CONTROLLABILITY AND STABILIZABILITY OF THE LINEARIZED ...

SIAM J. CONTROL OPTIM. Vol. 50, No. 5, pp. 2959–2987

c 2012 Society for Industrial and Applied Mathematics 

CONTROLLABILITY AND STABILIZABILITY OF THE LINEARIZED COMPRESSIBLE NAVIER–STOKES SYSTEM IN ONE DIMENSION∗ S. CHOWDHURY† , M. RAMASWAMY‡ , AND J.-P. RAYMOND§ Abstract. In this paper we consider the one-dimensional compressible Navier–Stokes system linearized about a constant steady state (Q0 , 0) with Q0 > 0. We study the controllability and stabilizability of this linearized system. We establish that the linearized system is null controllable for regular initial data by an interior control acting everywhere in the velocity equation. We prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control or by a boundary control acting only in the velocity equation. On the other hand, we show that the system is approximately controllable. We also show that the system is not stabilizable with a decay rate e−ωt for ω > ω0 , where ω0 is an accumulation point of the real eigenvalues of the linearized operator. Key words. linearized compressible Navier–Stokes system, null controllability, boundary control, localized interior control, approximate controllability, stabilizability AMS subject classifications. 93C20, 93B05, 35B37 DOI. 10.1137/110846683

1. Introduction. Control of fluid flow has been an important area of research in recent years. While there has been considerable work on the incompressible model [8, 9, 3, 14, 15, 21], there has been much less work on the compressible fluid flow model [18, 19, 11]. One of the reasons for this is that the compressible Navier–Stokes system is much less tractable theoretically since it is a coupled system of hyperbolic and parabolic equations [13]. Setting Iπ = (0, π), let us consider the full system in Iπ × (0, T ) for the density ρ(x, t) and velocity field u(x, t) of a compressible isothermal barotropic fluid ∂t ρ(x, t) + (ρu)x (x, t) = 0, ∂t (ρu)(x, t) + (ρu2 )x (x, t) + (p(ρ))x (x, t) − νuxx (x, t) = 0. Here ν > 0 is the fluid viscosity and the pressure p is assumed to satisfy the constitutive law p(ρ) = a ργ for a > 0, γ > 0. The one-dimensional Navier–Stokes system is a model for a fluid flow in a thin tube or a narrow channel and it is interesting to study the properties of the fluid, like velocity, ∗ Received

by the editors September 2, 2011; accepted for publication (in revised form) July 23, 2012; published electronically October 2, 2012. This work was completed under CEFIPRA Project 3701-1, “Control of Systems of Partial Differential Equations.” http://www.siam.org/journals/sicon/50-5/84668.html † T.I.F.R. Centre for Applicable Mathematics, Bangalore, 560065, India (shirshendu@ math.tifrbng.res.in). ‡ Corresponding author. T.I.F.R. Centre for Applicable Mathematics, Bangalore, 560065, India ([email protected]). § Institut de Math´ ematiques de Toulouse, Universit´ e Paul Sabatier, and CNRS, 31062 Toulouse Cedex, France ([email protected]). The author is partially supported by the ANR project CISIFS 09-BLAN-0213-03. 2959

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density, and energy change along the tube or the channel. These models can be viewed as one-dimensional approximations of two-dimensional or three-dimensional models. In this paper we are interested in control and stabilization of the one-dimensional linearized compressible Navier–Stokes system. This is the first step in studying the full nonlinear system and getting some local or global controllability and stabilizability results near a stationary solution. For that purpose, in this paper we consider the control system linearized about a constant steady state (Q0 , v0 ) with Q0 > 0, (1.1)

∂t ρ(x, t) + v0 ρx (x, t) + Q0 ux (x, t) = 0, ∂t u(x, t) −

ν Q0 uxx (x, t)

+ v0 ux (x, t) + aγ Qγ−2 ρx (x, t) = f χO , 0

where χO is the characteristic function of an open subset O ⊂ Iπ . When v0 = 0, system (1.1) may be completed by the following boundary and initial conditions: (1.2)

∀ t > 0,

u(0, t) = q0 (t),

u(π, t) = qπ (t),

ρ(x, 0) = ρ0 (x)

and u(x, 0) = u0 (x).

In (1.1)–(1.2), f is a distributed control and q0 and qπ are boundary controls. When v0 = 0, for example, if v0 > 0, an additional boundary condition has to be specified for ρ. By setting U(t) = (ρ(·, t), u(·, t))T , the above linearized system (1.1)–(1.2), with an additional homogeneous boundary condition for ρ if v0 = 0, may be written in the form U (t) + AU(t) = 0,

U(0) = U0 = (ρ0 , u0 )T

in Z = L2 (Iπ ) × L2 (Iπ ) or in Z0 = L2m (Iπ ) × L2 (Iπ ), when f = 0, q0 = 0, qπ = 0. (Here L2m (Iπ ) denotes the space of functions in L2 (Iπ ) with mean value 0.) When v0 = 0, the resolvent of −A, that is, R(λ, A) = (λI + A)−1 with λ > 0, is a compact operator in Z (see Proposition IV.13 in chapter 4 of [10]). This is no longer true if v0 = 0. Moreover, if v0 = 0, −A is a sectorial operator in Z (it is a consequence of Lemma 2.5). This seems to be not the case when v0 = 0. Thus we see that the properties of the two semigroups (e−tA )t≥0 (the one when v0 = 0 and the one when v0 = 0) are completely different. In this paper we only study the case when v0 = 0. The main results of the paper are stated in sections 5 and 6 where we obtain positive and negative results for the null controllability and approximate controllability of system (1.1)–(1.2) and in section 7 where we obtain negative stabilizability results for the same system. In Theorem 5.1, we state a null controllability for initial data in (H 1 (Iπ ) ∩ L2m (Iπ )) × L2 (Iπ ) by a distributed control f acting everywhere in Iπ in the velocity equation. We prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control (see Theorem 5.10) or by a boundary control (see Theorems 5.6 and 5.8). On the other hand, the system is approximately controllable (see Theorem 6.2). In Theorem 7.11, we show that system (1.1)–(1.2) is not stabilizable with a decay rate e−ωt if ω > ω0 = aγQγ0 /ν and ω is in the resolvent set of A. (The stabilizability for a decay rate e−ωt with ω < ω0 was already established in [2].) To the authors’ knowledge, these results are totally new. Our method is very much based on explicit eigenfunctions and the behavior of the spectrum. These techniques seem to extend to some other cases of fluid models, like the nonbarotropic case (i.e., p = p(ρ, θ)) in one dimension involving an additional

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

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equation for temperature θ and the barotropic case in a two-dimensional rectangular domain. Studying the spectrum of the linearized operator and the behavior of the semigroup in these cases, for certain special boundary conditions, leads to similar eigenfunctions but involves more tedious computations. This is because in both cases, the dimension of the invariant spaces (which are two-dimensional in our case; see section 2.1) are greater than 2 and hence calculations are much more complicated, at least for the analysis of interior controllability as in our case. Some of the other results, like approximate controllability and boundary null controllability (Theorem 6.2 and Theorem 5.8) for the linearized operator in one dimension for the perfect gas (with an extra equation for temperature θ and p = Rρθ), follow by similar techniques using only one boundary control for velocity. It also seems possible to extend Theorem 5.8 and Theorem 6.2 to two dimensions by similar methods. These works are in progress. There have been some results regarding control of different fluid models in recent years. Amosova in [1] considers compressible viscous fluid in one dimension in Lagrangian coordinates with zero boundary condition for the velocity on the boundaries of the interval (0, 1) and an interior control on the velocity equation. She proves local exact controllability to trajectories for the velocity, provided that the initial density is already on the “targeted trajectory” in [1]. Ervedoza et al. consider the compressible Navier–Stokes system in one space dimension in a bounded domain (0, L) in [7]. They prove in [7] local exact controllability to constant states (¯ ρ, u¯) with ρ¯ > 0, u ¯ = 0 using two boundary controls (both for density and velocity). Renardy in [16] proves exact controllability results for the linear viscoelastic fluids of the Maxwell kind using interior control in one dimension for a bounded interval (0, L) when time is sufficiently large. Doubova, Fern´andez-Cara, and Gonz´alez-Burgos in [6] prove large time null and approximate controllability results for the linear viscoelastic fluids of the Maxwell kind and for any time an approximate controllability result for fluids of the Jeffreys kind using interior control in a bounded domain of RN for N = 2, 3. See also [5] for more results in this direction. The plan of the paper is as follows. In section 2, we study the properties of the operator A and of the semigroup (e−At )t≥0 . We establish a controllability result of finite dimensional projection in section 3, and we estimate the control of minimal norm in section 4. As mentioned above, the positive and negative results on controllability and stabilizability are proved in sections 5, 6, and 7. The authors acknowledge the financial support from the Indo-French Centre for the Promotion of Advanced Research, Delhi, under project 3701-1. 2. Linearized operator. Let us recall that Z = L2 (Iπ ) × L2 (Iπ ) and introduce the positive constants (2.1)

b := aγQγ−2 , 0

ν0 :=

ν . Q0

Let Z be endowed with the inner product      π  π ρ σ := b ρ(x)σ(x) dx + Q0 u(x)v(x)dx. , u v 0 0 z The Lebesgue space L2m (Iπ ) contains all the square integrable functions with zero mean value    π 2 2 Lm (Iπ ) = f ∈ L (Iπ ), f (x) dx = 0 . 0

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We also introduce the space Z0 = L2m (Iπ ) × L2 (Iπ ), where the space Z0 will be equipped with the Z-scalar product defined above. We set 1 Hm (Iπ ) = H 1 (Iπ ) ∩ L2m (Iπ ), where H 1 (Iπ ) is the standard Sobolev space. We define 1 H{0} as the space of functions in H 1 (Iπ ) that vanish at x = 0

1 (Iπ ) = f ∈ H 1 (Iπ ), f (0) = 0 . H{0} We now define the unbounded operator (A, D(A)) in Z by D(A) = {U = (ρ, u)T ∈ Z | u ∈ H01 (Iπ ), (bρ − ν0 u ) ∈ H 1 (Iπ )} and (2.2)

 A =

0

d Q0 dx

d aγQγ−2 0 dx

−ν d2 Q0 dx2



 =

0

d Q0 dx

d b dx

d −ν0 dx 2

2

.

