NUMERICAL STABILIZATION OF A FLUID-STRUCTURE INTERACTION SYSTEM

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NUMERICAL STABILIZATION OF A

FLUID-STRUCTURE INTERACTION SYSTEM

Michel Fournié, Moctar Ndiaye, Jean-Pierre Raymond

To cite this version:

Michel Fournié, Moctar Ndiaye, Jean-Pierre Raymond. NUMERICAL STABILIZATION OF A FLUID-STRUCTURE INTERACTION SYSTEM. 2018. �hal-01743781�

NUMERICAL STABILIZATION OF A FLUID-STRUCTURE INTERACTION SYSTEM

FOURNI ´E MICHEL^∗, NDIAYE MOCTAR^∗, AND RAYMOND JEAN-PIERRE^∗

Abstract. We study the numerical stabilization of a fluid-structure interaction system in a wind tunnel, around an unstable stationary solution. The goal is to find a feedback control, acting only in the structure equation, able to stabilize locally the full coupled nonlinear system. The Finite Element Approximation of this coupled system leads to a Differential Algebraic Equation (D.A.E.) for the fluid velocity, the structure displacement and its velocity, and a Lagrange multiplier taking into account together the incompressibility condition and the equality of the fluid velocity and the displacement velocity of the structure at the fluid-structure interface. We determine a feedback control law based on a spectral decompo- sition of the D.A.E. But the D.A.E. is not standard because the operator acting on the Lagrange multiplier in the differential equation is not the transpose of the operator involved in the algebraic constraints. We overcome these difficulties by proving new relationships between eigenvalue problems involving Lagrange multipliers and those without Lagrange multipliers. In numerical simulations, we show how the calculation of the degree of stabilizability of different invariant subspaces (gener- alized eigenspaces) may be helpful to determine an efficient control strategy. In particular, the determined feedback law is able to reject a perturbation (leading to a complete stabilization of the nonlinear fluid-structure system) whose magnitude is of order 15% of the inflow velocity.

Key words. Fluid-structure interaction, feedback control, stabilization, Navier-Stokes equations, beam equations, Finite element approximation.

AMS subject classifications. 93B52, 37M99, 37N10, 76D55, 74F10, 76D05

1. Introduction. We are interested in stabilizing a fluid-structure interaction model in a wind tunnel whose geometrical configuration is represented in Figure 1.1. The inflow velocity U_∞ generated by a powerful fan system is assumed to be constant and horizontal U_∞ = (u_∞,0)^T, withu_∞ > 0. A thick plate, with a straight base, is located in the center of the wind tunnel, and the fluid flows around this obstacle. The wind tunnel is formally divided into two subdomains. In the left hand side of the wind tunnel (LHS for short), the fluid flow is not precisely modeled. At the inflow boundary of the right hand side of the wind tunnel (RHS for short), the flow is assumed to be a known Blasius type profilegs plus an unknown perturbationgp. This unknown perturbation, relatively small with respect togs, takes into account the fact that the fluid flow in the LHS of the wind tunnel is not precisely known.

U∞

computational domain

LHS RHS

Fig. 1.1: Configuration of the wind tunnel.

In the absence of perturbation, that is whengp= 0, assuming that the fluid flow is governed by the Navier-Stokes equations with some boundary conditions, the flow in the RHS is a stationary solution (us, ps). For a Reynolds number Re = 200, for which the numerical simulations are done, the stationary solution u_s is unstable. When g_p 6= 0, we observe a shedding of vortices behind the thick plate, see the vorticity profile in Figure 1.2. Our goal is to use a control device able to stabilize the fluid flow around the stationary solution u_s, in the presence of perturbations. This type of problem has been studied theoretically in [4, 21, 22] and numerically in [1, 2]. Here the novelty is that we would like to use the displacement of elastic structures, located at the upper and lower boundaries of the thick plate, to stabilize the coupled fluid-structure interaction system. Let us describe our model more precisely. The geometrical domain Ω, corresponding to the RHS of the wind tunnel, is defined by

Ω = ([0, L]×[−`, `])\([0, `s]×[−e, e]),

∗Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e Paul Sabatier Toulouse III & CNRS, 31062 Toulouse Cedex, France (michel.fournie@math.univ-toulouse.fr, moctar.ndiaye@math.univ-toulouse.fr, jean- pierre.raymond@math.univ-toulouse.fr)

1

Fig. 1.2: Vorticities of the steady flow (left) and the perturbed flow (right).

where L > 0 is the length of the domain, 2` is its height, `_s is the length of the thick plate in the computational domain, and 2e is the thickness of the plate. The boundary of Ω, denoted by Γ, is split into different parts Γ = Γ_s∪Γ_i∪Γ_e∪Γ_n where

Γs= (0, `s)× {−e} ∪(0, `s)× {e}, Γe= (0, L)× {−`} ∪(0, L)× {`} and Γn ={L} ×(−`, `), Γ_i= Γ_i,1∪Γ_i,2∪Γ_i,3 with Γ_i,1={0} ×(−`,−e), Γ<sub>i,2</sub>={0} ×(e, `), Γ_i,3={`_s} ×(−e, e).

