Controllability of a simplified model of fluid-structure interaction

DOI:10.1051/cocv/2013075 www.esaim-cocv.org

CONTROLLABILITY OF A SIMPLIFIED MODEL OF FLUID-STRUCTURE INTERACTION

S. Ervedoza

and M. Vanninathan

Abstract. This article aims at studying the controllability of a simplified fluid structure interaction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan,RAM: Res. Appl. Math.

John Wiley & Sons Ltd., Chichester (1995); J.-P. Raymond and M. Vanninathan,ESAIM: COCV11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. This interaction is modeled by a wave equation surrounding a harmonic oscillator. Our main result states that, in the radially symmetric case, this system can be controlled from the outer boundary. This improves previous results [J.-P. Raymond and M. Vanninathan, ESAIM: COCV11 (2005) 180–203;

M. Tucsnak and M. Vanninathan,Systems Control Lett.58(2009) 547–552]. Our proof is based on a spherical harmonic decomposition of the solution and the so-called lateral propagation of the energy for 1d waves.

Mathematics Subject Classification. 93B05, 93B07, 93C20, 74F10.

Received March 26, 2013. Revised August 13, 2013.

Published online March 28, 2014.

1. Introduction

In this work, we consider a simplified fluid structure interaction model derived and developed in [6,18,20], constituted by a wave equation surrounding an harmonic oscillator.

The model.Let ω and Ω be smooth bounded simply connected domains of R² such that ω ⊂ Ω, and set O=Ω\ω. This corresponds to the geometric configuration represented in Figure1.

Keywords and phrases.Controllability, observability, fluid-structure interaction.

∗ The first author is partially supported by the Agence Nationale de la Recherche (ANR, France), Project CISIFS number NT09-437023. The second author is supported by CEFIPRA within the Project 3701-1 Control of Systems of Partial Differential Equations. He also thanks French CNRS and Institut de Math´ematiques de Toulouse for hospitality and support for his stay in France during which this work was carried out. Both authors acknowledge the support of the Indo-French Centre for Applied Mathematics (IFCAM).

1 Institut de Math´ematiques de Toulouse ; UMR5219; Universit´e de Toulouse ; CNRS; UPS IMT, F-31062 Toulouse Cedex 9, France.[email protected];[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

O

ω ∂Ω

Figure 1. The geometric setting.

We are considering the following model:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

y−Δy=f, (t, x)∈(0, T)× O, y(t, x) = 0, (t, x)∈(0, T)×∂Ω,

∂_ny(t, x) =s·n, (t, x)∈(0, T)×∂ω, s+s=−

∂ω

yndσ, t∈(0, T), (y(0,·), y(0,·)) = (y₀, y₁),

(s(0), s(0)) = (s₀, s₁).

(1.1)

Here,ndenotes the unit outward normal vector onO. The function y represents the velocity potential of the fluid, the function s (∈ R²) corresponds to the displacement of an oscillator located in ω, and the function f(t, x) is an external force, acting on the fluid and thus localized in O. Here, the prime denotes differentiation with respect to the time variable.

Details on the derivation of (1.1) can be found in [6], p. 96. In particular, model (1.1) corresponds to an inviscid slightly inhomogeneous compressible fluid when the velocity of the fluid is small and assumed to derive from a potential y. Accordingly, it may be more relevant to consider Neumann boundary conditions for y on the external boundary∂Ω. This issue will be discussed in Section3.6.

The interaction between the wave equation and the oscillator takes place through the boundary conditions on ∂ω and the source term in the equation in s. Actually, we could also handle an equation of the form s+A₀s=−

∂ωyndσwithA₀ a self-adjoint positive definite 2×2 matrix for the oscillator, but this would not yield any particular difficulty and we thus chooseA₀=Ito simplify the notations.

Before going further, let us briefly present the Cauchy theory for system (1.1). In order to do that properly, we define the spaceH_ext¹ (O) by

H_ext¹ (O) ={ y∈H¹(O), y_|∂Ω= 0}.

Then, according to [18], Theorem 2.1, given an initial data (y₀, y₁, s₀, s₁) ∈ H_ext¹ (O)×L²(O)×(R²)² and source term f ∈ L¹(0, T;L²(O)), system (1.1) has a unique solution (y, s) lying in C⁰([0, T];H_ext¹ (O))∩ C¹([0, T];L²(O))×C¹([0, T];R²).

Note that this is the natural space since, at least formally, solutions (y, s) of (1.1) have an energy E(t) =1

2

O

|∇y(t)|²+|∂ty(t)|² dx+1 2

|s(t)|²+|s(t)|² , (1.2)

which satisfies

dE dt =

O∂_ty(t)f(t) dx. (1.3)

In particular, whenf = 0, the energy of solutions (y, s) of (1.1) is constant in time.

