Optimal Control in Coupled Systems with Moving Interfaces

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This project addresses the control of turbulence inside fluid flow in the case of free boundary interaction between a viscous fluid and a moving and deforming elastic body. Reducing and controlling turbulence flow is particularly relevant in the design of small-scale unmanned aircrafts and morphing aircraft wings (e.g. improving the flight of a robobee), and is also of great interest in the medical community (for example, blood flow in a stenosed or stented artery). The proposed problem presents new challenges since it treats the case of a moving and deforming elastic body coupled with a viscous fluid, building on the existing literature on control problems in fluid-structure interactions, which is predominantly focused on the assumption of small but rapid oscillations of the solid body, and therefore assumes that the common interface is static. Due to the presence of the free boundary and the fully nonlinear coupled system, the issue of existence and uniqueness of an optimal control will involve strategies from sensitivity and shape differentiability analysis, on top of techniques from control theory of partial differential equations and well-posedness analysis for fluid-structure interactions. Therefore, the project will be of interest to a broad mathematical and engineering audience. Mathematically, the proposed research will lead to: (i) the construction of quasilinear theory arising in Navier-Stokes equations coupled with waves, (ii) the development of the theory of strong shape derivatives for hyperbolic problems with non-smooth Neumann boundary conditions, which is challenging due to the failure of the Lopatinski condition, and (iii) the study of well-posedness analysis for the first linear model of fluid-elasticity interaction that takes into account the common interface and its curvatures, which are critical for a correct physical interpretation of the coupling. The project will launch new research in the field, since the approach and the techniques introduced for the minimization of drag in the fluid-elasticity interaction can be adjusted and used to investigate inverse or control problems in many other free boundary coupled physical systems, and with different types of controls. The principal investigator has partnered with the North Carolina Museum of Natural Sciences in an effort to create a "Math Day at the Museum", in order to promote and present her research to a broad audience, showcase students' research through posters and presentations, and encourage the participation of women and minorities in the study of math. During these events, the principal investigator will give public presentations on her research, develop and set up demonstrations of real world phenomena illustrating applications of partial differential equations and their control. In addition, the principal investigator plans to promote women and minorities in math, by involving several student groups from North Carolina State University, including the Association for Women in Mathematics, Women in Science and Engineering, and the Society of African American Physical and Mathematical Sciences.