Periodic Oscillations in Electrostatic Actuators Under Time Delayed Feedback Controller
Periodic Oscillations in Electrostatic Actuators Under Time Delayed Feedback Controller
In this paper, we prove the existence of two positive $T$-periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation\[\ddot{x}(t)+f_{D}(x(t),\dot{x}(t))+ x(t)=1- \dfrac{e \mathcal{V}^{2}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))}{x^2(t)}, \qquad x(t)\in\,]0,\infty[\]where $\displaystyle{x_{d}(t)=x(t-d)}$ and $\displaystyle{\dot{x}_{d}(t)=\dot{x}(t-d),}$ denote position and velocity feedback respectively, and\[\mathcal{V}(t,x(t),x_{d}(t),\dot{x}(t),\dot{x}_{d}(t))=V(t)+g_{1}(x(t)-x_{d}(t))+g_{2}(\dot{x}(t)-\dot{x}_{d}(t)),\] is the feedback voltage with positive input voltage $V(t)\in C(\mathbb{R}/T\Z)$ for $e\in \mathbb{R}^{+}, …