The control problem.In this article, we focus on the null-controllability property of system (1.1) when the control is an external forcef localized in some open subsetCofO. To be more precise, we consider the following

controlled system: ⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

y−Δy=χ_Cf, (t, x)∈(0, T)× O, y(t, x) = 0, (t, x)∈(0, T)×∂Ω,

∂_ny(t, x) =s·n, (t, x)∈(0, T)×∂ω, s+s=−

∂ω

yndσ, t∈(0, T), (y(0,·), y(0,·)) = (y0, y₁),

(s(0), s(0)) = (s₀, s₁),

(1.4)

wheref is the control function, andχ_C denotes the indicator function ofC.

More precisely, we ask if there exists a timeT >0 such that for any (y₀, y₁, s₀, s₁)∈H_ext¹ (O)×L²(O)×(R²)², one can find a control functionf ∈L²((0, T)× C) such that the corresponding solution (y, s) of (1.4) satisfies

(y(T), y(T), s(T), s(T)) = (0,0,0,0)∈H_ext¹ (O)×L²(O)×(R²)². (1.5) In that case, we say that system (1.4) is null-controllable at timeT.

Note that, due to the time-reversibility of system (1.4), one easily checks that the null-controllability of (1.4) at timeT is equivalent to the exact controllability of (1.4) in timeT, which states that for any initial and target states (y₀, y₁, s₀, s₁), (y^T₀, y₁^T, s^T₀, s^T₁) inH_ext¹ (O)×L²(O)×(R²)², there exists a control functionf ∈L²((0, T)×C) such that the corresponding solution of (1.4) satisfies

(y(T), y(T), s(T), s(T)) = (y₀^T, y₁^T, s^T₀, s^T₁)∈H_ext¹ (O)×L²(O)×(R²)². (1.6) Hence, in the sequel, we will not distinguish between the null and exact controllability properties.

Of course, the null-controllability property of (1.4) depends on the timeT and the setCin which the control is active. Our goal then is to derive conditions on the time T and the set C under which we can guarantee null-controllability.

Before stating our results, let us now briefly review previous results on system (1.4) and its controllability property.

In [18], it was proved that when the control function acts as a source term in all the fluid (i.e. C=O), then system (1.4) is exactly controllable in any time T >0. Later on in [20], it was proved that when the control is localized in space in the fluid part O in a neighborhood of both the boundaries ∂Ω and ∂ω, there exists a time T such that system (1.4) is controllable. Though, this time T >0 is not given explicitly in [20]. This is a drawback of the method used in [20], which relies on resolvent estimates [5,8,16,21]. This result is not completely satisfactory because one still needs the control region to be localized near the entire outer boundary

∂Ω and also near the entire interior boundary∂ω. But such a control may not be possible in real situations.

We also refer to [19], which considers a similar coupling between a structure and a fluid but with a fluid modeled by the Stokes equations. Of course, this yields a completely different model, and the well-posedness and controllability issues are completely different.

Case of a control region in a neighborhood of both boundaries ∂Ω and ∂ω.Our first main result handles the case in which the control region contains a neighborhood of both boundaries ∂Ω and ∂ω. Note that this is precisely the setting dealt with in [20] but we are going to provide new estimates on the time of controllability.

We suppose that there existsε >0 such that

{x∈ O, d(x, ∂O)≤ε} ⊂ C. (1.7) This geometric condition is represented in Figure2.

O

ω ∂Ω

Figure 2.Condition (1.7) means that the control set Ccontains the set represented in black, i.e. a neighborhood of both boundaries∂Ω and∂ω.

We also assume that the wave equation in domain O with homogeneous Dirichlet boundary conditions is null-controllable at timeT₀>0 when the control acts onC.

This means the following: for all (ϕ₀, ϕ₁)∈H₀¹(O)×L²(O), there exists a control functiong∈L²((0, T₀)×C) so that the solutionϕof

⎧⎨

⎩

ϕ−Δϕ=χ_Cg, (t, x)∈(0, T₀)× O, ϕ(t, x) = 0, (t, x)∈(0, T₀)×∂O, (ϕ(0,·), ϕ(0,·)) = (ϕ₀, ϕ₁).

(1.8) satisfies (ϕ(T₀), ϕ(T₀)) = (0,0).

Note that system (1.8) does not contain any rigid body anymore and simply is a wave equation inO.

According to [1,4], this system is null-controllable in timeT₀ if and only if (C,O, T0) satisfies the Geometric Control Condition of Bardos Lebeau and Rauch. Roughly speaking, this condition states that all the rays of Geometric Optics should meet the control regionCin a time less thanT₀. In our context, these rays of Geometric Optics simply are straight lines bouncing on the boundary ∂O according to Descartes–Snell’s law (Actually, since in the present setting the control set C contains a neighborhood of both boundaries∂Ω and ∂ω, we do not need here to describe the behavior of the rays on the boundary).

Note that, due to Assumption (1.7) and the geometry under consideration, the timeT₀always exists and is bounded by diam(Ω). For instance in the caseΩ=B(0, R₀) andω=B(0, r₀), the timeT₀ can be taken to be any time larger than 2

R₀²−r₀².