We also set Γ0 = Γs∪Γi ∪Γe and Γi,e = Γi∪Γe. The two elastic beams, used to stabilize the fluid flow, are located on Γs. The displacementηof the beams is assumed to be normal to Γs, and it satisfies an Euler-Bernoulli damped beam equation with clamped boundary conditions. Since the structure is deformed, the domain occupied by the fluid at timetdepends on the displacementη(t) of the structure, see Figure 1.3.

Γ

Γ

Γ

Γ

Γ

Ω Γ

Γ

Γ

Γ

Γ

Γ

Γ

Ω

Γ

Γ

Fig. 1.3: Reference configuration (left) and deformed configuration (right).

The fluid domain at timetis denoted by Ωη(t)and the fluid-structure interface by Γη(t). We use the notations

Q^∞_η = [

t∈(0,∞)

{t} ×Ω_η(t)

, Σ^∞_η = [

t∈(0,∞)

{t} ×Γ_η(t)

, e₁= (1,0) e₂= (0,1),

Q^∞= (0,∞)×Ω, Σ^∞_s = (0,∞)×Γs,Σ^∞_i = (0,∞)×Γi, Σ^∞_e = (0,∞)×Γe, and Σ^∞_n = (0,∞)×Γn. The Eulerian-Lagrangian system describing the evolution of the fluid-structure system [21] is

ut−divσ(u, p) + (u· ∇)u= 0 inQ^∞_η ,

divu= 0 inQ^∞_η , u=ηtnon Σ^∞_η , u=gs+gp on Σ^∞_i , u·n= 0 on Σ^∞_e , ε(u)n·τ= 0 on Σ^∞_e , σ(u, p)n= 0 on Σ^∞_n, ηtt−β∆sη−γ∆sηt+α∆²_sη=−σ(u, p)|Γ_η(t)n_η(t)p

1 +η_x²·n+fs+f on Σ^∞_s , η = 0 on (0,∞)×∂Γ_s, η_x= 0 on (0,∞)×∂Γ_s,

u(0) =u⁰on Ω, η(0) = 0 on Γs, ηt(0) =η₂⁰on Γs,

(1.1)

whereuandpstand for the fluid velocity and pressure, σ(u, p) is the Cauchy stress tensor σ(u, p) = 2νε(u)−pI, ε(u) =1

2(∇u+ (∇u)^T),

2

ν is the fluid viscosity,α >0,β ≥0 andγ >0 are parameters of the structure,nη(t)(resp. n) is the unit normal to Γη(t)(resp. Γs) exterior to Ωη(t)(resp. Ω), andu⁰,η₂⁰are initial data. Here ∆s=∂xxstands for the Laplace-Beltrami operator on Γ_s, the term −σ(u, p)|_Γη(t)n_η(t)p

1 +η_x²·nrepresents the force exerted by the fluid on the structure,f_sis a stationary force defined below,g_sis a stationary boundary condition andf is the control function which will be used to stabilize the fluid-structure interaction system.

We notice that a Neumann boundary condition is prescribed on Σ^∞_n while a Navier boundary condi- tion is prescribed on Σ^∞_e . What is done in the present paper can be also adapted without any difficulty to the case where a Neumann boundary condition is imposed on Σ^∞_e . In that case, the transformation used to rewrite the system in the reference configuration must be defined differently. Other geometries could also be considered.

Let (us, ps) be a stationary solution of the Navier-Stokes equations in the reference configuration Ω

−divσ(us, ps) + (us· ∇)us= 0 in Ω,

divus= 0 in Ω, us= 0 on Γs, us=gs on Γi,

us·n= 0 on Γe, ε(us)n·τ = 0 on Γe, σ(us, ps)n= 0 on Γn.

