Section: Research Program

Control and stabilization of heterogeneous systems

Fluid-Structure Interaction Systems (FSIS) are present in many physical problems and applications. Their study require to solve 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 [98] , [90] , [69] , and almost all mathematical results on such FSIS have been obtained in the last twenty years.

The most studied issue concerns the well-posedness of the problem modeling a rigid body moving into a viscous incompressible fluid. If the fluid fills the unbounded domain surrounding the structure, the free boundary difficulty can be overcome by using a simple change of variables that makes the rigid body fixed. One can then use classical tools for the Navier-Stokes system and obtain the existence of weak solutions (see, for instance, [57] , [58] , [91] ) or strong solutions for the case of a ball [95] . When the rigid body is not a ball, the additional terms due to the change of variables modify the nature of the system and only partial results are available for strong solutions [59] , [45] , [92] . When the fluid-solid system is confined in a bounded domain, the above strategy fails. Several papers have developed interesting strategies in order to obtain the existence of solutions. Since the coupling is strong, it is natural to consider a variational formulation for both the fluid and the structure equations (see [48] ). One can then solve the FSIS by considering the Navier-Stokes system with a penalization term taking into account the structure ( [42] , [89] , [53] ) or using a time discretization in order to fix the rigid body during some time interval ( [63] ). Using an appropriate change of variables has also been used (see [62] , [94] ), but of course, its construction is more complex than in the case where the FSIS fills the whole space. Most of the above results only hold up to a possible contact between two structures or between a structure and the exterior boundary. If the considered configuration excludes contacts, some authors also investigated the large time behavior of this system and the existence of time periodic solutions [97] , [79] , [60] .

Many other FSIS have been studied as well. Let us mention, for instance, rigid bodies immersed in an incompressible perfect fluid ( [81] , [66] , [61] ), in a viscous compressible fluid ( [47] , [35] , [52] , [36] ), in a viscous multipolar fluid or in an incompressible non-Newtonian fluid ( [54] ). 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). Several models for the dynamics of the deformable structure exist: one can use the plate equations or the elasticity equations. The obtained coupled FSIS is a complex system and the study of its well-posedness raises several difficulties. The main one comes from the fact that we gather two systems of different nature, as the linearized problem couples a parabolic system with a hyperbolic one. Theoretical studies have been performed for approximations of the complete system, using two strategies: adding a regularizing term in the linear elasticity equations (see [40] , [35] , [72] ) or approximate the equations of linear elasticity by a finite dimensional system (see [49] , [38] ). For strong solutions, the coupling between hyperbolic-parabolic systems leads to seek solutions with high regularity. The only known results [43] , [44] in this direction concern local (in time) existence of regular solutions, under strong assumptions on the regularity of the initial data. Such assumptions are not very satisfactory but seem inherent in this coupling between two systems of different natures. Another option is to consider approximate models, but so far, the available approximations are not obtained from a physical model and deriving a more realistic model is a difficult task.

In some particular important physical situations, one can also consider a simplified model. For instance, 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 ( [87] ). Although simplified, this model already contains many difficulties and allows starting the mathematical study of a challenging problem: understanding the locomotion mechanism of aquatic animals.

Using the above results and the corresponding tools, we aim to consider control or stabilization problems for FSIS. Some control problems have already been considered: using an interior control in the fluid region, it is possible to control locally the velocity of the fluid together with the velocity and the position of the rigid body (see [67] , [37] ). The strategy of control is similar to the classical method for a fluid (without solid) (see, for instance, [55] ) but with the tools developed in [94] . A first result of stabilization was obtained in [83] and concerns a fluid contained in bounded cavity where a part of the boundary is modeled by a plate system. The feedback control is a force applied on the whole plate and it allows to obtain a local stabilization result around the null state.

To extend these first results of control and stabilization, we first have to make some progress in the analysis of FSIS:

  • Contact: It is important to understand the behavior of the system when two structures are close, and in particular to understand how to deal with contact problems;

  • Deformable structures: To handle such structures, we need to develop new ideas and techniques in order to couple two infinite-dimensional dynamics of different nature.

At the same time, we can tackle control problems for simplified models. For instance, in some regimes, the Navier-Stokes system can be replaced by the Stokes system and the Euler system by Laplace's equation