Setting U(t) = (ρ(·, t), u(·, t))T , the system (1.1) with homogeneous boundary conditions in (1.2) and f = 0 can be written as U (t) + AU(t) = 0,

(2.3)

U(0) = U0 ∈ Z.

We now show that (−A, D(A)) is the infinitesimal generator of a C 0 semigroup of contractions on Z. Lemma 2.1 (see [2]). The operator A is maximal monotone in Z. Thus, (−A, D(A)) is the infinitesimal generator of a strongly continuous semigroup of contractions on Z, denoted by (S(t))t≥0 . For every U0 ∈ Z, there is a unique solution U of (2.3) in C([0, ∞); Z) and

ρ

U(t)Z ≤ U0 Z ,

∀ t ≥ 0.

Proof. For all u ∈ D(A), we have           π Q0 u ρ ρ ρ = = ν |u (x)|2 dx , , A   (bρ − ν u ) u u u 0 0 z z Thus (A, D(A)) is monotone. It is also maximal. Indeed for (g, h)T equation     ρ g (I + A) = u h

 0.

∈ Z, the

is equivalent to find ρ ∈ L2 (Iπ ), u ∈ H01 (Iπ ) such that ρ + Q0 u = g

and u + bρ − ν0 u = h

in (0, π).

Simplification leads to an equation for u ∈ H01 (Iπ ), u − Q0 bu − ν0 u = (h − bg  ) , where the right-hand side is in H −1 (Iπ ) and hence the equation admits a solution u in H01 (Iπ ). From the equality (bρ − ν0 u ) = h − u, it follows that (ρ, u) is in D(A). Thus (A, D(A)) is maximal monotone.

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CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

We easily verify that Ker(A) = {U = (e, 0)T | e ∈ R}. Remark 2.2. The adjoint (A∗ , D(A∗ )) of the operator (A, D(A)) in Z is defined by D(A∗ ) = {(σ, v)T ∈ Z | v ∈ H01 (Iπ ), (bσ + ν0 v  ) ∈ H 1 (Iπ )} and  A∗

σ



 =

v

0

d −Q0 dx

d −b dx

d −ν0 dx 2



2

σ



v

for (σ, v)T ∈ D(A∗ ). Moreover (−A∗ , D(A∗ )) is the infinitesimal generator of a strongly continuous semigroup of contractions (S ∗ (t))t≥0 on Z. 2.1. Spectrum of the linearized operator. In the case of the stationary solution (Q0 , v0 ) with velocity v0 > 0, Girinon [10] has shown that the resolvent R(λ, A) is a compact operator from Z to Z and that the semigroup (S(t))t≥0 is exponentially stable on Z, i.e., there exists M > 0 such that S(t)L(Z) ≤ M e−ωt for some ω > 0. In our case, v0 = 0, and the resolvent is no more compact. Yet, we can show the exponential stability by following the approach of [2], using Fourier analysis. For that we define a Fourier basis {Φn }n≥0 in Z as follows: Φ0 (x) = Φ2n (x) =

√1 (1, 0)T , bπ





2 T bπ (cos(nx), 0) ,

Φ2n−1 (x) =

2 T Q0 π (0, sin(nx))

for n ≥ 1.

Let us define the following spaces of one and two dimensions: V0 = span {Φ0 },

Vn = span {Φ2n , Φ2n−1 },

n ≥ 1,

where “span” stands for the vector space generated by those functions. One can verify that Z is the orthogonal sum of all these subspaces {Vn }n≥0 . Observe that V0 = Ker(A). This motivates us to introduce the space  (2.4) Z0 := (ρ, u)T ∈ Z

    

π

 ρ(x)dx = 0 .

0

Then it follows that the space Z0 is the orthogonal sum of the subspaces {Vn }n≥1 . Lemma 2.3. For all n ≥ 1 , Vn is invariant under A and An = A|Vn ∈ L(Vn ) has the matrix representation 

√0 − bQ0 n

in the basis {Φ2n , Φ2n−1 } of Vn .

 √ bQ0 n ν0 n 2

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S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Proof. Notice that   2 (0, −bn sin(nx))T = − Q0 b nΦ2n−1 , AΦ2n = bπ   2 (nQ0 cos(nx), ν0 n2 sin(nx))T = b Q0 n Φ2n + ν0 n2 Φ2n−1 . AΦ2n−1 = Q0 π Thus the matrix representation of A|Vn in this basis is   √ bQ0 n √0 (2.5) An = . − bQ0 n ν0 n2 Remark 2.4. In a similar manner, we can check that for all n ≥ 1, Vn is invariant under A∗ and A∗n = A∗ |Vn ∈ L(Vn ) has the matrix representation 

√ 0 bQ0 n

 √ − bQ0 n ν0 n 2

in the basis {Φ2n , Φ2n−1 } of Vn . Lemma 2.5. The spectrum σ(A) of A except zero, i.e., σ(A) \ {0}, lies on the right side of the complex plane. It consists of a finite number of pairs of complex eigenvalues and an infinite number of pairs of real eigenvalues, {λn , μn }n≥n0 . For 1 ≤ k < n0 , the complex eigenvalues {λk , μk = λk } satisfy |Re(λk )| ≥

ν0 , 2

|Im(λk )| ≤ 2ω0 , where ω0 :=

bQ0 . ν0

For n ≥ n0 , the real eigenvalues satisfy lim λn = ω0 ,

(2.6)

n→∞

μn → ∞ as n → ∞.

The eigenfunctions of A in Z, corresponding, respectively, to λn and μn , are (2.7) ξn (x) =

T T   λn μn sin(nx) sin(nx) cos(nx), and ζn (x) = cos(nx), . Q0 n Q0 n

Proof. Let us set α := bQ0 . The eigenvalues of An are then given by the roots of λ2 − λν0 n2 + αn2 = 0, and they are       ν0 n 2 4α ν0 n 2 4α λn = 1 − 1 − 2 2 , μn = 1+ 1− 2 2 . 2 ν0 n 2 ν0 n If

√ 2 α ν0

is an integer, then define γ+1 √ √ 2 aγQ0 2 2 α . = n0 := ν0 ν

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

If not, define n0 to be the smallest integer bigger than are real. In that case, since α = bQ0 , ω0 = να0 , we have  ω0 − λn = −

(2.8)

ω02 ν0 n 2



 +o

√ 2 α ν0 .

1 n2

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For n ≥ n0 , eigenvalues

 ,

and hence ω0 < λn < 2ω0 ,

(2.9)

ν0 n 2 < ν0 n2 − 2ω0 < μn < ν0 n2 . 2

For 1 ≤ n < n0 , the two complex eigenvalues are λn = δn − iηn ,

μn = δn + iηn ,

where ν0 n 2 ηn = 2



√ 2α 4α ν0 n 2 2 α ≤ − 1 ≤ = 2ω0 2 2 ν0 n 2 ν0 n ν0

and δn =

ν0 n 2 ν0 ≥ . 2 2

The limiting behavior can be determined using the above expressions and estimates. Eigenfunctions can be calculated using the operator A. Remark 2.6. Observe that the spectrum of A∗ is the same as that of A. The eigenfunctions of A∗ can be calculated as ∗ ξn (x) =

T  −λn sin(nx) cos(nx), , Q0 n

∗ ζn (x) =

T  −μn sin(nx) cos(nx), , Q0 n

corresponding to λn and μn , respectively. 2.2. The behavior of the semigroup. The semigroup (S(t))t≥0 is exponentially stable on the subspace Z0 , defined in (2.4). For details, see Corollary 1 in [2]. We now explore if the system is null controllable using interior control for the velocity component. For that we plan to take the finite dimensional projection of the controlled system onto Vn , using the semigroup generated by −An e−An t := Sn (t) = S(t)|Vn ,

n ≥ 0.

Clearly S0 (t) = IV0 . For n ∈ [1, n0 ), the matrix representation for Sn (t) in the basis {Φ2n , Φ2n−1 } is   √ − αn sin(ηn t) ηn cos(ηn t) + δn sin(ηn t) 1 −δn t , e (2.10) Sn (t) = √ ηn αn sin(ηn t) ηn cos(ηn t) − δn sin(ηn t) and for n ∈ [n0 , ∞), we have  μn e−λn t − λn e−μn t 1 (2.11) Sn (t) = √ μn − λn αn(e−λn t − e−μn t )

 √ − αn(e−λn t − e−μn t ) μn e−μn t − λn e−λn t

.

