Exponential stability of a second order delay differential equation without damping term

On local ISS of nonlinear second‐order time‐delay systems without damping

“…In References 20,21, the problem of asymptotic stability for the scalar equation $$$$ \ddot{x}(t)+ ax\left(t-h\right)- bx\left(t-g\right)=0 $$$$ was studied, where $$$ x(t)\in \mathbb{R} $$$, $$$ a $$$ and $$$ b $$$ are constant positive coefficients, $$$ h $$$ and $$$ g $$$ are constant positive delays (all these parameters can also be time‐varying). The system (3) has no damping proportional to the velocity $$$ \dot{x}\in \mathbb{R} $$$, and in the delay‐free case (when $$$ h=g=0 $$$) under the restriction $$$ a>b $$$ it has purely oscillating trajectories.…”

“…In addition, we will extend our result obtained for constant delays to time‐varying ones, and next apply it to the rotation stabilization of rigid body using delayed feedback. Note that since $$$ \mu >1 $$$, the local asymptotic stability of (4) cannot be derived from References 20,21 using the linearization techniques.…”

“…A benchmark scenario, where introduction of a small delay ensures asymptotic stability for a neutrally stable planar system, was studied in Reference 1: $$$ \ddot{x}(t)=-{a}_1x(t)+{a}_2x\left(t-\tau \right) $$$ with $$$ {a}_1,{a}_2,\tau >0 $$$ and $$$ x(t)\in \mathbb{R} $$$, and an extension of that result was obtained in the works, 20,21 where a second‐order linear time‐varying system is considered including a delayed position term with a negative gain, and it is proven that it can be stabilized by a position feedback with positive gain and with a sufficiently small delay.…”

Delay-Induced Resonance in the Time-Delayed Duffing Oscillator

“…Thus, the steady-state peak to peak amplitude in this region is zero. The time-delay acts as a damping term in the sense that it damps out the oscillations caused by the second derivative [Berezansky et al, 2015]. The system falls into the positive well, that is x ≈ 3.6 (see Figs.…”

Stabilization by delay distributed feedback control

“…where p, q, τ are positive constants, results on exponential stability were obtained in [11,12,21,29]. First results on stability of the second order delay equation without a damping term were obtained in the paper [15] and then developed in [8,17]. Stability of second order equations with dampling terms was studied in [1, 2, 5, 7, 9-13, 21, 29].…”