Also note that the Hilbert Uniqueness Method provides a control function g in L²((0, T₀)× C) of minimal L²((0, T₀)× C)-norm which moreover satisfies

g_L2((0,T0)×C)≤C(ϕ0, ϕ₁)_H1

0(O)×L²(O). (1.9)

We then prove the following:

Theorem 1.1. Assume that C satisfies Assumption (1.7) and that system (1.8) is null-controllable at time T₀>0. Then system (1.4)is exactly controllable at timeT₀.

Our argument consists mainly of two things:

• the control result obtained in [18] when the control acts everywhere in the fluid partO;

• the null-controllability of the wave equation (1.8).

The proof of Theorem1.1is then reduced to a suitable gluing argument between the two controlled trajectories.

Details of the Proof of Theorem1.1 are given in Section2.

O ω

Figure 3.The geometric setting in the radial configuration given by (1.10). Here, the control set C is represented in black: it is a ring of the formC =B(0, R₀)\B(0, R₁) localized close to the external boundary.

The radial case.We then focus on the radial case: we assume that there existsr₀, R₀, R₁positive real numbers with 0< r₀< R₁< R₀ and

ω=B(0, r₀), Ω=B(0, R₀), C=B(0, R₀)\B(0, R1), (1.10) whereB(0, r) denotes the ball centered at the origin of radiusr >0. This corresponds to the geometric setting given in Figure3.

We then prove the following:

Theorem 1.2. Under Assumption (1.10), system (1.4)is exactly controllable in any time T >2

R²₁−r²₀. (1.11)

To our knowledge, Theorem1.2is the first control result for system (1.4) which does not require the control to be supported in a neighborhood of∂ω. The main issue here is to ensure that no energy is trapped near the boundary∂ω as the coupled system vibrates according to the model under consideration.

Note however that Theorem 1.2 is restricted to the radial case. As we shall explain in Section 3.4, this assumption can be slightly weakened and the form ofΩandC can be relaxed. However, the radial assumption onω is completely necessary to our arguments.

Let us also emphasize that condition (1.11) is necessarily sharp due to the speed of propagation of waves.

Indeed, following the construction of Ralston in [17], if T < 2

R₁²−r₀², one can construct solutions of the system (1.1) with source term 0, known as Gaussian beams, which are localized outside (0, T)× C. By duality, this provides an impediment for the controllability of (1.4), see Section4.

As explained in Section3.1, following the Hilbert Uniqueness Method of Lions [13], the control result stated in Theorem1.2is based on the observability of the adjoint equation

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ϕ−Δϕ= 0, (t, x)∈(0, T)× O, ϕ(t, x) = 0, (t, x)∈(0, T)×∂Ω,

∂_nϕ(t, x) =w·n, (t, x)∈(0, T)×∂ω, w+w=−

∂ω

ϕndσ, t∈(0, T), (ϕ(0,·), ϕ(0,·)) = (ϕ₀, ϕ₁),

(w(0), w(0)) = (w₀, w₁).

(1.12)

Namely, we are going to study the following observability inequality: there exists a constant C >0 such that for all (ϕ₀, ϕ₁, w₀, w₁)∈H_ext¹ (O)×L²(O)×(R²)², the solution (ϕ, w) of (1.12) satisfies

(ϕ0, ϕ₁, w₀, w₁)²_H1

ext(O)×L²(O)×(R²)²≤C_obs² _T

0

Cχ²_C|ϕ(t, x)|²dtdx. (1.13) In order to prove that observability inequality, we strongly use the radial symmetry of the problem, which allows us to introduce polar coordinates and to do a spherical harmonic decomposition. In particular, we show somewhat surprisingly that the interaction between the waves and the oscillator appears only on the first mode of the Laplace–Beltrami operator on the disk, the other ones being completely decoupled from the displacement sof the solid. That way, we are able to derive a very precise description of the fluid structure interaction.

The main issue is then to study a 1d wave equation coupled with an oscillator and its observability property.

This study is done using the lateral propagation of the energy of 1d waves, which, roughly speaking, consists in exchanging the role played by the time and the space. This technique is rather classical and has been developed extensively in the context of 1d semilinear wave equation (seee.g.[22]).

Note also that our method gives a very precise description of the interaction between the fluid and the solid parts, which partly explains the difficulty of removing the radially symmetric assumption on the solidω. As we shall explain in Section3.4, using microlocal defect measures as in [3,4], we can remove the spherical assumption onΩandC. However, our method fails to apply when the solid partω is not a disk.