(1.2)

We choose fs = −ps|Γ_s in (1.1). Thus, (u, p, η) = (us, ps,0) is an unstable stationary solution of sys- tem (1.1), and it is the solution around which we want to stabilize (1.1). We choose the functionf of the form

f(t, x, y) =

nc

X

i=1

fi(t)wi(x, y), (1.3)

where the functions wi belongs to L²(Γs) and f(·) = (f1(·),· · ·, fn_c(·)) is the control variable. The functionswi must be appropriately chosen so that the linearized system around (us, ps,0) is stabilizable, see Section 7.3, where the stabilizability is studied. The goal of the paper is to find a control f, in feedback form, able to stabilize system (1.1) around the stationary solution (us, ps,0), with any prescribed exponential decay rate−ω <0, provided thatgp,u⁰−usandη⁰₂are small enough in appropriate functional spaces.

The analysis of this stabilization problem has been done in [11]. Here we would like to develop a similar strategy for the semi-discrete system obtained by approximating by a finite element method the system (1.1) rewritten in the reference configuration.

To rewrite system (1.1)₁₋₄in the reference configurationQ^∞, for allt≥0, we introduce the mapping T_η(t): (x, y)7−→(x, z) =

x,(`−e)y−`Eη(t, x, s(y)e)

`−e−s(y)Eη(t, x, s(y)e)

,

where the sign functionsis defined bys(y) =−1 ify <0,s(y) = 1 ify≥0 and E is a continuous linear operator such thatEη(t, x) = 0 for (t, x)∈(0,∞)×((`_s+L)/2, L), see [11]. In the following, to simplify the notation, we writeη in place ofEη.

For allt≥0,T_η(t)transforms Ω_η(t)into Ω. We make the change of unknowns

˜

u(t, x, z) :=u(t,T_η(t)⁻¹(x, z)), p(t, x, z) :=˜ p(t,T_η(t)⁻¹(x, z)), η1(t, x) :=η(t, x), η2(t, x) :=ηt(t, x), and u˜⁰=u⁰.

(1.4) The quadruplet (˜u,p, η˜ ₁, η₂) satisfies the system

˜

ut−divσ(˜u,p) + (˜˜ u· ∇)˜u=Ff[˜u,p, η˜ 1, η2] inQ^∞,

div ˜u= divF_div[˜u, η₁] inQ^∞, u˜=η₂non Σ^∞_s , u˜=g_p+g_s on Σ^∞_i ,

˜

u·n= 0 on Σ^∞_e , ε(˜u)n·τ= 0 on Σ^∞_e , σ(˜u,p)n˜ = 0 on Σ^∞_n, η1,t−η2= 0 on Σ^∞_s ,

η2,t−β∆sη1−γ∆sη2+α∆²_sη1=−σ(˜u,p)n˜ ·n+Fs[˜u, η1] +f+fson Σ^∞_s , η₁= 0 on (0,∞)×∂Γ_s, η_1,x= 0 on (0,∞)×∂Γ_s,

˜

u(0) = ˜u⁰in Ω, η1(0) = 0 on Γs, η2(0) =η₂⁰ on Γs,

(1.5)

3

where the nonlinear termsFf, Fdiv andFs are given in Appendix A. The linearization of system (1.5) around the stationary solution (us, ps,0,0) is

v_t−divσ(v, q) + (u_s· ∇)v+ (v· ∇)u_s−A₁η₁−A₂η₂= 0 inQ^∞, divv=A3η1in Q^∞, v=η2non Σ^∞_s , v=gp on Σ^∞_i ,

v·n= 0 on Σ^∞_e , ε(v)n·τ = 0 on Σ^∞_e , σ(v, q)n= 0 on Σ^∞_n , η1,t−η2= 0 on Σ^∞_s ,

η_2,t−β∆_sη₁−γ∆_sη₂+α∆²_sη₁−A₄η₁=−σ(v, q)n·n+f on Σ^∞_s , η1= 0 on (0,∞)×∂Γs, η1,x = 0 on (0,∞)×∂Γs,

v(0) =v⁰ in Ω, η₁(0) = 0 on Γ_s, η₂(0) =η⁰₂ on Γ_s.

(1.6)

The linear differential operatorsA₁,A₂,A₃ andA₄ are defined in Appendix A.

For the finite element approximation of systems (1.5) and (1.6), we shall take into account the Dirichlet boundary condition of the fluid flow on Γ_s, Γ_i and Γ_e by Lagrange multipliers. We introduce finite-dimensional subspacesXh⊂H¹(Ω;R²) for the velocity,Ph⊂L²(Ω) for the pressure,Sh⊂H₀²(Γs) for the displacement and its velocity, Dh ⊂ {g= (g1, g2)∈ L²(Γ0;R²), g = 0 on Γs∪Γi, g2 = 0 on Γe} for the multiplier associated to the Dirichlet boundary conditions on Γ0= Γs∪Γi∪Γe.

