Recent Advances on Periodic Motions in Parallel-Plate Electrostatic Actuators

This review shows some recent results on the existence, multiplicity, and stability of periodic solutions for a large class of micro-electro-mechanical parallel plate actuators, for which the equation of motion is given by a Liénard differential equation like \(\displaystyle \ddot {x}(t)+c_{d}(x(t))\dot {x}(t)+ h_{s}(x(t)) =f_{e}(t,x(t)), \) where \(c_{d}(x)\dot {x}\) represents a damping force, \(h_{s}(x)\) is a linear or nonlinear restoring force, and \(f_{e}:\mathbb {R}\times ]a,b[\to \mathbb {R}\) is a positive function, which is periodic in t, with one or two singularities in x and represents an electrostatic force between the plates due to a voltage load. In addition, we present a novel result in the case when the actuator is acting under the presence of a time-delay controller. In such a case, \(\displaystyle f_{e}(t,x)\, \rightarrow \, f_{e}(t,\sigma (s(t),s_{d}(t),g)), \quad \text{with} \quad \sigma (s,s_{d},g)=g(s_{d}-s), \) with \(s_{d}(t)=s(t-d)\) , where \(s=s(t)\) is the output signal ( \(s(t)=x(t)\) or \(s(t)=\dot {x}(t)\) ) measured by the controller with gain \(g\in \mathbb {R}\) and delay \(d\in \mathbb {R}^{+}\) as parameters of control. A brief introduction of the mathematical tools used to show the results in this document is presented and finally, some open mathematical problems related to the dynamics of parallel-plate actuators are listed.