The proposed research is aimed at better understanding the controllability structure of coupled fluid-elastic systems. Specifically, we focus on the situation of a fluid in a three-dimensional domain with a flexible boundary that is described by a plate or membrane equation. We suggest several problems which serve to address our fundamental question: How much control in one or more components of the coupled system is needed to obtain (i) controllability or approximate controllability of the coupled system or (ii) partial controllability (approximate or exact) of one or more components of the system? The proposed research extends the state-of-the-art in several ways. First, the partial controllability problem in the context of a fluid-elastic system has not been addressed for the types of physically-based models we are considering. Secondly, many of the models we propose to analyze have not been studied even for well-posedness. This is a nontrivial step even in the linear case for some of the models (e.g., when considering free boundary conditions). However we also hope to address local controllability for the nonlinear problems associated with the free boundary problem for the Stokes system associated with a moving flexible boundary. Likewise we hope to make some progress in the case of a compressible Navier-Stokes fluid coupled with a plate. Our main analytical tool to attack these problems will be the method of Carleman estimates together with microlocal analysis. Development of Carleman estimates for these types of coupled systems might turn out to be be very useful in a broader context, as it is likely that a similar approach may be useful in the analysis of a variety of other types of coupled systems. We also propose to investigate the closely related inverse problem of determination of plate coefficients through knowledge of the appropriately defined Dirichlet-to-Neumann maps associated with solutions of the coupled plate-fluid system.
The fluid-elastic structures we aim to study arise in many practical applications. For example,the problem of minimizing the engine noise in a portion of an aircraft fuselage through controllers placed on and inside the aircraft is closely related to the partial controllability problem (ii) mentioned above. One specific fluid-elastic structure we plan to investigate is the cochlea, which is the auditory organ in the inner ear that translates sound vibrations to electrical impulses transmitted to the brain. A better understanding of the controllability structure of the cochlea could lead to an improved design of cochlear implants and hearing aids.
