Abstract
Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation can be oscillating and asymptotically unstable, the delay equation , where , can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. MSC:34K20.
Highlights
1 Introduction Let us start with one of motivations of this study considering a simplified model for motion of a single mass point
A standard approach is to construct a feedback control u(t) which depends on the difference X(t) – Y (t) or more exactly on the difference X(t – τ (t)) – Y (t – τ (t)) since in real systems delay in receiving signal and in reaction on this signal arises
With constant delay τ and coefficients p and p and without damping term were obtained in the paper [ ]
Summary
The asymptotic stability of the equation x (t) + p x(t) + p x(t – τ ) = , with constant delay τ and coefficients p and p and without damping term were obtained in the paper [ ] These results were based on Pontryagin’s technique for analysis of the roots of quasi-polynomials [ ] and could not be used in the case of In order to formulate several simple corollaries of the main results proven below in Section , let us consider the equation x (t) + a(t)x t – τ (t) – b(t)x t – θ (t) = , t ∈ [ , +∞), x(ξ ) = for ξ < , where a(t), b(t), τ (t), and θ (t) are measurable essentially bounded nonnegative functions.
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