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 <(x)double over dot>(t) +f(D)(x(t), <(x)over dot>(t) = 1 - e nu(2)(t, x(t), x(d)(t), <(x)over dot>(t), <(x)over dot>(d)(t))/x2(t), x(t) is an element of]0,infinity[ where xd(t) = x(t - d) and <(x)over dot>(d)(t) = <(x)over dot>(t - d), denote position and velocity feedback respectively, and nu(t, x(t), x(d)(t), <(x)over dot>(t), <(x)over dot>(d)(t)) = V(t) + g(1)(x(t)) - x(d)(t)) + g(2)(<(x)over dot>(t) - <(x)over dot>(d)(t)), is the feedback voltage with positive input voltageV(t) is an element of C(R/TZ) for e is an element of R-+,R- g(1), g(2) is an element of R,d is an element of 0, T. The damping force f(D)(x, <(x)over dot>) can be linear, i.e., f(D)(x, <(x)over dot>) =c<(x)over dot>, c is an element of R+ or squeeze film type, i.e f(D)(x, <(x)over dot>) = gamma<(x)over dot>/x(3), gamma is an element of 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 Lienard 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.