Outline.Section2 aims at proving Theorem 1.1. Radially symmetric domains are considered in Section3. In Section 3.1, we reduce the null controllability problem for (1.4) to the observability inequality (1.13) for the adjoint system (1.12). Working with polar coordinates and spherical harmonics decomposition of the system, we show in Section3.2that only the first harmonic mode couples with the solid, while other modes are free of couplings. The problem is thus reduced to proving observability estimates for the individual components of the decomposition, a task we perform in Section3.3. In this process, we use the principle of lateral propagation of the energy for 1d waves to treat the first harmonic mode. Section3.4 then provides the corresponding control results and some generalizations of it when the control set does not contain a neighborhood of the entire outer boundary. Section 3.5 is devoted to the boundary controllability of the model. Section 3.6 discusses control results when the potentialy is assumed to satisfy the Neumann boundary conditions on the external boundary

∂Ω. Finally, in Section4, we provide the reader with comments and open problems.

2. Proof of Theorem 1.1

The goal of this section is to prove Theorem 1.1. In particular, in this section, we will always assume Condition (1.7).

The Proof of Theorem1.1is based on the following result, obtained in [18], Theorem 1.1:

Theorem 2.1([18]). For any T >0, for any(z₀, z₁, σ₀, σ₁)∈H_ext¹ (O)×L²(O)×(R²)², there exists a control function u∈L²((0, T)× O)such that the solution of

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

z−Δz=u, (t, x)∈(0, T)× O, z(t, x) = 0, (t, x)∈(0, T)×∂Ω,

∂_nz(t, x) =σ·n, (t, x)∈(0, T)×∂ω, σ+σ=−

∂ω

zndσ, t∈(0, T), (z(0,·), z(0,·)) = (z₀, z₁),

(σ(0), σ(0)) = (σ₀, σ₁)

(2.1)

satisfies the null-controllability requirement

(z(T), z(T), σ(T), σ(T)) = (0,0,0,0). (2.2)

Besides, there exists a constant C_T such that

u_L2((0,T)×O)+z_L∞(0,T;H¹_ext(O)) ≤C_T(z₀, z₁, σ₀, σ₁)_H1

ext(O)×L²(O)×(R²)². (2.3) We are now in position to prove Theorem1.1.

Proof of Theorem 1.1. LetC satisfying (1.7) for someε >0, and assume that T₀ is so that (C,O, T0) satisfies the Geometric Control Condition.

Fix (y₀, y₁, s₀, s₁) ∈ H_ext¹ (O)×L²(O)×(R²)². Then, according to Theorem 2.1, setting (z₀, z₁, σ₀, σ₁) = (y₀, y₁, s₀, s₁), there exists a control functionu∈L²((0, T₀)×O) so that the solution (z, σ) of (2.1) satisfies (2.2).

Besides, one has

u_L2((0,T0)×O)+z_L∞(0,T0;H_ext¹ (O))≤C(y0, y₁, s₀, s₁)_H1

ext(O)×L²(O)×(R²)². (2.4) Now, letη₁=η₁(x) be a smooth function inO taking value one on{x∈ O, d(x, ∂ω)≤ε/2} and vanishing identically on{x∈ O, d(x, ∂ω)≥ε}.

Then (η₁z, σ) satisfies the system

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

(η₁z)−Δ(η₁z) =η₁u−2∇η₁∇z−Δη₁z, (t, x)∈(0, T₀)× O,

η₁z(t, x) = 0, (t, x)∈(0, T₀)×∂Ω,

∂_n(η₁z)(t, x) =σ·n, (t, x)∈(0, T₀)×∂ω, σ+σ=−

∂ω

(η₁z)ndσ, t∈(0, T₀), (η₁z(0,·),(η₁z)(0,·)) = (η₁y₀, η₁y₁),

(σ(0), σ(0)) = (σ₀, σ₁),

(η₁z(T₀,·),(η₁z)(T₀,·)) = (0,0), (σ(T₀), σ(T₀)) = (0,0),

(2.5)

We then define η₂ = η₂(x) as a smooth function in O taking value one on {x ∈ O, d(x, ∂ω) ≤ ε/4} and vanishing identically on{x∈ O, d(x, ∂ω)≥ε/2}.

Since the wave equation (1.8) is assumed to be null-controllable in timeT₀, setting (ϕ₀, ϕ₁) = ((1−η₂)y₀,(1−

η₂)y₁)∈H₀¹(O)×L²(O) there exists a control function g∈L²((0, T₀)× C) that steers the solutionϕof (1.8) from (ϕ₀, ϕ₁) to (0,0) in a timeT₀satisfying (1.9) and, following,

g_L2((0,T0)×C)+ϕ_L∞(0,T0;H₀¹(O)) ≤C(ϕ0, ϕ₁)_H1

0(O)×L²(O)

≤C(y0, y₁)_H1

ext(O)×L²(O). (2.6)

Moreover, (1−η₁)ϕsatisfies

⎧⎪

⎨

⎪⎩

((1−η₁)ϕ)−Δ((1−η₁)ϕ) = (1−η₁)χ_Cg+ 2∇η₁∇ϕ+Δη₁ϕ, (t, x)∈(0, T₀)× O,

((1−η₁)ϕ)(t, x) = 0, (t, x)∈(0, T₀)×∂O,

∂_n((1−η₁)ϕ)(t, x) = 0, (t, x)∈(0, T₀)×∂ω,

(2.7)

with initial and final data

((1−η₁)ϕ(0,·),(1−η₁)ϕ(0,·)) = ((1−η₁)y₀,(1−η₁)y₁),

((1−η₁)ϕ(T₀,·),(1−η₁)ϕ(T₀,·)) = (0,0), (2.8) where we have used that, due to the conditions on the support ofη₁ andη₂, (1−η₁)(1−η₂) = (1−η₁).