The finite dimensional approximation of the linearized system (1.6) is

Findv∈H_loc¹ ((0,∞);Xh), q∈L²_loc((0,∞);Ph), τ ∈L²_loc((0,∞);Dh), η1∈L²_loc((0,∞);Sh), η2∈L²_loc((0,∞);Sh), such that

d dt

Z

Ω

v(t)φ dx=a_f(v(t), η₁(t), η₂(t), φ) +b(φ, q(t)) +hτ(t), φiΓ0, ∀φ∈X_h, b(v(t), ψ) =

Z

Ω

A3η1(t)ψ dx, ∀ψ∈Ph, hµ, v(t)i_Γ0 =

Z

Γ_i

g_p· µ dx+ Z

Γ_s

η₂(t)n·µ dx, ∀µ∈D_h, d

dt Z

Γ_s

η₁(t)ζ dx= Z

Γ_s

η₂(t)ζ dx, ∀ζ∈S_h, d

dt Z

Γ_s

η2(t)ζ dx=as(η1(t), η2(t), ζ)− hτ(t), ζ niΓs, ∀ζ∈Sh,

(1.7)

where

af(v, η1, η2, φ) = Z

Ω

−2νε(v) :ε(φ)−(us· ∇)φ·v−(φ· ∇)us·v+ Z

Ω

A1η1·φ+ Z

Ω

A2η1·φ

dx, b(φ, q) =

Z

Ω

divφ q dx, hµ, φiΓ₀= Z

Γs∪Γi

µ·φ dx+ Z

Γe

µ2φ2dx, a_s(η₁, η₂, ζ) =

Z

Γ_s

(−β∇η1· ∇ζ+α∆η₁·∆ζ+A₄η₁ζ−γ∇η2· ∇ζ)dx.

System (1.7) has to be completed by initial conditions. The Lagrange multiplier τ is introduced to take into account the boundary conditionsv=η₂non Γ_s,v=g_p on Γ_i andv·n= 0 on Γ_e.

In order to construct a linear feedback law, which is easy to compute, able to locally stabilize the nonlinear system (1.5), with any prescribed exponentially decay rate −ω <0, we are going to follow a strategy similar to that used in [11] for the continuous model. It can be summarized in several steps corresponding to each section.

– In section 2, we present the matrix formulation of the semi-discrete Finite Element approximations.

– In section 3, we reformulate the finite-dimensional linear system as a control system by eliminating the multiplier from the equations using a projector which plays a role similar to that of the Leray projector for the infinite-dimensional system.

– In section 4, we study the relationships between the eigenvalue problems involving Lagrange multipliers and those without Lagrange multipliers. Using these relationships, we are able to construct the feedback law without having to call the projector which is difficult to construct numerically.

4

– In section 5, we use the spectral decomposition to bring back the stabilization problem to the stabi- lization of a finite-dimensional linear system. Then, the feedback law is obtained by solving an Algebraic Riccati Equation of small dimension and then easy to solve.

– Finally, in section 6, thanks to numerical tests, we prove that the linear feedback law is able to stabilize the discrete nonlinear system. By choosing conveniently different parameters used to determine the feedback control law, we are able to stabilize perturbation amplitudes that are of order 15% of the stationary inflow boundary condition.

2. Semi-discrete approximations. In this section, we show that the matrix formulation of system (1.7) corresponds to system (2.3).

We denote by (φi)_1≤i≤nv a basis ofXh, (pi)_1≤i≤nq a basis ofPh, (ζi)_1≤i≤ns a basis ofSh, (µi)_1≤i≤nτ a basis ofDh. We set

v=Pn_v

i=1viφi, v⁰=Pn_v

i=1v_i⁰φi, η1=Pn_s

i=1ηⁱ₁ζi, η2=Pn_s

i=1η₂ⁱζi, η⁰₂=Pn_s i=1ηⁱ_2,0ζi, q=Pⁿq

i=1q_ip_i, g_p=Pn_τ

i=1g_pⁱµ_i, τ=Pn_τ

i=1τ_iµ_i, w_j=Pn_s i=1w_jⁱζ_i. If we denote with boldface letters the corresponding coordinate vectors, we have

v= (v₁,· · ·, v_nv)^T, v⁰= (v₁⁰,· · · , v_n⁰

v)^T, η₁= (η¹₁,· · ·, η₁^ns)^T, η₂= (η₂¹,· · ·, η₂^ns)^T, η⁰₂= (η¹_2,0,· · ·, η_2,0^ns)^T, f = (f₁,· · ·, f_nc)^T, g_p= (g¹_p,· · ·, gⁿ_p^τ)^T, q= (q₁,· · · , q_nq)^T,

τ = (τ₁,· · ·, τ_nτ)^T, θ= (q₁,· · ·, q_nq, τ₁,· · · , τ_nτ)^T, w_j= (w¹_j,· · · , wⁿ_j^s)^T, W= [w₁· · · w_nc].

