27] in mechanical engineering. Equations of neutral type also arise in boundary controlled hyperbolic partial differential equations subjected to small feedback delays [22,6] and in implementation schemes of predictive controllers for time-delay systems [7,25]. In this paper we discuss stability properties of the linear neutral equatioṅ

Positive trigonometric polynomials for strong stability of difference equations

“…However, even though these methods can handle the problem of nonsmoothness, they converge to local extrema as a rule. As suboptimal solutions are not sufficient (the global maximum of the spectral radius is needed) a brute force method has been used to solve the task so far, see [21,23,28]. In the first step, each dimension of [0, 2π] m is discretized to N points.…”

“…In order to handle this hypersensitivity of the stability of the difference equation with respect to delay values, the concept of strong stability was introduced by [10]. Let us remark that the strong stability concept has recently been generalized by [23] toward difference equations with dependencies in the delays.…”

Positive trigonometric polynomials for strong stability of difference equations

“…However, even though these methods can handle the problem of nonsmoothness, they converge to local extrema as a rule. As suboptimal solutions are not sufficient (the global maximum of the spectral radius is needed) a brute force method has been used to solve the task so far, see [18,20,25]. In the first step, each dimension of [0, 2π] m is discretized to N points.…”

“…In order to handle this hypersensitivity of the stability of the difference equation with respect to delay values, the concept of strong stability was introduced by [8]. Let us remark that the strong stability concept has recently been generalized by [20] towar d difference equations with dependencies in the delays.…”

A comparison of possible exponential polynomial approximations to get commensurate delays

“…Compared to retarded ones, NTDS are much more advanced, tricky and intricate regarding spectral properties [1,4,9]. Namely, positions of vertical strips of poles are sensitive to infinitesimal delay changes, which give rise to the notion of strong stability [10,11] that is affected i.a. by the rational dependence of delays [9].…”