Abstract
In this paper the exponential stability of linear neutral second order differential equations is studied. In contrast with many other works, coefficients and delays in our equations can be variable. The neutral term makes this object essentially more complicated for the study. A new method for the study of stability of neutral equation based on an idea of the Azbelev W-transform has been proposed. An application to stabilization in a model of human balancing has been described. New stability tests in explicit form are proposed.
Highlights
IntroductionThe principal development of the results presented in [3] is in considering cases of oscillation behavior of solutions to neutral equation (1.1), where positivity-based approach does not work (see Rem. 3.3)
The main object of this paper is the second order neutral delay differential equation m m m x (t) + qi(t)x (t − φi(t)) + ai(t)x (t − τi(t)) + bi(t)x(t − θi(t)) = f (t), t ∈ [0, ∞)i=1 i=1 i=1 with the corresponding initial functions (1.1)x(ξ) = φ(ξ), x (ξ) = ψ(ξ), x (ξ) = η(ξ) for ξ < 0, defining what should be set into the equation instead of x(t − θi(t)), x (t − τi(t)) and x (t − φi(t)) in the case of t − θi(t) < 0, t − τi(t) < 0 or t − φi(t) < 0 respectively
I=1 i=1 i=1 with the corresponding initial functions x(ξ) = φ(ξ), x (ξ) = ψ(ξ), x (ξ) = η(ξ) for ξ < 0, defining what should be set into the equation instead of x(t − θi(t)), x (t − τi(t)) and x (t − φi(t)) in the case of t − θi(t) < 0, t − τi(t) < 0 or t − φi(t) < 0 respectively
Summary
The principal development of the results presented in [3] is in considering cases of oscillation behavior of solutions to neutral equation (1.1), where positivity-based approach does not work (see Rem. 3.3). The results of [3] cannot be used for the model of human balancing, see, for example, equation (4.10), in which the coefficient of φ(t) is negative and the assumption of the main result of [3] about exponential stability of the ordinary part φ + Aφ + Bφ = 0 is not fulfilled.
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