Note that (1−η₁)ϕsatisfies null Dirichlet and Neumann boundary conditions on the interface ∂ω. In par- ticular, settingy=η₁z+ (1−η₁)ϕands=σ, one immediately gets that (y, s) satisfies (1.4) with initial data (y₀, y₁, s₀, s₁), and control function

χ_Cf =η₁u−2∇η1∇z−Δη₁z+ (1−η₁)χ_Cg+ 2∇η1∇ϕ+Δη₁ϕ. (2.9) Indeed, the right hand side terms in (2.9) are all supported in C due to the conditions on the support of η₁ and (1.7), and each term belongs toL²((0, T₀)× C) thanks to (2.4) and (2.6).

This concludes the Proof of Theorem1.1since by construction, (y, s) solves the null controllability require-

ment (1.5) withT =T₀.

3. The radially symmetric case

In this section, we consider the radially symmetric case and assume the geometry given in (1.10):

ω=B(0, r₀), Ω=B(0, R₀), C=B(0, R₀)\B(0, R₁), with 0< r₀< R₁< R₀.

To simplify notations, we set A(ra, r_b) the annulus B(0, r_b)\B(0, ra). We also define S(0, r) as the circle of radiusr >0.

Note that our assumptions also concern the control region C, which is assumed to be an annulus C = A(R1, R₀). In particular, the control region is a neighborhood of the whole external boundary, but not of the internal boundaryS(0, r₀).

Now, system (1.4) is rewritten as:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

y−Δy=χ_R1<|x|<R0f, (t, x)∈(0, T)× A(r₀, R₀), y(t, x) = 0, (t, x)∈(0, T)×S(0, R₀),

∂_ny(t, x) =s·n, (t, x)∈(0, T)×S(0, r₀), s+s=−

S(0,r0)yndσ, t∈(0, T), (y(0,·), y(0,·)) = (y0, y₁),

(s(0), s(0)) = (s₀, s₁).

(3.1)

3.1. Link between observability and controllability

In this section, the fact that we are considering the radially symmetric case is irrelevant. We therefore consider the control problem (1.4)–(1.5) and the correspondingadjoint system (1.12).

Let us begin by explaining why equation (1.12) can be seen as the adjoint of (1.4). To this end, let us remark that system (1.4) can be rewritten as a first-order system

⎛

⎜⎝ y y s s

⎞

⎟⎠

=A

⎛

⎜⎝ y y s s

⎞

⎟⎠+Bf,

where

A

⎛

⎜⎝ y₀ y₁ s₀ s₁

⎞

⎟⎠=

⎛

⎜⎜

⎜⎝

y₁ Δy₀

s₁

−

∂ω

y₁ndσ−s₀

⎞

⎟⎟

⎟⎠, B=

⎛

⎜⎝ 0 χ_C

0 0

⎞

⎟⎠.

The operatorAis an unbounded operator defined on the space

X =

⎧⎪

⎨

⎪⎩

⎛

⎜⎝ y₀ y₁ s₀ s₁

⎞

⎟⎠, with

y₀∈H_ext¹ (O), y₁∈L²(O), s₀∈R², s₁∈R²

⎫⎪

⎬

⎪⎭

and its domain is

D(A) =

⎧⎪

⎨

⎪⎩

⎛

⎜⎝ y₀ y₁ s₀ s₁

⎞

⎟⎠, with

y₀∈H²(O)∩H_ext¹ (O),

∂_ny₀=s₁·non ∂ω, y₁∈H_ext¹ (O), s₀∈R², s₁∈R²

⎫⎪

⎬

⎪⎭.

Of course,X is a Hilbert space when endowed with its natural scalar product

⎛⎜⎝ y₀ y₁ s₀ s₁

⎞

⎟⎠,

⎛

⎜⎝ z₀ z₁ σ₀ σ₁

⎞

⎟⎠

X

=

O∇y₀· ∇z₀+

Oy₁z₁+s₀·σ₀+s₁·σ₁.

Besides, one easily checks thatA is skew-adjoint with respect to this scalar product. Therefore, system (1.12) is the adjoint of system (1.4) when identifying the dual ofX with itself.

Let us also note that the adjoint system (1.12) is conservative: The energy of solutions (ϕ, w) of (1.12), defined by

E(t) = 1

2(ϕ(t), ϕ(t), w(t), w(t))²_H1

ext(O)×L²(O)×(R²)²

= 1 2

O

|∇ϕ(t)|²+|ϕ(t)|² +1 2

|w(t)|²+|w(t)|² ; (3.2)

is constant in time.

