Section: Research Program

Control and stabilization of heterogeneous systems

Fluid-Structure Interaction Systems (FSIS) are present in many physical problems and applications. Their study involves solving several challenging mathematical problems:

  • Nonlinearity: One has to deal with a system of nonlinear PDE such as the Navier-Stokes or the Euler systems;

  • Coupling: The corresponding equations couple two systems of different types and the methods associated with each system need to be suitably combined to solve successfully the full problem;

  • Coordinates: The equations for the structure are classically written with Lagrangian coordinates whereas the equations for the fluid are written with Eulerian coordinates;

  • Free boundary: The fluid domain is moving and its motion depends on the motion of the structure. The fluid domain is thus an unknown of the problem and one has to solve a free boundary problem.

In order to control such FSIS systems, one has first to analyze the corresponding system of PDE. The oldest works on FSIS go back to the pioneering contributions of Thomson, Tait and Kirchhoff in the 19th century and Lamb in the 20th century, who considered simplified models (potential fluid or Stokes system). The first mathematical studies in the case of a viscous incompressible fluid modeled by the Navier-Stokes system and a rigid body whose dynamics is modeled by Newton's laws appeared much later [93], [88], [68], and almost all mathematical results on such FSIS have been obtained in the last twenty years.

The most studied FSIS is the problem modeling a rigid body moving into a viscous incompressible fluid ( [51], [47], [87], [57], [62], [90], [92], [76], [60]). Many other FSIS have been studied as well. Let us mention [78], [65], [61], [50], [40], [56], [41], [58] for different fluids. The case of deformable structures has also been considered, either for a fluid inside a moving structure (e.g. blood motion in arteries) or for a moving deformable structure immersed in a fluid (e.g. fish locomotion). The obtained coupled FSIS is a complex system and its study raises several difficulties. The main one comes from the fact that we gather two systems of different nature. Some studies have been performed for approximations of this system: [45], [40], [71], [52], [43]). Without approximations, the only known results [48], [49] is done with very strong assumptions on the regularity of the initial data. Such assumptions are not satisfactory but seem inherent to this coupling between two systems of different natures. In order to study self-propelled motions of structures in a fluid, like fish locomotion, one can assume that the deformation of the structure is prescribed and known, whereas its displacement remains unknown ( [85]). This permits to start the mathematical study of a challenging problem: understanding the locomotion mechanism of aquatic animals. This is related to control or stabilization problems for FSIS. Some first results in this direction were obtained in [66], [42], [81].