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For null controllability, we will need the behavior of the semigroup Sn (t) as n tends to ∞. Lemma 2.7. For n large, the semigroup Sn (t) has the form   sn,1 (t) −sn (t) 1 Sn (t) = , (μn − λn ) sn (t) sn,2 (t) where (2.12)

sn,1 (t) = O(ν0 n2 e−ω0 t ),

sn,2 (t) = O(2ω0 e−ω0 t ),

√ sn (t) = O( αne−ω0 t ),

uniformly for all t ∈ [ε, T ], for any 0 < ε < T . Proof. Using the previous estimates, we get ν0 n2 t 2 ν0 n2 −2ω0 t e ≤ μn e−λn t ≤ ν0 n2 e−ω0 t , ω0 e−ν0 n t ≤ λn e−μn t ≤ 2ω0 e− 2 . 2

From (2.11), we have sn,1 (t) = μn e−λn t − λn e−μn t and hence ν0 n2 t 2 ν0 n2 −2ω0 t e − 2ω0 e− 2 ≤ sn,1 (t) ≤ ν0 n2 e−ω0 t − ω0 e−ν0 n t . 2 √ Again from (2.11), sn (t) = αn(e−λn t − e−μn t ) implies that

(2.13)

(2.14)

√

αn(e−2ω0 t − e−

ν0 n2 t 2

) ≤ sn (t) ≤

√ 2 αn(e−ω0 t − e−ν0 n t ).

Similarly we have ν0 n2 t ν0 n2 −ν0 n2 t e ≤ μn e−μn t ≤ ν0 n2 e− 2 , ω0 e−2ω0 t ≤ λn e−λn t ≤ 2ω0 e−ω0 t . 2

Thus for sn,2 (t) = (μn e−μn t − λn e−λn t ), we obtain (2.15)

ν0 n2 t ν0 n2 −ν0 n2 t − 2ω0 e−ω0 t ≤ sn,2 (t) ≤ ν0 n2 e− 2 − ω0 e−2ω0 t . e 2

Hence (2.12) follows. Remark 2.8. The semigroup (Sn∗ (t))t≥0 , generated by −A∗n , is defined by   sn,1 (t) sn (t) 1 ∗ Sn (t) = . (μn − λn ) −sn (t) sn,2 (t) 3. Controllability of finite dimensional projections. Now we look for null controllability in Z0 of the system below using interior control f ∈ L2 (0, ∞; L2 (Iπ )) for the velocity component, (3.1)

U (t) + AU(t) = F(t),

U(0) = U0 ∈ Z0 ,

with F(t) = (0, f (·, t))T . Recall that the weak solution in Z0 of this system is given by  t U(t) = S(t)U0 + S(t − s)F(s)ds ∀ t ≥ 0. 0

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CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

Using the Fourier basis in Z0 , we set F(t) =

∞ 

gn (t)Φ2n +

n=1

∞ 

fn (t)Φ2n−1 .

n=1

For all n, the Fourier coefficients are given by gn (t) = F(t), Φ2n z = 0,



fn (t) = F(t), Φ2n−1 z =

2Q0 π



π

f (x, t) sin(nx)dx. 0

Thus the projection of the control F(t) on the two-dimensional space Vn is Fn (x, t) = fn (t)Φ2n−1 (x). Defining Un = U|Vn and U0,n = U(0)|Vn , the finite dimensional system obtained by projecting (3.1) on Vn is   0 (3.2) Un (t) = −An Un (t) + fn (t)B, Un (0) = U0,n with B = . 1 Lemma 3.1. For any given T > 0, the finite dimensional system (3.2) is controllable. Proof. One can verify that the Kalman rank condition holds and hence the finite dimensional system is controllable. Indeed, the matrix [B | An B] is of rank two. 4. Estimates for minimum norm control. For a given T > 0, the finite dimensional system (3.2) is controllable and hence there exists a control which brings this system to rest in time T . Among all such controls, the one of minimal norm is given by ∗ −1 −T An e )U0,n , fn (t) = −(B ∗ e−(T −t)An Wn,T

(4.1) where

 (4.2)

Wn,T =

T

∗

e−tAn BB ∗ e−tAn dt.

0

(See [22, Part IV, Chapter 2, Theorem 2.3].) Let us set ∗

−1 −T An e . Dn (t) = −B ∗ e−(T −t)An Wn,T

(4.3)

We will need a few estimates for Wn,T and its inverse. Lemma 4.1. If An , B are as in (2.5), (3.2) and if we set  n,T , where W n,T = wn,1 (4.4) Wn,T := (μn − λn )−2 W wn

wn wn,2

 ,

then wn,1 =

n2 α (1 − e−2λn T ) + O(1), 2λn

wn,2 =

λn n 2 ν0 + + O(1). 2 2

wn = −

 √  n α e−2λn T + o(1), 2

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S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Furthermore the inverse of Wn,T is given by  (μn − λn )2 wn,2 −1 Wn,T = n,T ) −wn det (W

−wn wn,1

 ,

n,T satisfies the estimate and the determinant of W n2 α −2λn T αν0 n4 n,T (e (1 − e−2λn T ) + − 3) + O(1) ≤ det W 8λn 2 αν0 n4 n2 α (1 − e−4λn T ) + O(1). (4.5) ≤ + 4λn 4 Proof. Using the expression of Sn (t) and Lemma 2.7, we have      1 1 0 sn,1 (t) −sn (t) −sn (t) Sn (t)B = = . sn (t) sn,2 (t) 1 (μn − λn ) (μn − λn ) sn,2 (t) Similarly, we have B

∗

Sn∗ (t)

1 [0 1] = (μn − λn )



sn,1 (t) −sn (t)

Putting these two together yields Sn (t)BB ∗ Sn∗ (t) = (μn − λn )−2

sn (t) sn,2 (t) 

 =

1 [−sn (t) sn,2 (t)]. (μn − λn )

s2n (t) −sn (t)sn,2 (t)

−sn (t)sn,2 (t) s2n,2 (t)

 .

Using the definition of Wn,T given by (4.4), we find  T  T √ wn = − (sn sn,2 )(t)dt = − αn(e−λn t − e−μn t )(μn e−μn t − λn e−λn t )dt 0

√ = − αn

 0

0

T

((μn + λn )e−(λn +μn )t − μn e−2μn t − λn e−2λn t )dt

 T √ e−2μn t e−2λn t = − αn −e−(λn +μn )t + + 2 2 0   √ 1 −2λn T −2μn T −(λn +μn )T (e = − αn +e )−e . 2 Thus the behavior of wn for large n is given by √ √ n α −2λn T n α −2λn T (4.6) − + o(1) < wn < − − e−(λn +μn )T ) + o(1). e (e 2 2 Similarly, we find  T  2 2 wn,1 = sn (t)dt = n α 0



0

T

(e−2λn t + e−2μn t − 2e−(λn +μn )t )dt

T e−2λn t e−2μn t 2e−(λn +μn )t + + −2λn −2μn (λn + μn ) 0     1 1 2 1 e−2λn T e−2μn T 2e−(λn +μn )T 2 =n α + − − + + . 2λn 2μn λn + μn 2 λn μn λn + μn = n2 α

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CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

Thus, we have n2 α n2 α (1 − e−2λn T ) + O(1) ≤ wn,1 ≤ 2λn 2λn

(4.7)



1 − e−2λn T +

λn μn

 + O(1)

and  wn,2 = 

T

0

s2n,2 (t)dt =

 0

T

(μ2n e−2μn t + λ2n e−2λn t − 2μn λn e−(λn +μn )t )

μn e−2μn t λn e−2λn t 2μn λn −(λn +μn )t e = − − + 2 2 μn + λn =

T 0

2μn λn 1 μn + λn 2μn λn −(λn +μn )T − − (μn e−2μn T + λn e−2λn T ) + e . 2 (μn + λn ) 2 μn + λn

Hence, we can write ν0 n 2 λn λn ν0 n 2 + (1 − e−2λn T ) + O(1) ≤ wn,2 ≤ + + O(1). 2 2 2 2

(4.8)

Further, we have −1 Wn,T =

(μn − λn )2 wn,1 wn,2 − wn2



wn,2 −wn

−wn wn,1

 .

From the expression of wn,1 , wn,2 , and wn , it follows that αν0 n4 n2 α −4λn T (e (1 − e−2λn T ) + + 2e−2λn T − 6) + O(1) 8λn 4 αν0 n4 n2 α + O(1), + ≤ wn,1 wn,2 ≤ 4λn 4 n2 α −4λn T n2 α −4λn T e (e + o(1) ≤ wn2 ≤ + e−2(λn +μn )T ) + o(1). 4 4 n,T ) satisfies (4.5). Thus det(W Now we can prove the following. Theorem 4.2. If system (3.2) is driven to rest in time T > 0 by using the minimal norm control fn determined in (4.1), setting fn (t) = Dn (t)U0,n with Dn (t) = [dn,1 (t) dn,2 (t)] as defined in (4.3), then for n large we have dn,1 (t) = O(ne−ω0 (T −t) )

and

dn,2 (t) = O(e−ω0 (T −t) )

uniformly with respect to t ∈ [0, T − ε] for any 0 < ε < T . Proof. Using the expression for the control of minimum norm, we have ∗

−1 −T An Dn (t) = −B ∗ e−(T −t)An Wn,T e

=− =

1 [−sn (T − t) sn,2 (T − t)] n,T ) det(W

1 [dˆn,1 (t) dˆn,2 (t)],  det(Wn,T )



wn,2 −wn

−wn wn,1



sn,1 (T ) −sn (T ) sn (T ) sn,2 (T )



2970

S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

where we have denoted dˆn,1 (t) := (wn,2 sn (T − t) + wn sn,2 (T − t))sn,1 (T ) − sn (T )(wn sn (T − t) + wn,1 sn,2 (T − t)), dˆn,2 (t) := −(wn,2 sn (T − t) + wn sn,2 (T − t))sn (T ) − sn,2 (T )(wn sn (T − t) + wn,1 sn,2 (T − t)). Recall from Lemma 2.7 that for all n and all t ∈ (0, T ], sn,2 (T − t) is bounded. Hence for large n, the dominant terms in these two expressions are d˜n,1 (t) := wn,2 sn (T −t)sn,1 (T )−sn (T )sn (T −t)wn ,

d˜n,2 (t) := −wn,2 sn (T −t)sn (T ).