For the fluid, we introduce the stiffness matrices Avv, Avq, Avτ, Avθ, Aθg and the mass matrix Mvv

defined for all 1≤i≤nv, 1≤j ≤nv, 1≤k≤nq, 1≤l≤nτ, 1≤m≤nτ by (Avv)ij=−2ν

Z

Ω

ε(φi) :ε(φj)− Z

Ω

(us· ∇)φj·φi− Z

Ω

(φj· ∇)us·φi, (Mvv)ij = Z

Ω

φj·φi, (A_vq)_ik=

Z

Ω

p_kdivφ_i, (A_vτ)_il= Z

Γ_s∪Γ_i

µ_l·φ_i+ Z

Γ_e

µ_l,2φ_l,2, (M_{τ τ})_ml = Z

Γ_m

τ_l·τ_m,

Avθ=h

Avq Avτ

i

, Aθg= 0

Mτ τ

.

For the structure, we introduce the stiffness matrices A_η1η2,A_η2η1, A_η2η2 and the mass matrixM_ηη de- fined for all 1≤i≤n_sand 1≤j≤n_sby

(Aη₁η₂)ij= (Mηη)ij = Z

Γs

ζj·ζi, (Aη₂η₁)ij =− Z

Γs

(β∇ζj· ∇ζi+α∆ζj·∆ζi) + Z

Γs

A4ζj·ζi

(Aη₂η₂)ij=−γ Z

Γs

∇ζj· ∇ζi,

We also introduce the coupling matricesAvη₁,Avη₂,Aη₁q,Aη₂τ,Aη₁θandAη₂θdefined for all 1≤i≤nv, 1≤j≤ns, 1≤k≤ns, 1≤l≤nq, 1≤m≤nτ by

(Avη₁)ij = Z

Ω

A1ζj·φi, (Avη₂)ij = Z

Ω

A2ζj·φi, (Aη₁q)kl=− Z

Ω

A3ζk·pl, (Aη₂τ)km=− Z

Γs

ζkn·µm, A_η1θ=h

A_η1q 0 0i

, A_η2θ=h

0 A_η2τ 0i . We set

nz=nv+ 2ns, nθ=nq+nτ and n=nz+nθ.

We introduce the mass matrixM, the stiffness matrixA, and the control operatorB∈ L(R^nc,R^nz)

M =

"

M_zz 0

0 0

#

, A=

"

A_zz Ae_zθ Ab^T_zθ 0

#

and B=



 0 0 MηηW



, (2.1)

where Mzz=





Mvv 0 0

0 Mηη 0

0 0 Mηη



, Azz=





Avv Avη₁ Avη₂

0 0 Aη₁η₂

0 Aη₂η₁ Aη₂η₂



, Aezθ=



 Avθ

0 Aη₂θ



, Abzθ=



 Avθ

Aη₁θ

Aη₂θ



. (2.2)

5

Thus, the matrix formulation of system (1.7) is

M d dt







 v η₁ η₂ θ









=A







 v η₁ η₂ θ







 +







 0 0 M_ηηWf

0







 +







 0 0 0

−Aθggp







 ,



 v η₁ η₂



(0) =



 v⁰

0 η⁰₂



, (2.3)

or, equivalently

M_zz d dt



 v η₁ η2



=A_zz



 v η₁ η2



+Ae_zθθ+Bf, Ab^T_zθ



 v η₁ η2



=A_θgg_p,



 v η₁ η2



(0) =



 v⁰

0 η⁰₂



, (2.4) where v, η1, η2, v⁰, η⁰₂ and gp are the coordinate vectors of v, η1, η2, v⁰, η₂⁰ and gp respectively, θ= (q,τ)^T,qandτ are the coordinate vectors ofqandτrespectively. The initial conditionsv⁰andη₂⁰ are such that (v⁰,0,η₂⁰) belongs to Ker(Ab^T_zθ).

