Exponential Decay Properties of a Mathematical Model for a Certain Fluid-Structure Interaction

Abstract

In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. In the case that the PDE plate component is rotational inertia-free, one will have that solutions of this fluid-structure PDE system exhibit an exponential rate of decay. By way of proving this decay, an estimate is obtained for the resolvent of the associated semigroup generator, an estimate which is uniform for frequency domain values along iR. Subsequently, we proceed to discuss relevant point control and boundary control scenarios for this fluid-structure PDE model, with an ultimate view to optimal control studies on both finite and infinite horizon. (Because of said exponential stability result, optimal control of the PDE on time interval (0, ∞) becomes a reasonable problem for contemplation.)

Rational rates of uniform decay for strong solutions to a fluid-structure PDE system

“…With the exception of the aforementioned [19], along with [4] -each which deals with the same kind of FSI studied in the present work, and both providing a proof of exponential rates of uniform decay for finite energy solutions -and of this contribution, the studies on the stability properties of solutions to PDE systems which describe actual FSI, in the absence of any form of additional dissipation, are ongoing as conclusive answers are still lacking.…”

“…Exponential decay for solutions of the present FSI model does not seem likely: owing to the coupling mechanism of the disparate PDE dynamics, via the matching of structural and fluid velocities, control of the mechanical velocity solution variable in H 1 ( )-norm is quite problematic. By contrast, exponential decay of the FSI model for ρ = 0 is possible, inasmuch as the mechanical velocity solution component can be readily controlled in (energy) L 2 -norm, via the Dirichlet trace of the dissipative fluid velocity; see [19] and [4]. Given the uniform decay rate of order O(1/t 1− ) which was obtained in [22] for a 'simplified' FSI model (see also [9]), the rational rate obtained in Theorem 1.5 appears optimal.…”

“…The reason for Assumption 1.1 is that, if [O, ] obey either (G. 1) or (G.2), then for smooth initial data [u 0 , w 1 , w 2 ] one is assured of sufficiently smooth solutions [u, w, w t ] to the fluid-structure system (1.2) below. (See [4,Appendix], and the definition of associated fluid-structure generator A ρ : H ρ → H ρ in (2.9)-(2.10) of Section 2.) Such higher regularity will justify the PDE multiplier computations, which are requisite for our derivation of rational decay rates.…”

“…Well-posedness of the IBVP (1.2)-(1.3) has been fully discussed in [5] and [4,Appendix] for both cases ρ > 0 and ρ = 0. The proof of well-posedness provided in [5] hinges upon demonstrating the existence of a modeling semigroup e A ρ t t≥0 ⊂ L(H ρ ), for an appropriate generator A ρ : H ρ → H ρ .…”

“…Aiming as we are to obtain sharp decay rate estimates for the solutions to the initial/boundary value problem (IBVP) (1.2)-(1.3), with the preliminary analysis of [4] in hand, which considered the case that the elastic wall dynamics is modeled via the Euler-Bernoulli equation (that is, when ρ = 0 in (1.2e)), we choose as in [4] to operate here in the "frequency domain". To wit, we will invoke an energy method with respect to a formally 'Laplace transformed' version of the system (1.2)-(1.3), with the ultimate objective of applying the sharp resolvent criterion in [12] (see a penultimate version of this resolvent criterion in [39]).…”