Using all the earlier estimates from Lemma 2.7, we see that   ν n2 T ν0 n2 (T −t) √ ν0 n2 −2ω0 T − 02 2 e − 2ω0 e αn(e−2ω0 (T −t) − e− ) 2 √ ≤ sn (T − t)sn,1 (T ) ≤ (ν0 n2 e−ω0 T )( αne−ω0 (T −t) ). Thus, we can write  2 5√  ν0 n α −2ω0 (T −t) −2ω0 T e + O(n3 ) ≤ wn,2 sn (T − t)sn,1 (T ) e 4  2 5√  ν0 n α −ω0 T −ω0 (T −t) ≤ e + O(n3 ). e 2 Similarly, the other terms satisfy  √ 3 α αn e−2ω0 T e−2ω0 (T −t) e−2λn T + o(1) ≤ −wn sn (T − t)sn (T ) 2  √ 3 α αn ≤ e−ω0 T e−ω0 (T −t) e−2λn T + o(1), 2   ν0 αn4 − e−ω0 (T −t) e−ω0 T + O(n2 ) ≤ −wn,2 sn (T − t)sn (T ) 2   ν0 αn4 ≤− e−2ω0 (T −t) e−2ω0 T + O(n2 ). 2 From (4.5) for sufficiently large n, we have 4 αν0 n4 n,T ≤ α(ν0 + ω0 )n . (1 − e−2ω0 T ) ≤ det W 32ω0 2ω0

From these estimates, we see that

    d˜n,1 ν02 ω0 n   −2ω0 T −2ω0 (T −t) √ e e + o(1) ≤   n,T )   det(W 2 α(ω0 + ν0 ) 32ν0 ω0 n ≤ √ e−ω0 (T −t) + o(1), α(eω0 T − e−ω0 T )     ν0 ω 0 d˜n,2   −2ω0 T −2ω0 (T −t) e e + o(1) ≤   n,T )   det(W (ν0 + ω0 ) ≤

(eω0 T

16ω0 e−ω0 (T −t) + o(1), − e−ω0 T )

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

2971

so that Dn (t) = [dn,1 (t)

dn,2 (t)] = [O(ne−ω0 (T −t) ) O(e−ω0 (T −t) )]

uniformly with respect to t ∈ [0, T − ε] for any 0 < ε < T . 5. Null controllability. We will explore first interior and then boundary and localized interior null controllability. 5.1. Interior control. Now we are in a position to answer the interior null controllability question for our system, using the earlier estimates. 1 2 1 Theorem π ), where Hm (Iπ ) = {ρ ∈  π 5.1. Let us denote Y := Hm (Iπ ) × L (I 1 T H (Iπ ) | 0 ρ(x)dx = 0}. Assume that U0 = (ρ0 , u0 ) ∈ Z0 and that the system (3.1) is null controllable in time T > 0 by an interior control f ∈ L2 (0, T ; L2 (Iπ )) for the velocity. Then U0 = (ρ0 , u0 )T belongs to Y. Conversely, assume that U0 = (ρ0 , u0 )T ∈ Y. Then, for every T > 0, the system (3.1) is null controllable in time T by an interior control f ∈ L2 (0, T ; L2(Iπ )) for the velocity. Proof. If the system (3.1) were null controllable in time T > 0, using an interior control for the velocity, then there will exist a minimum norm control F(x, t) = (0, f (x, t))T which brings the system to rest in time T . Then the projections of this control F into the space Vn , say, Fn (t) = (0, fn (t))T , will bring the finite dimensional system on Vn to rest in time T . Since F2L2 (0,T ;Z0 ) =

∞ 

fn 2L2 (0,T ) ,

n=1

the controls fn are also with minimal norm. Moreover, we have (5.1)

fn (t) = Dn (t)U0,n = dn,1 (t)U0,n,1 + dn,2 (t)U0,n,2

and (5.2)

∞   n=1

0

T

|dn,1 (t)U0,n,1 + dn,2 (t)U0,n,2 |2 dt < ∞.

Since U0 = (ρ0 , u0 )T ∈ Z0 , we also know that ∞ 

|U0,n,1 |2 < ∞ and

n=1

∞ 

|U0,n,2 |2 < ∞.

n=1

From Theorem 4.2 and (5.2), it follows that ∞ 

n2 |U0,n,1 |2 < ∞.

n=1 1 (Iπ ) × L2 (Iπ ) is the orthogonal sum of the spaces {Vn }n≥1 , one can Since Y = Hm verify that

(5.3)

(ρ0 , u0 )T 2Y =

∞  n=1

2 n2 U0,n,1 +

∞ 

2 U0,n,2 .

n=1

Thus if U0 belongs to Z0 and the system (3.1) is null controllable in time T > 0, by f ∈ L2 (0, T ; L2(Iπ )), then U0 belongs to Y.

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S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Conversely, let us assume that U0 = (ρ0 , u0 )T ∈ Y. The control fn given in (5.1) brings the finite dimensional system on Vn to zero in time T > 0. Our estimates derived from Theorem 4.2 for the coefficients dn,1 and dn,2 , together with the condition (ρ0 , u0 )T 2Y =

∞ 

2 n2 U0,n,1 +

n=1

∞ 

2 U0,n,2 < ∞,

n=1

imply that ∞ 

fn 2L2 (0,T ) < ∞.

n=1

Thus the control F=

∞ 

fn Φ2n−1

n=1

belongs to L2 (0, T ; Z0 ) and it brings the system to zero in time T . This completes the proof. 5.2. Boundary control. Here we will answer in the negative the question of null controllability in time T of the system      d 0 Q0 dx ρ 0 ρ ∂t = , + d d2 u u 0 b dx −ν0 dx 2 (5.4) ρ(x, 0) = ρ0 (x), u(x, 0) = u0 (x), x ∈ Iπ , u(0, t) = 0,

u(π, t) = q(t),

t > 0,

using a regular boundary control q ∈ H 1 (0, T ) at π for the velocity component. The 1 1 1 initial condition (ρ0 , u0 ) is in the space Hm (Iπ ) × H{0} (Iπ ). (Here H{0} (Iπ ) is the 1 space of functions in H (Iπ ) vanishing at 0.) First we recall that the solution of (5.4) is also regular enough. The following proposition can be proved on the same lines as in [2, Theorem 3.1]. 1 1 (Iπ ) × H{0} (Iπ ) and q ∈ H 1 (0, ∞), Proposition 5.2. Given (ρ0 , u0 ) in Hm satisfying the compatibility condition u0 (π) = q(0), the system (5.4) has a unique solution (ρ, u) with ρ ∈ H 1 (0, ∞; H 1 (Iπ )) and u ∈ H 1 (0, ∞; L2 (Iπ )) ∩ L2 (0, ∞; H 2 (Iπ ) ∩ 1 H{0} (Iπ )) satisfying ρH 1 (0,∞;H 1 ) + uH 1 (0,∞;L2 ) + uL2 (0,∞;H 2 ) ≤ C(U0 Hm + qH 1 (0,∞) ). 1 ×H 1 {0} To discuss the boundary null controllability, the idea is, as in [20] and [17], to use the adjoint equation to derive certain identity equivalent to the boundary null controllability. For that, we consider the adjoint problem      d  0 −Q0 dx σ 0 σ −∂t = , + 0 d d2 v v −b dx −ν0 dx2 (5.5) v(0, t) = 0, v(π, t) = 0, t > 0, σ(x, T ) = σT (x),

v(x, T ) = vT (x),

x ∈ Iπ ,

1 (Iπ ) × H01 (Iπ ). Existence and regularity of with the terminal condition (σT , vT ) in Hm

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

2973

the solutions of (5.5) are given in the proposition below. For details, see Theorem 2.3 in [2]. Proposition 5.3. The family of operators (S ∗ (t))t≥0 determines a strongly con1 1 tinuous semigroup on Hm (Iπ ) × H01 (Iπ ). For all VT = (σT , vT )T ∈ Hm (Iπ ) × H01 (Iπ ), ∗ the solution V(t) = S (t)VT satisfies V(t)Hm 1 ×H 1 ≤ CVT H 1 ×H 1 . m 0 0 In addition 1 ) + vH 1 (0,∞;L2 ) + vL2 (0,∞;H 2 ) ≤ CVT H 1 ×H 1 . σH 1 (0,∞;Hm m 0

We also need a regularity result for the nonhomogeneous adjoint system       d 0 −Q0 dx σ f σ −∂t = , + g d d2 v v −b dx −ν0 dx 2 (5.6) v(0, t) = 0, v(π, t) = 0, t > 0, σ(x, T ) = 0,

x ∈ Iπ .

v(x, T ) = 0,

Proposition 5.4. If (f, g)T ∈ L2 (0, T ; Z), then the solution V to (5.6) belongs to L2 (0, T ; D(A∗ )) and it satisfies VL2 (0,T ;D(A∗)) ≤ C(f, g)T L2 (0,T ;Z) . In particular, we have bσ(π, ·) + ν0 vx (π, ·)L2 (0,T ) ≤ C(f, g)T L2 (0,T ;Z) . This proposition follows from Theorem 3.1, Part II, in [4]. Let (ρ, u) and (σ, v) be the solutions of (5.4) and (5.5), respectively, as mentioned in the above propositions. Taking the inner product in Z of (5.4) with (σ, v)T and integrating, we obtain     T  ∂t ρ + Q0 ux σ dt = 0. , ∂t u + b ρx − ν0 uxx v 0 z An integration by parts and the use of (5.5) leads to (5.7)  Q0

T

0

(bσ(π, t) + ν0 vx (π, t))q(t) dt 

=b 0

π

 [ρ0 (x)σ(x, 0) − ρ(x, T )σT (x)]dx + Q0

0

π

[u0 (x)v(x, 0) − u(x, T )vT (x)]dx.