3. Reformulation of the finite-dimensional linear system.

3.1. The stationary finite-dimensional linear system. The goal of this subsection is to rewrite the system

(λM−A)







 v η₁ η₂ θ









=







 F_f F_s,1 F_s,2 0









, (3.1)

as a system satisfied by (v,η₁,η₂) in which the multiplier θ is eliminated, and to characterize the multiplierθ in terms of the solution (v,η1,η2) and of the data (Ff, Fs,1, Fs,2). We are going to consider system (3.1) either whenλ∈Cand (Ff, Fs,1, Fs,2)∈C^nz, in which case the solution (v,η1,η2) and the multiplierθ respectively belong toC^nz andC^nθ, or whenλ∈Rand (Ff, Fs,1, Fs,2)∈R^nz, in which case the solution (v,η1,η2) and the multiplierθ respectively belong toR^nz andR^nθ. In order to collect both the results stated for either real or complex solutions, below we state results valid for Kⁿ where either K=Cor K=R. Let us notice that the matrices involved in system (2.3) have real coefficients.

We first consider the following system involving only the velocityv and the multiplierθ

λM_vvv =A_vvv+A_vθθ+F_f, A^T_vθv=g. (3.2) To eliminate the multiplierθfrom equation (3.2)₁, we are going to introduce the projection into Ker(A^T_vθ) parallel to Im(M_vv⁻¹A_vθ).

Proposition 3.1. 1. The projectorP^Tv inK^nvontoKer(A^T_vθ)parallel toIm(M_vv⁻¹A_vθ)is defined by P^Tv =I−M_vv⁻¹A_vθ(A^T_vθM_vv⁻¹A_vθ)⁻¹A^T_vθ.

2. The projectorPv inK^nv ontoKer(A^T_vθM_vv⁻¹)parallel toIm(A_vθ)is Pv =I−A_vθ(A^T_vθM_vv⁻¹A_vθ)⁻¹A^T_vθM_vv⁻¹. Moreover, we have

PvAvθ= 0, P²v=Pv, PvMvv=MvvP^Tv, PvMvv =P²vMvv =PvMvvP^Tv, M_vv⁻¹Pv=P^TvM_vv⁻¹.

Proof. It is similar to that of [1, Proposition 3.1].

Remark 3.1. From the Inf-Sup condition (6.4), it follows that Avθ is of rank nθ. Thus, the matrix A^T_vθA⁻¹_vvAvθ is invertible. We set

A=P^TvM_vv⁻¹Avv, Aθθ= (A^T_vθA⁻¹_vvAvθ)⁻¹, Mθθ= (A^T_vθM_vv⁻¹Avθ)⁻¹ and L=A⁻¹_vvAvθAθθ.

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Lemma 3.1. We have (I−P^Tv)L=M_vv⁻¹AvθMθθ.

Proof. The proof follows from the fact thatI−P^Tv =M_vv⁻¹AvθMθθA^T_vθ.

Lemma 3.2. Let F_f belong to K^nv andg belong to K^nθ. The couple (v,θ) is a solution of system (3.2)if and only if

λP^Tvv=AP^Tvv−AP^TvLg+P^TvM_vv⁻¹F_f, (I−P^Tv)v=M_vv⁻¹A_vθM_θθg, θ=−Mθθ −λg+Nθθg+A^T_vθM_vv⁻¹AvvP^Tvv+A^T_vθM_vv⁻¹Ff

,

(3.3)

whereNθθ=A^T_vθM_vv⁻¹AvvM_vv⁻¹AvθMθθ.

Proof. The proof is similar to that of [1, Proposition 3.2].

In order to eliminate the multiplierθ in system (3.1), we introduce the following operators

P=





P^v 0 0 0 I_Kns 0 0 0 I_Kns



, A_zθ=



 Avθ

0 0



 and M_s=





Mvv 0 0

0 Mηη 0

0 Nη₂η₁ Mηη+Nη₂η₂



, (3.4) where

Nη2η2=Aη2θMθθA^T_η2θ and Nη2η1 =Aη2θMθθA^T_η1θ. (3.5) We have

M_s⁻¹=





M_vv⁻¹ 0 0

0 M_ηη⁻¹ 0

0 −(Mηη+Nη₂η₂)⁻¹Nη₂η₁M_ηη⁻¹ (Mηη+Nη₂η₂)⁻¹



.

Proposition 3.2. The operator P^T is the projector in K^nz ontoKer(A^T_zθ)parallel to Im(M_s⁻¹Azθ) and the operator Pis the projector in K^nz ontoKer(A^T_zθM_s⁻¹)parallel toIm(Azθ). Moreover, we have

PAzθ= 0, P²=P, PMs=MsP^T, PMs=P²Ms=PMsP^T, M_s⁻¹P=P^TM_s⁻¹. Proof. It follows from Proposition 3.1.

