Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

Abstract

We study control properties of a linearized fluid-structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that, as in the case without structure, the system is not exactly null-controllable in finite time.

Robust multi‐model controllers design without impulse in switching times for non‐linear vibrations suppression of sandwich plate

“…N is the number of sub‐regions and by the use of Taylor series expansion and neglecting all higher (higher than 1st) order terms, the linearized system is gained as stated in Equation (8) [33]: $$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {{{{\dot{\rm x}}}}}_{\rm{i}}\left( {\rm{t}} \right) \approx {\left. {\dfrac{{\partial {\rm{l}}}}{{\partial {\rm{x}}}}} \right|}_{ \def\eqcellsep{&}\begin...$$…”

“…N is the number of sub-regions and by the use of Taylor series expansion and neglecting all higher (higher than 1st) order terms, the linearized system is gained as stated in Equation ( 8) [33]:…”

On the exact controllability of a Galerkin scheme for 3D viscoelastic fluids with fractional Laplacian viscosity and anisotropic filtering

“…Linearizations emerge quite naturally as approximations for the description of viscoelastic fluids behavior, and related models have been analyzed in a number of papers (see, e.g., Refs. [21,22,46,47,52]) where, depending on the situation, the authors study and deduce various types (null, approximate, exact) of controllability results.…”