The above relation will lead us to the identity equivalent to the boundary null controllability. 1 1 (Iπ ) × H{0} (Iπ ), the Proposition 5.5. For each initial state (ρ0 , u0 ) ∈ Hm solution of the system (5.4) can be brought to rest in time T by a control q ∈ H 1 (0, T ) satisfying q(0) = u0 (π) if and only if      T ρ0 σ(·, 0) Q0 (5.8) [bσ(π, t) + ν0 vx (π, t)]q(t)dt = , u0 v(·, 0) 0 z 1 for all (σT , vT )T ∈ Hm (Iπ ) × H01 (Iπ ), where (σ, v)T is the solution of the adjoint system (5.5).

2974

S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Using this proposition, we now rule out the boundary null controllability. Theorem 5.6. For any T > 0, no nontrivial finite linear combination of eigenvectors can be driven to rest by a boundary control q ∈ H 1 (0, T ) at π for the velocity. 1 1 Proof. Recall that Hm (Iπ ) × H{0} (Iπ ) is the orthogonal sum of {Vn }n≥1 . Thus we can write (ρ0 , u0 )T =

∞ 

c˜n Φ2n +

n=1

∞ 

d˜n Φ2n−1 .

n=1

Since Φ2n , Φ2n−1 are linear multiples of (cos(nx), 0)T and (0, sin(nx))T , respectively, 1 1 (Iπ ) × H{0} (Iπ ) can be written as (ρ0 , u0 )T ∈ Hm (5.9)

ρ0 (x) =



cn cos(nx),

u0 (x) =

n≥1



dn sin(nx).

n≥1

∗ ∗ + bn ζn for some an and bn . With this terminal Let us choose (σT , vT )T = an ξn condition the solution of the adjoint system (5.5) is ∗ ∗ (σ, v)T = an e−λn (T −t) ξn + bn e−μn (T −t) ζn .

In particular, we have (5.10)

vx (x, t) = [an λn e−λn (T −t) + bn μn e−μn (T −t) ]

(−1) cos(nx) . Q0

Now applying Proposition 5.5, we get  T   an e−λn (T −t) + bn e−μn (T −t) b(−1)n Q0 0

   n+1 −λn (T −t) −μn (T −t) ν0 (−1) + an λn e + b n μn e q(t)dt Q0



π

= 0

(bcn cos2 (nx)(an e−λn T + bn e−μn T ))dx



−

0

π



dn sin2 (nx)(an λn e−λn T + bn μn e−μn T n

 dx.

After simplification this reduces to (−1) (bQ0 − ν0 λn )an e n

−λn T



T

eλn t q(t)dt

0

+ (−1)n (bQ0 − ν0 μn )bn e−μn T =



T

eμn t q(t)dt

0

(cn bπn − πdn λn ) (cn bπn − πdn μn ) an e−λn T + bn e−μn T . 2n 2n

A choice of bn = 0 and then an = 0 in this identity gives  T (cn bπn − πdn λn ) n (5.11) , eλn t q(t)dt = (−1) (bQ0 − ν0 λn ) 2n 0  T (cn bπn − πdn μn ) (−1)n (bQ0 − ν0 μn ) (5.12) . eμn t q(t)dt = 2n 0

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

2975

This can be done for each n ∈ N. Thus the null controllability is equivalent to the existence of a function q ∈ H 1 (0, T ) such that for each n ≥ 1, the identities (5.11) and (5.12) hold. Now we propose to show that no nontrivial finite linear combination of eigenvectors can be driven to zero in finite time. Let us take as initial conditions (ρ0 , u0 )T =

N 

cˆn ξn +

n=1

N 

dˆn ζn .

n=1

This can be rewritten in the form (5.9) with the following relation between the coefficients: λn μn ˆ (5.13) cn = cˆn + dˆn , dn = cˆn + dn . nQ0 nQ0 If the system starting from these initial conditions were null controllable, then there would exist a function q ∈ H 1 (0, T ) such that (5.11)–(5.12) holds true. For z in the complex plane, let us set  T F (z) = eizt q(t)dt. 0

Then using the Paley–Wiener theorem, F is an entire function of z. Since cn = dn = 0 for n > N , we have F (−λn i) = 0 = F (−μn i) ∀n > N. As −λn i −→ −ω0 i when n −→ ∞, −ω0 i is an accumulation point of zeros of the entire function F . Hence F ≡ 0. Then for n ≤ N also, F (−λn i) = 0 = F (−μn i), which from (5.11) and (5.12) implies that (cn bπn − πdn λn ) (cn bπn − πdn μn ) =0= . 2n 2n Hence cn = dn = 0 for each n ≥ 1. As λn and μn are different, we conclude from (5.13) that cˆn = dˆn = 0,

n ≥ 1.

This is a contradiction since the initial condition is nontrivial. Hence the theorem follows. Remark 5.7. The previous theorem rules out null controllability using regular boundary control q ∈ H 1 (0, T ). We can extend this negative result to less regular controls q ∈ L2 (0, T ). For that we need to interpret the solution of (5.4) by transposition. With Propositions 5.3 and 5.4, we can show that (5.4) has a unique solution (ρ, u)T ∈ L2 (0, T ; Z) in the sense of transposition when intial condition (ρ0 , u0 )T ∈ Z and boundary value q ∈ L2 (0, T ). Then as in Proposition 5.5, we can conclude that the system (5.4) is null controllable in Z in time T if and only if for each initial state (ρ0 , u0 )T ∈ Z there exists some control q ∈ L2 (0, T ) such that for any pair 1 1 (σT , vT )T ∈ Hm (Iπ ) × H01 (Iπ ), the solution (σ, v)T ∈ C([0, T ]; Hm (Iπ ) × H01 (Iπ )) of (5.5) satisfies  T (bσ(π, t) + ν0 vx (π, t)) q(t)dt Q0 0 (5.14)  π  π =b 0

ρ0 (x) σ(x, 0)dx + Q0

0

u0 (x) v(x, 0)dx.

2976

S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Thus using Remark 5.7 and the proof of Theorem 5.6, we will get the following generalization of Theorem 5.6. Theorem 5.8. For any T > 0, no nontrivial finite linear combination of eigenvectors can be driven to rest using a boundary control q ∈ L2 (0, T ) at π for the velocity. Thus the system (5.4) is not null controllable in Z in time T, using boundary control q ∈ L2 (0, T ), for the velocity component u at π. Remark 5.9. The above results established for system (5.4), stated in the interval (0, π), can also be proved in any arbitrary bounded open interval (L1 , L2 ), with −∞ < L1 < L2 < +∞, by a suitable analysis. 5.3. Localized interior control. We will use the negative boundary null controllability result of the previous section to rule out null controllability using a localized interior control. This will show that Theorem 5.1 is optimal in the sense that the system is not null controllable using any boundary control or an interior control acting on a subset of Iπ . Let us analyze the null controllability issue for the following system using localized interior control f ∈ L2 (0, ∞; L2 (O)) for the velocity component, where O is an open interval of Iπ : (5.15)

U (t) + AU(t) = F(t), U(0) = U0 ,

where F(t) = (0, f (·, t)χO )T , U0 = (ρ0 , u0 )T , and χO is the characteristic function of O. Theorem 5.10. For T > 0 and O an open interval of Iπ , there exists initial 1 condition U0 ∈ Hm (Iπ ) × H01 (Iπ ) such that for all f ∈ L2 (0, ∞; L2 (O)), the system (5.15) starting from U0 cannot be brought to rest in time T . Proof. We will prove this by contradiction. Let us assume that the system (5.15) is null controllable using localized interior control. Let us fix an initial condition 1 (Iπ ) × H01 (Iπ ). Using Proposition 3 of [2], the solution U = (ρ, u)T of (ρ0 , u0 )T ∈ Hm 1 system (5.15) is in H 1 (0, ∞; Hm (Iπ )) × [L2 (0, ∞; H 2 (Iπ )) ∩ H 1 (0, ∞; L2 (Iπ ))]. Hence the trace of u at x = L, for any L, 0 < L < π lies in L2 (0, ∞). Let us choose L ∈ (0, π) such that O ⊂ (L, π). By our assumption, the solution U vanishes in time T on Iπ . Hence if we take as the boundary control q(t) = u(L, t), the system (5.4) on (0, L) is null controllable in time T for this initial condition (ρ0 , u0 )T . By Remark 5.9, after a change of variable, this will contradict Theorem 5.8, since the initial condition is arbitrary. Therefore the theorem follows. 6. Approximate controllability. As the system is not null controllable, except in the particular case considered in Theorem 5.1, we explore here if it is at least approximately controllable using a boundary control q ∈ H 1 (0, T ) for the velocity component at π. Definition 6.1. The system (5.4) is approximately controllable in time T > 0 when for any U0 = (ρ0 , u0 )T ∈ Z0 and any other UT = (ρT , uT )T ∈ Z and any ε > 0, there exists a control q ∈ H 1 (0, T ) such that the solution to system (5.4) satisfies U(T ) − UT Z ≤ ε. The following theorem positively answers the question of approximate controllability. Theorem 6.2. The system (5.4) is approximately controllable in time T > 0 by a control q ∈ H 1 (0, T ).