We introduce the matrixAinR^nz×nz defined by

A=P^TM_s⁻¹Aezz, (3.6)

where

Aezz =





Avv Avη₁+AvvP^TvLA^T_η

1θ Avη₂+AvvP^TvLA^T_η

2θ

0 0 Aη₁η₂

Aη₂v Aeη₂η₁ Aeη₂η₂



, (3.7)

and

Aη2v=−Aη2θMθθA^T_vθM_vv⁻¹Avv,

Aeη₂η₁ =Aη₁η₂−Aη₂θMθθ(A^T_vθM_vv⁻¹Avη₁−NθθA^T_η

1θ), Ae_η2η2 =A_η2η2−A_η2θM_θθ(A^T_vθM_vv⁻¹A_vη2−N_θθA^T_η

2θ).

(3.8)

Proposition 3.3. Let (F_f, F_s,1, F_s,2)be in K^nz. A quadruplet(v,η₁,η₂,θ)∈Kⁿ is solution of the equation

(λM−A)







 v η1

η2

θ









=







 Ff

Fs,1

Fs,2

0









, (3.9)

7

if and only if

(λM_s−M_sA)P^T



 v η₁ η2



=P





F_f Fs,1

Fs,2−Aη₂θMθθA^T_vθM_vv⁻¹Ff



,

(I−P^Tv)v=−M_vv⁻¹AvθMθθA^T_η1θη1−M_vv⁻¹AvθMθθA^T_η2θη2, θ=−Mθθ(λA^T_η

1θη1+λA^T_η

2θη2−Nθθ(A^T_η

1θη1+A^T_η

2θη2))

−M_θθA^T_vθM_vv⁻¹A_vvP^Tvv−M_θθA^T_vθM_vv⁻¹(A_vη1η₁+A_vη2η₂+F_f).

(3.10)

If in addition,

A^T_vθM_vv⁻¹Ff+A^T_η1θM_ηη⁻¹Fs,1+A^T_η2θM_ηη⁻¹Fs,2= 0, then

(λMs−MsA)P^T



 v η1

η2



=MsP^TM_zz⁻¹



 Ff

Fs,1

Fs,2



,

andθ in (3.10)satisfies θ=−Mθθ(λA^T_η

1θη1+λA^T_η

2θη2−Nθθ(A^T_η

1θη1+A^T_η

2θη2))

−MθθA^T_vθM_vv⁻¹(A_vvP^Tvv+A_vη1η₁+A_vη2η₂) +M_θθA^T_η

1θM_ηη⁻¹F_s,1+M_θθA^T_η

2θM_ηη⁻¹F_s,2. Proof. Step 1. Let (v,η1,η2,θ) be a solution of (3.9). The couple (v,θ) is solution of the system

λMvvv=Avvv+Avθθ+Avη₁η1+Avη₂η2+Ff, A^T_vθv=−(A^T_η

1θη1+A^T_η

2θη2).

Thanks to Lemma 3.2, we have

λP^Tvv=AP^Tvv+AP^TvL(A^T_η1θη1+A^T_η2θη2) +P^TvM_vv⁻¹(Avη1η1+Avη2η2) +P^TvM_vv⁻¹Ff, (I−P^Tv)v=−M_vv⁻¹AvθMθθA^T_η

1θη1−M_vv⁻¹AvθMθθA^T_η

2θη2, θ=−M_θθ(λA^T_η

1θη₁+λA^T_η

2θη₂−N_θθ(A^T_η

1θη₁+A^T_η

2θη₂))−M_θθA^T_vθM_vv⁻¹(A_vvP^Tvv+A_vη1η₁+A_vη2η₂+F_f).

The system onη1 andη2is given by

λM_ηηη₁=A_η1η2η₂+F_s,1 and λM_ηηη₂=A_η2θθ+A_η2η1η₁+A_η2η2η₂+F_s,2. We have

Aη₂θθ=−Aη₂θMθθ(λA^T_η

1θη1+λA^T_η

2θη2−Nθθ(A^T_η

1θη1+A^T_η

2θη2))

−A_η2θM_θθA^T_vθM_vv⁻¹(A_vvP^Tvv+A_vη1η₁+A_vη2η₂+F_f).

Or equivalently

Aη₂θθ=−λAη₂θMθθ(A^T_η

1θη1+A^T_η

2θη2) + (Aη₂θMθθNθθA^T_η

1θ−Aη₂θMθθA^T_vθM_vv⁻¹Avη₁)η1

+(A_η2θM_θθN_θθA^T_η

2θ−A_η2θM_θθA^T_vθM_vv⁻¹A_vη2)η₂−A_η2θM_θθA^T_vθM_vv⁻¹A_vvP^Tvv

−Aη2θMθθA^T_vθM_vv⁻¹Ff.