Remark 3.1. Let us emphasize that here we made the choice of identifyingX with its dual. This will make our arguments easier since we are going to work only with finite-energy solutions. Other choices are possible, and in particular the one in [18], which consists in identifying the dual ofL²(O) with itself. The adjoint system should then be stated in the dual ofH_ext¹ (O)×L²(O)×(R²)², which is not completely obvious to characterize.

This requires a careful analysis done in [18] which we shortcut here.

The by-now classical Hilbert Uniqueness Method (HUM) of Lions [13,14] then yields the following result:

Theorem 3.2. Assume that T >0 is such that the observability property (1.13)holds for solutions of (1.12).

Then system (1.4) is exactly controllable in time T: given any initial data (y₀, y₁, s₀, s₁)∈H_ext¹ (O)×L²(O)× (R²)², there exists a control functionf ∈L²((0, T)× C)such that the solution of (1.4) solves (1.5).

Besides, it can be computed as follows. Let us define the functionalJ onH_ext¹ (O)×L²(O)×(R²)² by J(ϕ₀, ϕ₁, w₀, w₁) =1

2 _T

0

Cχ²_C|ϕ(t, x)|²dtdx+

O∇ϕ0· ∇y0+

Oϕ₁y₁+w₀·s₀+w₁·s₁, (3.3) where(ϕ, s)is the solution of (1.12)with initial data (y₀, y₁, s₀, s₁)∈H_ext¹ (O)×L²(O)×(R²)².

The functional J has a unique minimizer ( ˜ϕ₀,ϕ˜₁,w˜₀,w˜₁)in H_ext¹ (O)×L²(O)×(R²)². Denoting by ( ˜ϕ,w)˜ the corresponding solution of (1.12), the function

f =χ_Cϕ˜, (t, x)∈(0, T)× C (3.4)

is an admissible control function for (1.4)and (1.5).

Moreover, it has minimal L²((0, T)× C)-norm among all admissible controls for (1.4)–(1.5).

Proof. Multiplying the equation of y in (1.4) by ϕ, ϕ being a smooth solution of (1.12) and integrating by parts, we obtain

_T

0

C

χ_Cf ϕ=− _T

0

O

yϕ− _T

0

O∇y· ∇ϕ+

O∇ϕ· ∇y ^T

0

+

O

ϕy ^T

0 − _T

0

∂ω

∂_nyϕ.

But

− _T

0

O∇y· ∇ϕ= _T

0

OyΔϕ− _T

0

∂ω

y∂_nϕ.

Besides, the equations on the boundary∂ωin (1.12) and in (1.4) yield

−

∂ω

∂_ny ϕ−

∂ω

y∂_nϕ=−

∂ω

s·nϕ−

∂ω

yw·n=−s·

∂ω

ϕn

−w·

∂ω

yn

=s·(w+w) +w·(s+s) = d

dt(s·w+s·w). Combining these last identities, we get

_T

0

Cχ_Cf ϕ=

O∇ϕ· ∇y ^T

0

+

Oϕy ^T

0

+ (s·w+s·w)^T

0. (3.5)

Therefore,f is an admissible control function for (1.4) if and only if for all (ϕ₀, ϕ₁, w₀, w₁)∈H_ext¹ (O)×L²(O)×

(R²)²,

0 = _T

0

Cχ_Cf ϕ+

O∇ϕ0· ∇y0+

Oϕ₁y₁+s₁·w₁+s₀·w₀. (3.6) If ( ˜ϕ₀,ϕ˜₁,w˜₀,w˜₁)∈ H_ext¹ (O)×L²(O)×(R²)² is a minimizer of the functionalJ defined in (3.3), then the corresponding Euler–Lagrange equation coincides with (3.6) for f = χ_Cϕ˜, hence f = χ_Cϕ˜ is an admissible control function for (1.4).

One should then explain that such a minimizer exists. This is precisely due to the observability inequal- ity (1.13) which easily implies that the functionalJ is strictly convex and coercive. Therefore it has a unique minimizer (sinceJ is convex, this also is the only critical point) ( ˜ϕ₀,ϕ˜₁,w˜₀,w˜₁).

We then indicate how to prove thatf =χ_Cϕ˜is the control function of minimalL²((0, T)× C)-norm. For any control functionf₁, and in particular also forf as in (3.4), one can choose (ϕ, w) = ( ˜ϕ,w) in (3.6). Subtracting˜ the identities withf₁and with f=χ_Cϕ˜, one obtains

_T

0

Cχ_Cfϕ˜= _T

0

Cχ_Cf₁ϕ˜, hence

f²_L2((0,T)×C)= _T

0

C

f₁f ≤ f_1L2((0,T)×C)f_L2((0,T)×C).