CONTROL OF LINEARIZED NAVIER–STOKES SYSTEM

Proof. Setting

 LT (q) =

0

T

2977

S(T − s) B q(s) ds,

it is well known that the system (5.4) is approximately controllable when ImLT is dense in Z. (See [22, Part IV, Chapter 2, Theorem 2.5].) Then using the identity (5.7), it is enough to prove that if the solution to the 1 (Iπ ) × H01 (Iπ ), obeys adjoint system (5.5), with terminal condition (σT , vT ) ∈ Hm  T (6.1) (bσ(π, t) + ν0 vx (π, t)) q(t)dt = 0 0

for all q ∈ H 1 (0, T ), then (σT , vT ) = (0, 0). Let us decompose the solution (σ, v) to system (5.5) as follows:   ∗ ∗ cn,T e−λn (T −t) ξn (x) + dn,T e−μn (T −t) ζn (x), (σ, v)T (x, t) = n≥1

n≥1

∗ ∗ + n≥1 dn,T ζn . Hence, where cn,T and dn,T are defined by (σT , vT )T = n≥1 cn,T ξn it follows that  (−1)n (cn,T e−λn (T −t) + dn,T e−μn (T −t) ), σ(π, t) = n≥1

    −λn −λn (T −t) (−1) cn,T vx (π, t) = e Q0 n≥1      −μn + (−1)n dn,T e−μn (T −t) . Q0 

n

n≥1

Inserting these expressions in identity (6.1), we obtain  T   (−1)n cn,T (b Q0 − ν0 λn ) e−λn (T −t) 0

n≥1

+ dn,T (b Q0 − ν0 μn ) e

−μn (T −t)



 q(t)dt = 0

for every q ∈ H 1 (0, T ) and hence for every q ∈ L2 (0, T ) by density. It follows that    (6.2) (−1)n cn,T (bQ0 − ν0 λn ) e−λn (T −t) + dn,T (bQ0 − ν0 μn ) e−μn (T −t) = 0 n≥1

for 0 < t < T . Let us set pn = (−1)n {cn,T (bQ0 − ν0 λn )e−

λn T 2

} and qn = (−1)n {dn,T (bQ0 − ν0 μn ) e−

μn T 2

}.

Notice that the sequences {pn } and {qn } belong to 1 (C). Indeed, we have in view of (2.6) and (2.9), 1/2  1/2    1    − λn2 T  2 (cn,T n) < ∞, cn,T (bQ0 − ν0 λn ) e ≤c n2 n≥1 n≥1 n≥1 1/2  1/2       2 − μn2 T  2 −μn T |dn,T | μn e < ∞. dn,T (bQ0 − ν0 μn ) e ≤c n≥1

n≥1

n≥1

2978

S. CHOWDHURY, M. RAMASWAMY, AND J.-P. RAYMOND

Now, we consider the function ϕ(s) =



pn e−λn s + qn e−μn s .

n≥1

In view of (6.2), ϕ(s) = 0 for 0 < s < T2 . Since this function is analytic in the half complex plane  ∞ Re s > 0, we have ϕ(s) = 0 for Re s > 0. The Laplace transform of ϕ, Lϕ(z) = 0 e−sz ϕ(s)ds, is well defined for Re z > − min(ν0 /2, ω0 ), and by a direct calculation, we have  ∞   pn qn Lϕ(z) = + z + λn z + μn n=1 =

n 0 −1  n=1

qn pn + z + λn z + μn



+

∞  

n=n0

qn pn + z + λn z + μn



= 0.

For 1 ≤ m < n0 , −λm and −μm are isolated poles of Lϕ and Lϕ is identically zero in a neighborhood of each of those poles, thus pm = 0 and qm = 0 for 1 ≤ m < n0 . Now, applying Lemma 1 in [17] to the function  pn e−λn s + qn e−μn s = 0, n≥n0

for 0 < s < (6.3)

T 2,

we obtain cn,T (b Q0 − ν0 λn ) = 0 and dn,T (b Q0 − ν0 μn ) = 0

for all n ≥ n0 . The proof is complete. 7. Stabilizability. From the results of section 5, the systems (3.1) and (5.4) are not null controllable, except in the case of (3.1), with interior control everywhere in Iπ and only for more regular initial conditions. Thus it is natural to ask if we can at least stabilize the system for all initial values with a prescribed decay rate. This question has already been explored for the case of boundary control by Arfaoui et al. in [2]. They show that a similar system is stabilizable using boundary control with decay rate e−ωt for 0 < ω < ω0 , where ω0 = bνQ00 is the accumulation point for the real eigenvalues of A (see Theorem 4.1 in [2]). Here we explore further the stabilizability for the system (5.4) with decay rate e−ωt , with ω > ω0 , using boundary control for the velocity component. Since we are looking for decay rate e−ωt , it is convenient to consider the shifted system      d 0 Q0 dx ρ ρ ρ 0 + −ω = , ∂t d d2 u u u 0 b dx −ν0 dx 2 (7.1) ρ(x, 0) = ρ0 (x), u(x, 0) = u0 (x), x ∈ Iπ , u(0, t) = 0, u(π, t) = q(t),

t > 0.

Since (ρ, u, q) obeys system (7.1) if and only if (! ρ, u !, q!) = e−ωt (ρ, u, q) solves system (5.4), system (7.1) is stabilizable by a boundary feedback control qρ0 ,u0 ∈ L2 (0, ∞) if and only if system (5.4) is stabilizable with the exponential decay rate e−ωt by the corresponding feedback boundary control. Thus studying the stabilizability of system (5.4) with the exponential decay rate e−ωt is equivalent to studying the stabilizability of system (7.1). Throughout this section, ω > ω0 is in the resolvent set of A and ω is fixed.

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7.1. The shifted system. Here we establish the existence and regularity of a solution for (7.1) and then rewrite the system as an operator equation. This system is the same as the one considered in (5.4), except for the new term ω (ρ, u)T appearing in (7.1). Then as in Proposition 5.2, we have the following existence result. 1 1 Proposition 7.1. For any T > 0, U0 = (ρ0 , u0 )T ∈ Hm (Iπ ) × H{0} (Iπ ) 1 and q ∈ H (0, ∞) satisfying the compatibility condition q(0) = u0 (π), there exists a unique solution UU0 ,q = (ρ, u)T of (7.1), with ρ ∈ H 1 (0, T ; H 1 (Iπ )) and 1 u ∈ H 1 (0, T ; L2(Iπ )) ∩ L2 (0, T ; H 2(Iπ ) ∩ H{0} (Iπ )), satisfying (7.2)

ρH 1 (0,T ;H 1 ) + uH 1 (0,T ;L2 ) + uL2 (0,T ;H 2 ) ≤ C(T )(U0 Hm + qH 1 (0,T ) ). 1 ×H 1 {0}

Proof. The proof is easy and is left to the reader. Now we show the existence of a solution with less regular initial conditions. Proposition 7.2. For U0 = (ρ0 , u0 )T ∈ Z0 and q ∈ H 1 (0, ∞), there exists a unique solution (ρ, u) of (7.1) in C([0, ∞); Z). Proof. Let us set (7.3)

u ˜0 (x) =

q(0) x, ∀ x ∈ [0, π]. π

So u ˜0 (π) = q(0). Then with Proposition 7.1 we get U(0,˜u0 ),q ∈ C([0, ∞); Z). With Lemma 2.1, we already have U(ρ0 ,u0 −˜u0 ),0 ∈ C([0, ∞); Z). Therefore using linearity we conclude that U(0,˜u0 ),q + U(ρ0 ,u0 −˜u0 ),0 , the solution of (7.1), belongs to C([0, ∞); Z). In order to find the projections of (7.1) on the finite dimensional subspaces, we need to write this evolution equation with inhomogeneous boundary condition as an inhomogeneous operator equation using the usual lifting procedure and the Dirichlet operator corresponding to A. For that, for each t > 0, we consider the stationary problem with inhomogeneous boundary condition for the second component at π    d −ω Q0 dx w1 0 = , d d2 w2 0 (7.4) b dx −(ν0 dx 2 + ω) w2 (0, t) = 0,

w2 (π, t) = q(t).

Proposition 7.3. Assume that q ∈ H 1 (0, ∞). Then, for all t > 0, (7.4) admits 1 a unique solution W(t) = (w1 (t), w2 (t))T belonging to H 1 (Iπ ) × H 2 (Iπ ) ∩ H{0} (Iπ ). Moreover, as a function of t ∈ (0, ∞), W satisfies the following estimates: (7.5)

w1 H 1 (0,∞;H 1 ) + w2 H 1 (0,∞;H 2 ) ≤ CqH 1 (0,∞)

and (7.6)

w1 L2 (0,∞;H 1 ) + w2 L2 (0,∞;H 2 ) ≤ CqL2 (0,∞) .

Proof. The proof is standard using the fact that ω belongs to the resolvent set of A. Also it relies on the estimate w1 (t)H 1 (Iπ ) + w2 (t)H 2 (Iπ ) ≤ C|q(t)| for the solution W(t) = (w1 (t), w2 (t))T to (7.4).

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Let H denote the Banach space D(A∗ ) endowed with the graph norm of A∗ . Let H denote the dual of H with Z as pivot space. Set Aω := A − ωI. Observe that the unbounded operator A∗ω can be viewed as an isomorphism from H to Z since ω (> ω0 ) belongs to the resolvent set of A. Thus A˜ω = (A∗ω )∗ , the adjoint of A∗ω ∈ isom(H, Z), belongs to isom(Z, H ). But it can also be viewed as an unbounded operator in (D(A∗ )) with domain Z, and therefore it is also an extension to (D(A∗ )) of the unbounded operator Aω . We equip H with the inner product

f, gH = (A˜ω )−1 f, (A˜ω )−1 gZ .

