Feedback stabilization of a simplified 1d fluid–particle system

Abstract

We consider the feedback stabilization of a simplified 1d model for a fluid-structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ < σ0. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.Mathematics Subject Classification (2010): 74F10, 35Q35, 76D55, 93C20, 93D15.

On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems

“…We prove local Carleman estimates that permit to check (UC) for general coupled Navier-Stokes systems and we deduce feedback and dynamical stabilization of nonlinear MHD and micropolar systems (see Corollary 26, Theorem 21 and Theorem 24). Concerning the use of Fattorini's criterion for feedback stabilization of fluid-structure system we refer to [7,6]. Finally, let us underline that a stabilizability property can also be used to tackle some controllability issues: we refer for instance to the work [13] where the authors prove the global controllability to steady trajectories of a 1d nonlinear heat equation by using a stabilization procedure involving only the unstable modes.…”

Eight(y) mathematical questions on fluids and structures

“…To our knowledge, multiple symmetric solutions of (10)- (11) have not even been detected numerically.…”

“…). For any ðU; V Þ a H 1=2 ðGÞ satisfying (12) there exists a weak solution ðu 1 ; u 2 ; pÞ a H 1 ðSÞ Â L 2 0 ðSÞ of (10)- (11). Moreover:…”

“…Theorem 5 impliess that large lifts may occur only when uniqueness of the solution is no longer true. If the smallness assumption of ðU; V Þ is violated one may obtain multiplicity results for (10)- (11), see [158,244,295,296]. In particular, Velte [296] showed that at a certain Reynolds number there is more than one solution of (10) in the domain between two concentric rotating cylinders, see Section 4 and [285, Section 4 in Chapter II].…”

Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid

“…Taking the L ∞ (F 0 ) norm of all the derivatives of {v} and {w} in the remaining terms in (6.4), then using once again (4.7) and (4.10) with the right Sobolev embeddings, we get 2,5,3 , where C > 0 does not depend on any other parameter. We conclude the proof of (ii) by using Theorem 3.1, Proposition 4.1 and Theorem 5.1.…”