Replacing the above expression ofA_η2θθ in the system satisfied by (η₁,η₂), we obtain λM_ηηη₁=A_η1η2η₂+F_s,1,

λ(Mηη+Aη₂θMθθA^T_η

2θ)η2+λAη₂θMθθA^T_η

1θη1

= (Aη₂η₁+Aη₂θMθθNθθA^T_η

1θ−Aη₂θMθθA^T_vθM_vv⁻¹Avη₁)η1

+(A_η2η2+A_η2θM_θθN_θθA^T_η

2θ−A_η2θM_θθA^T_vθM_vv⁻¹A_vη2)η₂ +Fs,2−Aη₂θMθθA^T_vθM_vv⁻¹AvvP^Tvv−Aη₂θMθθA^T_vθM_vv⁻¹Ff.

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Thus we have

(λM_s−M_sP^TM_s⁻¹Ae_zz)P^T



 v η₁ η₂



=P





F_f F_s,1

F_s,2−A_η2θM_θθA^T_vθM_vv⁻¹F_f



,

and (3.10) is proved. The converse can be proved by similar arguments.

Step 2. If in addition

A^T_vθM_vv⁻¹Ff+A^T_η1θM_ηη⁻¹Fs,1+A^T_η2θM_ηη⁻¹Fs,2= 0, then

−M_θθA^T_vθM_vv⁻¹F_f =M_θθA^T_η

1θM_ηη⁻¹F_s,1+M_θθA^T_η

2θM_ηη⁻¹F_s,2 and

Fs,2−Aη₂θMθθA^T_vθM_vv⁻¹Ff =Fs,2+Aη₂θMθθA^T_η1θM_ηη⁻¹Fs,1+Aη₂θMθθA^T_η2θM_ηη⁻¹Fs,2. Hence

(λMs−MsP^TM_s⁻¹Aezz)P^T



 v η1

η2



=P





F_f F_s,1 F_s,2+A_η2θM_θθA^T_η

1θM_ηη⁻¹F_s,1+A_η2θM_θθA^T_η

2θM_ηη⁻¹F_s,2



,

θ=−Mθθ(λA^T_η

1θη1+λA^T_η

2θη2−Nθθ(A^T_η

1θη1+A^T_η

2θη2))

−MθθA^T_vθM_vv⁻¹(A_vvP^Tvv+A_vη1η₁+A_vη2η₂) +M_θθA^T_η

1θM_ηη⁻¹F_s,1+M_θθA^T_η

2θM_ηη⁻¹F_s,2. We notice that

M_s⁻¹





F_f F_s,1 Fs,2+Aη₂θMθθA^T_η

1θM_ηη⁻¹Fs,1+Aη₂θMθθA^T_η

2θM_ηη⁻¹Fs,2



=M_zz⁻¹



 Ff

Fs,1

Fs,2



. Thus we can conclude because

(λ−P^TM_s⁻¹Aezz)P^T



 v η1

η2



=P^TM_zz⁻¹



 Ff

Fs,1

Fs,2



.

3.2. Instationary systems.

Proposition 3.4. Let(v⁰,η₂⁰,g_p)∈R^nv×R^ns×R^nτ such that(v⁰,0,η⁰₂)belongs toKer(Ab^T_zθ)and g_p(0) = 0. System (2.3)or (2.4)is equivalent to the system

d dtP^T



 v η1

η2



=AP^T



 v η1

η2



+Bf, P^T



 v η1

η2



(0) =P^T



 v⁰

0 η⁰₂



, (I−P^Tv)v =−M_vv⁻¹A_vθM_θθA^T_η

1θη₁−M_vv⁻¹A_vθM_θθA^T_η

2θη₂+M_vv⁻¹A_vθM_θθA_θgg_p, θ=−MθθA^T_η

1θη⁰₁−MθθA^T_η

2θη₂⁰ −Mθθ(A^T_vθM_vv⁻¹Avη₁−NθθA^T_η

1θ)η1

−Mθθ(A^T_vθM_vv⁻¹Avη₂−NθθA^T_η

2θ)η2−MθθA^T_vθM_vv⁻¹AvvP^Tvv+MθθAθgg⁰_p−MθθNθθAθggp,

(3.11)

where

B=M_s⁻¹B. (3.12)

Proof. The proof follows from Proposition 3.3.

9