On the asymptotic stability of equilibria of nonlinear mechanical systems with delay

Abstract

We consider some classes of nonlinear mechanical systems with retarded argument. It is assumed that, in the absence of delay, the systems in question have asymptotically stable equilibria. We analyze how the delay affects the stability of these equilibria. The Lyapunov function method and Razumikhin's approach are used to derive conditions under which asymptotic stability is preserved for arbitrary delay values. We suggest a method for stabilizing strongly nonlinear conservative systems by constructing a delay feedback control depending only on the generalized coordinates.

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

“…Let us choose a Lyapunov function for (5) as follows: $$$$ V\left(x,\dot{x}\right)=\frac{1}{2}{\dot{x}}^{\top}\dot{x}+\Pi (x)-\delta {\left\Vert \dot{x}\right\Vert}^{\lambda -1}{x}^{\top}\dot{x}+\varepsilon {\left\Vert x\right\Vert}^{\sigma -1}{x}^{\top}\dot{x}, $$$$ where $$$ \delta $$$ and $$$ \varepsilon $$$ are positive coefficients, $$$ \lambda \ge 1 $$$, $$$ \sigma \ge 1 $$$ are powers to be properly selected. Note that the full energy $$$ E\left(x,\dot{x}\right)=\frac{1}{2}{\dot{x}}^{\top}\dot{x}+\Pi (x) $$$ can be used to establish global asymptotic stability of the Liénard Equation (6), however, $$$ E $$$ is not a strict Lyapunov function in this case, then a variant of such a Lyapunov function $$$ V $$$ was proposed in Reference 37 being strict and guaranteeing asymptotic stability of the origin locally. Using properties of homogeneous functions (i.e., …”

“…which is asymptotically stable at the origin, [35][36][37] and the remaining terms d(t) + Δ(x t , ̇xt ) are considered as perturbations. For the rest of the proof, all computations are done for the case h > 0, and if h = 0, then the arguments stay unchanged by imposing the respective terms to be zero.…”

“…ẋ⊤ ẋ + Π(𝑥) can be used to establish global asymptotic stability of the Liénard equation ( 6), however, 𝐸 is not a strict Lyapunov function in this case, then a variant of such a Lyapunov function 𝑉 was proposed in 37 being strict and guaranteeing asymptotic stability of the origin locally. Using properties of homogeneous functions (i.e., 𝑎 1 ‖𝑥‖ 𝜇+1 ≤ Π(𝑥) ≤ 𝑎 2 ‖𝑥‖ 𝜇+1 for all 𝑥 ∈ ℝ 𝑛 for some real 𝑎 1 ≤ 𝑎 2 , and 𝜕Π (𝑥) 𝜕𝑥 is also a homogeneous vector function of degree 𝜇) we obtain…”

On Stability of Second-Order Nonlinear Time-Delay Systems Without Damping

“…which is asymptotically stable at the origin [27], [28], [29], and the remaining terms are considered as perturbations.…”

“…ẋ2 + β µ+1 x µ+1 can be used to establish global asymptotic stability of the Liénard equation ( 6), however, E is not a strict Lyapunov function in this case, then V was proposed in [29] being strict and guaranteeing asymptotic stability only locally.…”