This completes the Proof of Theorem3.2.

In the radially symmetric context we are considering, the adjoint system (1.12) becomes:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ϕ−Δϕ= 0, (t, x)∈(0, T)× A(r₀, R₀), ϕ(t, x) = 0, (t, x)∈(0, T)×S(0, R₀),

∂_nϕ(t, x) =w·n, (t, x)∈(0, T)×S(0, r₀), w+w=−

S(0,r0)ϕndσ, t∈(0, T), (ϕ(0,·), ϕ(0,·)) = (ϕ₀, ϕ₁),

(w(0), w(0)) = (w₀, w₁),

(3.7)

and the corresponding observability inequality is the following one: Find T > 0 for which there exists a con- stant C_obs > 0 such that all solutions (ϕ, w) of (3.7) with initial data (ϕ₀, ϕ₁, w₀, w₁) ∈ H_ext¹ (A(r₀, R₀))× L²(A(r₀, R₀))×(R²)²satisfy

(ϕ₀, ϕ₁, w₀, w₁)²_H1

ext(A(r0,R0))×L²(A(r0,R0))×(R²)² ≤C_obs² _T

0

A(R1,R0)χ²_R1<|x|<R0|ϕ(t, x)|²dtdx. (3.8) We now focus on the proof of the observability result (3.8) for the adjoint system (3.7).

3.2. A spherical harmonics decomposition

In this section, we introduce polar coordinates and use a decomposition of the solutionsϕof (3.7) in spherical harmonics.

Since we are working in two dimensions, thekth spherical harmonics are spanned by cos(kθ),sin(kθ). Hence, we can write

ϕ(t, x) =1 π

∞ k=1

(ψ_k,a(t, r) cos(kθ) +ψ_k,b(t, r) sin(kθ)) + 1

2πψ(t, r),˜ (3.9)

(t, x)∈(0, T)× A(r₀, R₀), wherex= (r, θ) in the polar coordinates,r∈(r₀, R₀), θ∈(0,2π).

With these notations, we have

−Δϕ=−1 π

∞ k=1

1

r∂_r(r∂_rψ_k,a)−k² r²ψ_k,a

cos(kθ)

−1 π

∞ k=1

1

r∂_r(r∂_rψ_k,b)−k² r²ψ_k,b

sin(kθ)

− 1 2π

1

r∂_r(r∂_rψ)˜

. (3.10)

Simple computations also show that

−

S(0,r0)ϕndσ=r_{0 2π}

0 ϕ(t, x)

cos(θ) sin(θ)

dθ=r₀

ψ_1,a (t, r₀) ψ_1,b(t, r₀)

. (3.11)

Finally, the condition∂_nϕ(t, x) =w(t)·non (0, T)×S(0, r₀) reads

∂_rϕ(t, r₀, θ) =w_a(t) cos(θ) +w_b(t) sin(θ), where we write

w(t) =

w_a(t) w_b(t)

.

Therefore, (ϕ, w) is a solution of (3.7) if and only if the functions ˜ψ(t, r), (ψ_k,a(t, r), ψ_k,b(t, r))_k≥1, w_a(t), w_b(t) satisfy:

• corresponding to the zero-mode term:

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩ ψ˜−1

r∂_r(r∂_rψ) = 0,˜ (t, r)∈(0, T)×(r₀, R₀),

∂_rψ(t, r˜ ₀) = 0 = ˜ψ(t, R₀), t∈(0, T), ψ(0, r) =˜

_2π

0 ϕ₀(r, θ)dθ, r∈(r₀, R₀), ψ˜(0, r) =

_2π

0 ϕ₁(r, θ)dθ, r∈(r₀, R₀),

(3.12)

• corresponding to thekth modes,k≥2,j∈ {a, b},

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ψ_k,j−1

r∂_r(r∂_rψ_k,j) +k²

r²ψ_k,j = 0,(t, r)∈(0, T)×(r₀, R₀),

∂_rψ_k,j(t, r₀) = 0 =ψ_k,j(t, R₀), t∈(0, T), ψ_k,j(0, r) =

_2π

0 ϕ₀(r, θ)e_k,j(θ)dθ, r∈(r₀, R₀), ψ_k,j(0, r) =

_2π

0 ϕ₁(r, θ)e_k,j(θ)dθ, r∈(r₀, R₀),

(3.13)

with

e_k,j(θ) =

cos(kθ) if j=a, k≥1,

sin(kθ) if j=b, k≥1, (3.14)

• corresponding to the first modes, forj∈ {a, b},

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

ψ_1,j−1

r∂_r(r∂_rψ_1,j) + 1

r²ψ_1,j= 0,(t, r)∈(0, T)×(r₀, R₀), ψ_1,j(t, R₀) = 0, t∈(0, T),

∂_rψ_1,j(t, r₀) =πw_j(t), t∈(0, T), w_j(t) +w_j(t) =r₀ψ_1,j (t, r₀), t∈(0, T), ψ_1,j(0, r) =

_2π

0 ϕ₀(r, θ)e_1,j(θ)dθ, r∈(r₀, R₀), ψ_1,j(0, r) =

_2π

0 ϕ₁(r, θ)e_1,j(θ)dθ, r∈(r₀, R₀), w_j(0) =w₀·e_j, w_j(0) =w₁·e_j,

(3.15)

withe_1,jas in (3.14) and

e_a= 1

0

, e_b= 0

1

.