(7.7)

Now we write system (7.1) as an evolution equation. For that, we define the Dirichlet operator D : R → Z,

(7.8)

Dq(t) = W(t),

where W(t) is the solution of (7.4). We define the control operator B ∈ L(R, H ) by B := A˜ω D, where D ∈ L(R, Z), as defined in (7.8). Theorem 7.4. Assume that q ∈ H 1 (0, ∞) and (ρ0 , u0 )T ∈ Z. Then U ∈ C([0, ∞); Z) is the unique solution of system (7.1) if and only if it is the weak solution to the evolution equation U (t) + A˜ω U(t) = Bq(t),

(7.9)

U(0) = U0 = (ρ0 , u0 )T .

If q ∈ L2 (0, ∞) and (ρ0 , u0 )T ∈ H , then (7.9) has a unique solution U in C([0, ∞); H ). Proof. Let U be the solution of (7.1) and W(t) be the solution of (7.4). Let us set X = U − W. Then, due to Propositions 7.2 and 7.3, X ∈ C([0, ∞); Z). Considering A˜ω as the extention of Aω to H , we notice that W(t) belongs to the domain of A˜ω and we have X + A˜ω X = −(W + A˜ω W) = −W , X(0) = U0 − W(0), X2 (0, t) = 0 = X2 (π, t), where X2 stands for the second component of X. As A˜ω is also the infinitesimal generator of a semigroup on H , by Duhamel’s formula we have  t ˜ ˜ e−(t−s)Aω W ds. (7.10) X(t) = e−tAω (U0 − W(0)) − 0

Using integration by parts and the equation for W, we obtain  t  t ˜ω ˜ ˜ −(t−s)A  e W ds = e−(t−s)Aω A˜ω W(s)ds − W(t) + e−tAω W(0). − 0

0

Using this in (7.10) yields (7.11)

˜

U(t) = e−tAω U0 +

This identity is equivalent to (7.9).

 0

t

˜

e−(t−s)Aω A˜ω W(s)ds.

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1 1 For q ∈ H 1 (0, ∞) and (ρ0 , u0 )T ∈ Hm (Iπ ) × H{0} (Iπ ) with q(0) = u0 (π), the solution given by formula (7.11) coincides with the solution given by Proposition 7.1. Hence, by density arguments it follows that the solution defined by (7.11) and the one given by Proposition 7.2 also coincide when U0 ∈ Z0 and q ∈ H 1 (0, ∞). So for q ∈ H 1 (0, ∞) and (ρ0 , u0 )T ∈ Z0 , (7.9) has a solution in C([0, T ]; Z) and

U (t) = −A˜ω U(t) + Bq(t) ∈ L2 (0, T ; H ) as A˜ω ∈ L(Z, H ), B ∈ L(R, H ). For q ∈ L2 (0, ∞), Bq ∈ L2 (0, T ; H ) for each T, 0 < T < ∞. Therefore using semigroup theory we conclude that for initial conditions (ρ0 , u0 )T ∈ H , (7.9) has a unique solution U in C([0, T ]; H ) for each T, 0 < T < ∞. 7.2. Estimates for stabilizing control. In this section we will show that if the system (7.9) is stabilizable by some control q ∈ L2 (0, ∞), then we can find a stabilizing control which depends continuously on the initial data of this system. First we recall the standard definition of stabilizability from [4, Part V, Chapter 1, section 2] when the control q ∈ L2 (0, ∞). Definition 7.5. (−A˜ω , B) is said to be stabilizable in H , by controls in 2 L (0, ∞), if for any U0 ∈ H there exists a control q ∈ L2 (0, ∞) such that the solution UU0 ,q of (7.9) satisfies  ∞ ||UU0 ,q (t)||2H dt < ∞. (7.12) 0

The continuous dependence of the stabilizing control on the initial data is proved in the following theorem. Theorem 7.6. Let ω > ω0 belong to the resolvent set of A. Assume that (−A˜ω , B) is stabilizable in H by a L2 -control. Then the Riccati equation (7.13)

X ∈ L(H , H), X = X ∗ ≥ 0, −A˜∗ω X − X A˜ω − XBB ∗ X + ((−A˜ω )∗ )−1 (−A˜ω )−1 = 0

∞ admits a unique solution that we denote by Xmin . The equation

(7.14)

∞ )U(t) in (0, ∞), U(0) = U0 , U (t) = (−A˜ω − BB ∗ Xmin

admits a unique solution in C([0, ∞); H ) and this solution satisfies (7.15)

||U(t)||H ≤ C1 e−θt ||U0 ||H ,

for some positive constants C1 and θ. ∞ U(t) satisfies −B ∗ Xmin (7.16)

∀ U0 ∈ H

Moreover, the stabilizing control f˜(t) =

||f˜||L2 (0,∞) ≤ C||U0 ||H .

˜ = (−A˜ω )−1 B ∈ L(R, Z). We first show that (−A˜ω , B) Proof. Step 1. We set B  ˜ is stabilizable is stabilizable in H by a control q in L2 (0, ∞) if and only if (−A˜ω , B) in Z by the same control q. Set V(t) = (−A˜ω )−1 U(t), where U ∈ C([0, ∞); H ) is the unique solution of (7.9) for U0 ∈ H and q ∈ L2 (0, ∞). Then V(t) ∈ C([0, ∞); Z) is the weak solution of (7.17)

˜ V (t) = −A˜ω V(t) + Bq(t),

V(0) = V0 = (−A˜ω )−1 U0 ∈ Z.

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Since A˜ω is an isomorphism from Z to H , we have C1 || − A˜ω V(t)||H ≤ ||V(t)||Z ≤ C2 || − A˜ω V(t)||H , where C1 , C2 are positive constants. So,  ∞  ∞  C1 ||U(t)||2H dt ≤ ||V(t)||2Z dt ≤ C2 0

0

∞

0

||U(t)||2H dt.

˜ Hence the equivalence between the stabilizability of (−A˜ω , B) in H and of (−A˜ω , B) in Z is proved. Step 2. We can easily verify that X ∈ L(H , H) is a solution to (7.13) if and only if P = (−A˜ω )∗ X(−A˜ω ) is a solution to the following equation: (7.18)

P ∈ L(Z), P = P ∗ ≥ 0,

˜B ˜ ∗ P + I = 0. −A˜∗ω P − P A˜ω − P B

˜ is stabilizable in Z. From [4, Part V, Due to Step 1, we know that (−A˜ω , B) ∞ Chapter 1, section 3], it follows that (7.18) admits a unique solution Pmin . Thus, ∞ ∗ −1 ∞ −1  Xmin = ((−A˜ω ) ) Pmin (−A˜ω ) ∈ L(H , H) is the unique solution of (7.13). More˜ ˜ ˜∗ ∞ over the semigroup (et(−Aω −BB Pmin ) )t≥0 is exponentially stable on Z and the semi∗ ∞ ˜ group (et(−Aω −BB Xmin ) )t≥0 is exponentially stable on H . Thus (7.15) is proved, and (7.16) follows from (7.15). 7.3. Projection on the eigenspace. Here we compute first the projection of (7.9) on the eigenspaces corresponding to the real eigenvalues of A˜ω . The eigenvalues of A˜ω are λn − ω and μn − ω. The corresponding eigenfunctions are respectively the functions ξn and ζ n defined in (2.7). The adjoint operator A˜∗ω has the same eigenvalues and its eigenvectors, after normalization, are respectively T T   −λn −μn ∗ ∗ ˜ sin(nx) sin(nx) , ζn (x) = θn cos(nx), , ξn (x) = θn cos(nx), Q0 n Q0 n where the normalizing constants θn and θ˜n are chosen suitably later on. As ω > ω0 and the real eigenvalues {λn } of A accumulate at ω0 , there are infinitely many λn which are less than ω. Our aim is to calculate the projection of (7.9) on the eigenspaces corresponding to such λn ’s. Let us set En = span{ξn }. Lemma 7.7. The projection Qn from H into En , for n ≥ n0 , is defined by Qn (ζ) = ζ, ξ ∗n H ξ n ,

ζ ∈ H .

Proof. As Z0 is the orthogonal sum of the spaces Vn , for n ≥ 1, we note that the eigenfunctions {ξn }, {ζ n } corresponding to the eigenvalues of A˜ω form a complete orthogonal system in H and similarly the eigenfunctions {ξ∗n }, {ζ ∗n } for A˜∗ω . We normalize these eigenfunctions in such a way that

ξ ∗n , ξ m H = δm,n and ζ ∗n , ζ m H = δm,n so that (7.19)

θn =

2(λn − ω)[Q0 ν0 ω 2 (ω0 − ω)n4 + Q0 ω 4 n2 ] π[bQ0 ν0 ω 2 n4 + bQ0 ω 2 (2λn − ω)n2 + ω 3 λ2n ]

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in view of the inner product in H . Thus these two sequences of eigenfunctions together form a biorthogonal system in H . Hence we can take the projection from H onto En as Qn . We will need the following lemma regarding the action of B ∗ . Lemma 7.8. Let B ∈ L(R, H ) be the control operator defined as B = A˜ω D with the Dirichlet operator D ∈ L(R, Z) defined in (7.8). Then for the adjoint operator B ∗ ∈ L(H, R) and ξ ∗n , the eigenfunction of A˜∗ω corresponding to (λn − ω) for n ≥ n0 , we have B ∗ ξ ∗n = (−1)n+1 θn {bQ0 − ν0 λn }. Proof. We have B ∗ = D∗ A˜∗ω . Here D∗ : Z −→ R is the adjoint of D given by ˜ R. ˜ z = q, D∗ V

Dq, V ˜ for V ˜ ∈ Z of the form Let us calculate D∗ V,

˜ = Then V

(7.20)

v˜1  v ˜2

˜ = A∗ω V V

with

V ∈ H 1 (Iπ ) × (H 2 (Iπ ) ∩ H01 (Iπ )).

is given by 

−ω

d −Q0 dx

d −b dx

d −(ν0 dx 2 + ω)



2

v1



v2

 =

v˜1 v˜2

,

v2 (0) = 0, v2 (π) = 0. If we take the inner product of (7.20) with W(t) and integrate by parts we get   ˜ z = − q(t), b Q0 v1 (π) + ν dv2 (π) .

