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)+f _D(x(t),ẋ(t))+x(t)=1−[Formula presented],x(t)∈]0,∞[where x _d(t)=x(t−d) and ẋ _d(t)=ẋ(t−d), denote position and velocity feedback respectively, and V(t,x(t),x _d(t),ẋ(t),ẋ _d(t))=V(t)+g ₁(x(t)−x _d(t))+g ₂(ẋ(t)−ẋ _d(t)),is the feedback voltage with positive input voltage V(t)∈C(R/TZ) for e∈R ⁺,g ₁,g ₂∈R, d∈0,T. The damping force f _D(x,ẋ) can be linear, i.e., f _D(x,ẋ)=cẋ, c∈R ⁺ or squeeze film type, i.e., f _D(x,ẋ)=γẋ/x ³, 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.