This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated with the stationary state. We prove again that for initial data that are close to the stationary state, we can stabilize the position and velocity of the rigid body and the velocity of the fluid. Moreover, this domain may differ from the domain F(hS, RS) of the stationary solution.
Since the feedback control is constructed independent of the initial date, it is more robust to initial model fluctuations or inaccuracies. As already explained above, in the 2D case, due to problems related to the time regularity of the pressure and time derivative of the velocity, we assume This means that at time t = 0 it is not allowed to disturb the initial positions, but only the velocity of the fluid and the object.
For strong solutions in the 2D case (initial data in H1), the initial trace and the initial value of the control must coincide. The proof is similar to the proof for the existence of the fixed point and we therefore skip it here. The last part of this section consists in showing that the feedback operator of the linear system allows to stabilize the non-linear system.
If Z is a vector-valued function space of the time variable>0, we use the subscriptσinZσ to denote Zσ.
A semigroup formulation
Note that the above Neumann problem is formal and must be interpreted in its weak form since w∈L2(F). The operatorAdefined by is densely defined with a compact solution and is the infinitesimal generator of an analytic semigroup opH. Since the first two statements are obvious, we provide only a brief proof of the last statement.
The bilinear form−a(·,·) is regularity accretive, i.e. there exists c0>0 andλ0>0 such that. 3.19) Consequently, standard arguments (see [8, Thm. 2.12]) guarantee that the operatorA0 defined froma(·,·) is the infinitesimal generator of an analytic semigroup on H. Thus, from regularity results for the Stokes problem with regular boundary data we deduce that D(A0) =D(A) (defined in (3.17)) and an integration by parts yields A0=Given by (3.18) (see [35] for a similar proof). The proof is analogous to the one that yields characterization when A is defined from the bilinear form(·,·), see the proof of Theorem 16.
To prove the last propositions, we introduce the Dirichlet map D0∈ L(R3,H2(F)) defined by D0(ℓ, ω) =z where z is the solution of. Next, we introduce the Dirichlet operator DF :V0(∂Ω)→L2(F)×R6 defined as follows: foru∈V0(∂Ω) we denote by DFudef= [wuℓuωuhuθu]∗the unique solution of. To obtain (3.23), it suffices to prove that fors=32 ands=−12 and then use an interpolation argument.
The fact that the above equality is satisfied for all Y ∈ D(A∗) precisely means thatPX satisfies the first equality of (3.11) in [D(A∗)]′.
Stabilizability of the linear system
The above implication can be proved as follows: combining (3.34) and a classical unique result for Stokes-type systems (see e.g. [16]) we derive thatϕ= 0 and π= 0 onF. Indeed, if a family (˜vj) is admissible, all families in a neighborhood of (˜vj) in (V0(∂Ω))Nσ are admissible. There is a wide choice for the family(vj)in(3.31), since it is proved in [5, 7] that a family(vj) is generically allowed provided that Nσ is greater than or equal to the maximum of the geometric multiplicities of eigenvalues with real part greater than−σ.
Once the family(vj) is determined, the family(ϕj, ξj, ζj, aj, bj) in (3.31) can for example be obtained from the solution of a finite dimensional Riccati equation of size Mσ×Mσ, where Mσ is the dimension of the subspace composed of eigenvectors related to eigenvalues with real part greater than−σ. First, since (3.27) implies that Fσ∗B∗ is relatively bounded with respect to A∗, the equality D(A∗σ) = D(A∗) follows from the expression A∗σ =A∗+Fσ∗B∗ with ' a disturbance argument.
The fixed point procedure
Finally, let us recall that Theorem 2 follows from Theorem 26 and, in particular, that (1.16) is satisfied. In fact, we conclude from Theorem 26 that the stabilized solution [w, ℓ, ω, h, θ, q] satisfies. 3.48) If we consider ev defined in Ω by formula (2.5) and expand v with a rigid rate ℓ+ωy⊥inS, and also consider vS defined in Ω by expanding it with zero in S, we can also assume that. From the application of Banach's fixed point theorem, we also have the uniqueness of the solution (w, q, ℓ, ω, h, θ) within the class of functions belonging to the neighborhood of the origin G. However, the uniqueness within the class of arbitrarily large (and not necessarily stable) functions is not given by Theorem 26.
In this section, C denotes a generic positive constant which can change from line to line and which is independent of,θ,ℓ,ωand of the variable sy,x, but which may depend on kηkW1,∞(Ω)or on the geometry.
Proof of Propositions 12 and 13
The fact that h0= 0 andθ0 = 0 implies that L−∆, K−IdandG− ∇ are of order and suitable estimates are obtained in terms of time L2 norms ofD2w, t∂tw and tp.
Estimates of the differences
It is similar to the proof of Theorem 2 and we only highlight the main differences. As in the 2D case, a change of variables must be performed to write the system in the fixed domain F(hS, RS). We use the same type of change of variables as in the 2D case, more precisely we replace (2.2) with.
A first step in proving the stabilization of the above system consists in - as in the 2D case - considering a linear system connected to. However, such a feedback law does not allow to construct a fixed-point solution for the nonlinear system in a similar way as in Section 3.3, because, as explained in the introduction, we do not necessarily have Fσ[w, ℓ, ω, H, Q ]∗equal to the trace ew0 in∂Ω, which is required to determine a solution of the 3D problem. It can be easily seen that this last condition is satisfied if v0 (i.e. the initial velocity of the system before the change of variables) belongs to H1(F(h0, R0)) and satisfies the compatibility conditions and (1.22).
Following the general framework of [5] for the construction of dynamic control (see also [7]), we introduce the "extended" space HE. Since the stabilization of (A+σ, B) by finite-dimensional control generated by the family (vj) thus implies the stabilizability of (AE+σ, BE) (see [5, Theorem 8]), we can find a family [ ϕj, ξj, ζj, aj, cj]∗ ∈ D(A∗), a matrix Λ of sizeNσ×Nσ and a corresponding feedback operatorFE,σ:HE→RNσ defined by. Decomposition of Vector Spaces and Application to the Stokes Problem in Arbitrary Dimension, Czechoslovak Math.
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