Analysis and control theory of some cochlear models
The standard 2-dimensional cochlear model consists of a 1-dimensional elastic structure, modeling the basilar membrane, surrounded by an incompressible 2-dimensional fluid within a 2-dimensional cochlear cavity. The dynamics are typically driven by a pressure differential across the basilar membrane transmitted through the round and oval windows (a portion of the boundary of the cochlear). First we describe a model in which the basilar membrane is modeled as an infinite array of oscillators and the fluid is described by Laplace's equation. In this setting, we show that the coupled system is approximately controllable with control acting on an arbitrary open set of the basilar membrane. Second we consider a cochlear model where the basilar membrane has a longitudinal elastic tension. In this case the differential equations describing the dynamics of the system have variable coefficients. We first change the variables and then use the method of multipliers to prove exact controllability result.