W(t), V dx R ˜ z = Dq(t), V ˜ z , it follows that Since W(t), V   ˜ R = −q(t) b Q0 v1 (π) + ν dv2 (π) .

q(t), D∗ V dx Using these we calculate for ξ ∗n = (ξn,1 , ξn,2 )T 

q(t), D∗ A∗ω ξ ∗n R = −q(t)(ν0 Q0 ξn,2 (π) + b Q0 ξn,1 (π)) = −q(t) θn cos(nπ) (b Q0 − ν0 λn ).

These now lead to the projected equation. Proposition 7.9. The projection of system (7.9) on the eigenspace En for n ≥ n0 is given by the scalar equation (7.21) U (t), ξ ∗n H + (λn − ω) U(t), ξ ∗n H = bn q(t),

U(0), ξ ∗n H = U0 , ξ ∗n H ,

where (7.22)

bn := B ∗ ξ ∗n = (−1)n+1 θn (bQ0 − ν0 λn ).

Proof. Let us define the projection of U(t) on En as Un (t) := Qn U(t) = U(t), ξ ∗n H ξ n .

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Observe that Qn (A˜ω U(t)) = (λn − ω) U(t), ξ ∗n H ξ n . Now the projection of the inhomogeneous term is, by the definition of Qn , Qn (Bq(t)) = Bq(t), ξ ∗n H ξ n = q(t), B ∗ ξ∗n R ξn = bn q(t)ξ n . Thus the projected system in the one-dimensional space En is (7.23)

Un (t) + (λn − ω)Un (t) = bn q(t)ξ n ,

Un (0) = Qn (U0 ),

After dropping ξn , we get the scalar equation (7.21). 7.4. Negative results for stabilizability. Here we first get the expression for the minimum norm control qn that stabilizes the one-dimensional projected system. Lemma 7.10. If the system (7.9) is stabilizable in H by boundary control q ∈ 2 L (0, ∞), then the projected scalar equation (7.21) is also stabilizable in En and the minimum norm control, for n ≥ n0 , is given by   ω − λn qn (t) = −2e−t(ω−λn)

U0 , ξ∗n H . bn Proof. We apply to our scalar equation (7.21) the results for infinite dimensional system from [12] (see Proposition 2, section 5), where the expression for minimum norm control is obtained using variational arguments. Denoting  ∞ b2n , e−t(ω−λn ) b2n e−t(ω−λn ) dt = W∞ := 2(ω − λn ) 0 we can write the expression for the minimum norm control that stabilizes the scalar equation (7.21) as (7.24)

2(ω − λn ) t((ω−λn )−2(ω−λn )) e

U0 , ξ∗n H b2n   ω − λn = −2e−t(ω−λn )

U0 , ξ ∗n H . bn

q¯n (t) = −bn

Using (7.24), we now answer in the negative the question of stabilizability. Theorem 7.11. The system (5.4) is not boundary stabilizable in H with an exponential decay rate ω, for ω > ω0 and ω in the resolvent set of A, by a boundary control q ∈ L2 (0, ∞) at x = π for the velocity component u. Proof. Let us assume that the system is stabilizable using boundary control f . Then, by Theorem 7.6, there exists a positive constant K1 such that for any initial condition U0 ∈ H , we have (7.25)

f L2(0,∞) ≤ K1 U0 H .

Then, for the initial condition Xn = ξξn Z ∈ H with n ≥ n0 , there exists a control fn n satisfying (7.25), which stabilizes the system (7.9). Since Z is continuously embedded in H , we have fn L2 (0,∞) ≤ K2 ,

∀ n ≥ n0 .

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On the other hand, if we consider the projected system (7.21) on En with the initial condition

Xn , ξ ∗n H =

1 , ξn Z

then, from the above calculations, the minimal norm control is qn (t) = −2e−t(ω−λn ) Thus the norm of qn is qn L2 (0,∞)

ω − λn . bn ξ n Z

 2(ω − λn ) = . |bn |ξ n Z

As fn also stabilizes the same equation, we have (7.26)

fn L2 (0,∞) ≥ qn L2 (0,∞) .

With (2.8) and (7.19), we obtain   ω02 1 , bQ0 − ν0 λn = − 2 + o n n2     1 2 (ω0 ω + 2λn ω − 2λn ω0 ) (λn − ω) (ω0 − ω) + θn = + o , 2 bπ ν0 n n2    λ2n π ξn Z = . b+ 2 2 n Q0 Using the expression for bn from (7.22) and the above estimates, it follows that   1 2ω02 (ω − ω0 )(ω − λn ) +o |bn | = . 2 bπn n2 Thus

 |bn |ξn z = O

1 n2

 −→ 0 as n −→ ∞.

Hence, the sequence {qn L2 (0,∞) }n is unbounded but {fn L2 (0,∞) }n is bounded. This contradicts (7.26). Thus the system is not boundary stabilizable for arbitrary U0 ∈ H . Let us end the paper by completing a result stated in Theorem 4.1 in [2]. It is 1 1 (Iπ )×H{0} (Iπ ) with an exponential shown there that system (5.4) is stabilizable in Hm decay rate −ω with 0 < ω < ω0 , by boundary controls in H 1 (0, ∞) satisfying (7.27)

1 ×H 1 , qU0 H 1 (0,∞) ≤ K3 U0 Hm

1 1 where qU0 is the control stabilizing the initial condition U0 ∈ Hm (Iπ ) × H{0} (Iπ ). Here, using the same arguments as in the proof of Theorem 7.11, we prove that the 1 1 (Iπ )× H{0} (Iπ ) with an exponential decay rate −ω when system is not stabilizable Hm ω > ω0 .

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1 Corollary 7.12. The system (5.4) with initial condition in U0 ∈ Hm (Iπ ) × 1 1 is not boundary stabilizable in Hm (Iπ ) × H{0} (Iπ ) with the exponential decay rate −ω, for ω > ω0 and ω belonging to the resolvent set of A, by a boundary control qU0 ∈ H 1 (0, ∞) satisfying (7.27) and the compatibility condition qU0 (0) = u0 (π). Proof. Let us argue by contradiction. We choose the initial condition Xn = ξn 1 ∈ Hm (Iπ ) × H 1 (Iπ ) for n ∈ N∗ , where ξ 1 1 1 H{0} (Iπ )

n Hm ×H

1 ×H 1 = ξ Z + ∇ξ Z . ξn Hm n n

Notice that  ∇ξ n (x) =

−n sin(nx) λn Q0 cos(nx)



 ,

∇ξn Z = n

π 2

  λ2n . b+ 2 n Q0

If (7.27) holds true, we have 1 ×H 1 = K3 , qn L2 (0,∞) ≤ qXn L2 (0,∞) ≤ qXn H 1 (0,∞) ≤ K3 Xn Hm

where qn is the L2 -control of minimal norm stabilizing Xn . Now we follow the same 1 1 ×H 1 is now of order O( ) and is still going arguments as before. Since |bn |ξn Hm n to zero, the control qn of minimum norm stabilizing Xn is such that qn L2 (0,∞) is unbounded. Thus, we obtain a contradiction and the proof is complete. REFERENCES [1] E. V. Amosova, Exact local controllability for equations of viscous gas dynamics, Differential Equations, 47 (2011), pp. 1776–1795. [2] H. Arfaoui, F. Ben Belgacem, H. El Fekih, and J.-P. Raymond, Boundary stabilizability of the linearized viscous Saint-Venant system, Discrete Cont. Dyn. Syst. Ser. B, 15 (2011), pp. 491–511. [3] V. Barbu, I. Lasiecka, and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 852, 2006. [4] A. Bensoussan, G. Da Prato, M. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd ed., in Systems Control Found. Appl., Birkh¨ auser Boston, Boston, MA, 2007. ´ ndez-Cara, and M. Gonza ´ lez-Burgos, Some control[5] J. L. Boldrini, A. Doubova, E. Ferna lability results for linear viscoelastic fluids, SIAM J. Control Optim., 50 (2012), pp. 900– 924. ´ ndez-Cara, and M. Gonza ´ lez-Burgos, Controllability results for [6] A. Doubova, E. Ferna linear viscoelastic fluids of the Maxwell and Jeffreys kinds, C. R. Acad. Sci. Paris Ser. I Math., 331 (2000), pp. 537–542. [7] S. Ervedoza, O. Glass, S. Guerrero, and J.-P. Puel, Local exact controllability for the onedimensional compressible Navier-Stokes equation, Arch. Ration. Mech. Anal., 206 (2012), pp. 189–238. ´ ndez-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel, Local exact control[8] E. Ferna lability of the Navier-Stokes system, J. Math. Pures Appl., 83 (2004), pp. 1501–1542. [9] A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete Cont. Dyn. Syst., 10 (2004), pp. 289–314. [10] V. Girinon, Quelques Problemes aux Limites pour les Equations de Navier-Stokes Compressibles, Ph.D. thesis, Universit´ e de Toulouse, 2008. [11] K. Ito and S. S. Ravindran, Optimal control of compressible Navier-Stokes equations, in Proceedings of the First International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, FL, 1996, pp. 289–294. [12] S. Kesavan and J.-P. Raymond, On the degenerate operator Riccati equations, Control Cybernet., 38 (2009), pp. 1393–1427. [13] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Volume 2, Compressible Models, Oxford Lecture Ser. Math. Appl. 10, Oxford University Press, New York, 1998.

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