At this stage, the advantages of using a spherical harmonic decomposition are already clear: we have cou- pling with the solid only in (3.15), and not in (3.12),(3.13). Furthermore, non-locality in space in the original system (3.7) (in the source term of the equation of the oscillator) has disappeared.

For eachk≥2, we define the energyE_k corresponding to thekth mode, solution of (3.13), E_k[ψ_k](t) =

_R0

r0

|ψ_k(t, r)|²rdr+ _R0

r0

|∂rψ_k(t, r)|²rdr+ _R0

r0

k²

r²|ψk(t, r)|²rdr. (3.16) Note that the energy of solutionsψ_k of (3.13) is constant.

We also defineE₀ the energy corresponding to the 0th harmonic:

E₀[ ˜ψ](t) = _R0

r0

|ψ˜(t, r)|²rdr+ _R0

r0

|∂rψ(t, r)|˜ ²rdr, (3.17)

which also is constant in timet∈(0, T) for solutions ˜ψof (3.12).

Similarly, corresponding to the 1st harmonic solutions, we introduce E₁[ψ₁, w](t) =

_R0

r0

|ψ₁(t, r)|²rdr+ _R0

r0

|∂rψ₁(t, r)|²rdr (3.18) +

_R0

r0

1

r|ψ1(t, r)|²dr+π|w(t)|²+π|w(t)|², which also is independent of time for solutions (ψ₁, w) of (3.15).

Moreover, using the L²(S(0,1)) orthogonality of the modes, the energy of solutions (ϕ, w) of (3.7) can be decomposed as follows

E(t) = 1

4πE₀[ ˜ψ](t) + 1 2π

j∈{a,b}

⎛

⎝E₁[ψ_1,j, w_j](t) +

k≥2

E_k[ψ_k,j](t)

⎞

⎠. (3.19)

And similarly, the observation operator can also be diagonalized with this spherical harmonic decomposition

A(R1,R0)|ϕ(t, x)|²dx= 1 2π

_R0

R1

|ψ˜(t, r)|²rdr+ 1 π

j∈{a,b}k≥1

_R0

R1

|ψ_k,j(t, r)|²rdr.

Therefore, system (3.7) satisfies (3.8) if and only if there exists a constant C_obs such that

• for allk∈N\{1}, any solutionψ_k of

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ ψ_k−1

r∂_r(r∂_rψ_k) +k²

r²ψ_k= 0, (t, r)∈(0, T)×(r₀, R₀),

∂_rψ_k(t, r₀) = 0 =ψ_k(t, R₀), t∈(0, T), ψ_k(0, r) =ψ_k,0(r) r∈(r₀, R₀), ψ_k(0, r) =ψ_k,1(r) r∈(r₀, R₀),

(3.20)

satisfies

E_k[ψ_k]≤C_obs² _T

0

_R0

R1

|ψ_k(t, r)|²rdr, (3.21)

• any solution (ψ, w) of

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ ψ−1

r∂_r(r∂_rψ) + 1

r²ψ= 0, (t, r)∈(0, T)×(r₀, R₀), ψ(t, R₀) = 0, t∈(0, T),

∂_rψ(t, r₀) =πw(t), t∈(0, T), w(t) +w(t) =r₀ψ(t, r₀), t∈(0, T), ψ(0, r) =ψ₀(r) r∈(r₀, R₀), ψ(0, r) =ψ₁(r) r∈(r₀, R₀), w(0) =w₀, w(0) =w₁,

(3.22)

satisfies

E1[ψ, w]≤C_obs² _T

0

_R0

R1

|ψ(t, r)|²rdr. (3.23)

Note that fork= 0, ˜ψsolves (3.20) with k= 0 and should satisfy (3.21) withk= 0, so that the modek= 0 is contained in the modesk∈N\{1}.

3.3. Observability

In the previous section, we have shown that the observability inequality (3.8) for system (3.7) is equiv- alent to uniform (with respect to k) observability properties (3.21) for systems (3.20) and the observability inequality (3.23) for system (3.22).

Based on that equivalence, we show the following result, which directly implies Theorem 1.2 according to Theorem3.2:

Theorem 3.3. Let T be such that

T >2

R²₁−r²₀= 2R₁

1− r²₀

R²₁· (3.24)

Then there exists a constant C_obs such that all solutions(ϕ, w)of (3.7)with initial data in H_ext¹ (A(r0, R₀))× L²(A(r₀, R₀))×(R²)² satisfy (3.8).