A 2D membrane MEMS device model with fringing field: Curvature-dependent electrostatic field and optimal control

An important problem in membrane micro-electric-mechanical-system (MEMS) modeling is the fringing-field phenomenon, of which the main effect consists of force-line deformation of electrostatic field E near the edges of the plates, producing the anomalous deformation of the membrane when external voltage V is applied. In the framework of a 2D circular membrane MEMS, representing the fringing-field effect depending on |u|2 with the u profile of the membrane, and since strong E produces strong deformation of the membrane, we consider |E| proportional to the mean curvature of the membrane, obtaining a new nonlinear second-order differential model without explicit singularities. In this paper, the main purpose was the analytical study of this model, obtaining an algebraic condition ensuring the existence of at least one solution for it that depends on both the electromechanical properties of the material constituting the membrane and the positive parameter δ that weighs the terms |u|2. However, even if the the study of the model did not ensure the uniqueness of the solution, it made it possible to achieve the goal of finding a stable equilibrium position. Moreover, a range of admissible values of V were obtained in order, on the one hand, to win the mechanical inertia of the membrane and, on the other hand, to ensure that the membrane did not touch the upper disk of the device. Lastly, some optimal control conditions based on the variation of potential energy are presented and discussed.