Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions
We consider a coupled system of a linearly elastic body immersed in a flowing fluid which is modeled by means of the incompressible Navier–Stokes equations with mixed Dirichlet–Neumann-type boundary conditions. For this system we formulate an optimal control problem which amounts to a minimization under constraints of a hydro-elastic pressure at the interface between the two environments. The corresponding functional lacks convexity and radial unboundedness — a serious obstacle to the solution of the optimization problem. The approach taken to solve it is based on transforming the variable domain occupied by the fluid into a fixed domain corresponding to the undeformed elastic inclusion. This leads to a free boundary elliptic problem. Mathematical challenge also results from the fact that the corresponding quasilinear elliptic model is equipped with mixed (Zaremba type) boundary conditions, which intrinsically lead to compromised regularity of elliptic solutions. It is shown that under the assumption of small strains, the controlled structure is wellposed in suitable Sobolev's spaces and the nonlinear control-to-state map is well defined and continuous. The obtained wellposedness result provides thus foundation for proving existence of an optimal control, where the latter is based on compensated compactness methods. The change in the boundary conditions makes the analysis different and substantially more challenging — particularly at the level of wellposedness of both uncontrolled and controlled dynamics. A key to the existence/uniqueness theory is a suitable localization of the spatial domain and of the resulting PDE. The geometry of the fluid domain plays a critical role in the arguments. The analysis of optimal control is nonstandard due to the lack of convexity and of radial unboundedness of the associated functional cost — main tools for proving weak lower-semicontinuity of the functional. This difficulty is dealt with the help of compensated compactness.
Nonlinear Analysis: Real World Applications
Lasiecka, I., Szulc, K., & Żochowski, A. (2018). Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions. Nonlinear Analysis: Real World Applications, 44, 54-85. https://doi.org/10.1016/j.nonrwa.2018.04.004