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
𝑥̈(𝑡) + 𝑓𝐷(𝑥(𝑡), 𝑥̇ (𝑡)) + 𝑥(𝑡) = 1 − 𝑒2(𝑡, 𝑥(𝑡), 𝑥𝑑 (𝑡), 𝑥̇ (𝑡), 𝑥̇ 𝑑 (𝑡))
𝑥2(𝑡) , 𝑥(𝑡) ∈]0, ∞[
where 𝑥𝑑 (𝑡) = 𝑥(𝑡 − 𝑑) and 𝑥̇ 𝑑 (𝑡) = 𝑥̇ (𝑡 − 𝑑), denote position and velocity feedback respectively,
and
(𝑡, 𝑥(𝑡), 𝑥𝑑 (𝑡), 𝑥̇ (𝑡), 𝑥̇ 𝑑 (𝑡)) = 𝑉 (𝑡) + 𝑔1(𝑥(𝑡) − 𝑥𝑑 (𝑡)) + 𝑔2(𝑥̇ (𝑡) − 𝑥̇ 𝑑 (𝑡)),
is the feedback voltage with positive input voltage 𝑉 (𝑡) ∈ 𝐶(R∕𝑇Z) for 𝑒 ∈ R+, 𝑔1, 𝑔2 ∈ R,
𝑑 ∈ 0, 𝑇 . The damping force 𝑓𝐷(𝑥, 𝑥̇ ) can be linear, i.e., 𝑓𝐷(𝑥, 𝑥̇ ) = 𝑐𝑥̇ , 𝑐 ∈ R+ or squeeze
film type, i.e., 𝑓𝐷(𝑥, 𝑥̇ ) = 𝛾𝑥̇ ∕𝑥3, 𝛾 ∈ R+. The fundamental tool to prove our result is a local
continuation method of periodic solutions from the non-delayed case (𝑑 = 0). Our approach
provides new insights into the delay phenomenon on microelectromechanical systems and can
be used to study the dynamics of a large class of delayed Liénard equations that govern the
motion of several actuators, including the comb-drive finger actuator and the torsional actuator.
Some numerical examples are provided to illustrate our results