The goal of this project is to develop a methodology for the study of controllability properties of complex coupled nonlinear systems of partial and ordinary differential (or integro-differential) equations modeling the swimming phenomena. These systems include the fluid equations (for example, Stokes or Navier-Stokes equations) and the equations describing the position of a swimming device (or a living organism) in the fluid. The swimming motion of the latter is the result of the action of "internal" forces generated by the device, which also serve as the "external" forces for the surrounding fluid. In a typical swimming model the aforementioned internal forces are either the forces, preserving the structure of the device at hand or the "propulsion" forces (for example, of rotation or rowing nature), which make it swim. The magnitudes of the latter forces are modeled as coefficients and serve as nontraditional (in the framework of the mathematical controllability theory) multiplicative or bilinear controls.
In this project we will investigate the mathematical nature of the swimming motion of a mechanical device (such as a robotic fish or a robotic eel) or a living organism in a fluid. What makes the latter to swim the way it does? Or more precisely, what kind forces are necessary to employ to achieve the desirable swimming motion? How can we design a mechanical device that can do the same? Answers to these questions are of great interest in biological and medical applications and in engineering applications dealing with propulsion systems in fluids.
