Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid–structure interaction

Systems & Control Letters 58 (2009) 499–509

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Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid–structure interaction Irena Lasiecka a,∗ , Amjad Tuffaha b a

Department of Mathematics, University of Virginia, Charlottesville, VA, United States

b

Department of Mathematics, University of Southern California, Los Angeles, CA, United States

article

info

Article history: Received 28 July 2008 Received in revised form 15 February 2009 Accepted 16 February 2009 Available online 29 March 2009 Keywords: Fluid–structure interaction Boundary control Singular estimate control system Riccati equation Feedback control Hyperbolic trace theory Maximal parabolic regularity

a b s t r a c t We consider a Bolza boundary control problem involving a fluid–structure interaction model. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The model considered consists of a linearized Navier–Stokes equation coupled on the interface with a dynamic wave equation. The model incorporates convective terms resulting from the linearization of the Navier–Stokes equation around equilibrium. The existence of the optimal control and its feedback characterization via a solution to a Riccati equation is established. The main mathematical difficulty of the problem is caused by unbounded action of control forces which, in turn, result in Riccati equations with unbounded coefficients and in singular behavior of the gain operator. This class of problems has been recently studied via the so-called Singular Estimate Control Systems (SECS) theory, which is based on the validity of the Singular Estimate (SE) [G. Avalos, Differential Riccati equations for the active control of a problem in structural acoustic, J. Optim. Theory, Appl. 91 (1996) 695–728; I. Lasiecka, Mathematical Control Theory of Coupled PDE’s, in: NSF- CMBS Lecture Notes, SIAM, 2002. with Unbounded Controls; I. Lasiecka, A. Tuffaha, Riccati Equations for the Bolza Problem arising in boundary/point control problems governed by c0 semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008) 229–246]. It is shown that the fluid–structure interaction does satisfy the Singular Estimate (SE) condition. This is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface. Thus, the established Singular Estimate allows for the application of recently developed general theory which, in turn, implies well-posedness of the feedback synthesis and of the associated Riccati Equation. Moreover, the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The physical model under consideration has been extensively treated in the literature (cf. [1–7]) and describes the elastic motion of a solid fully immersed in a viscous incompressible fluid. The mathematical model consists of a linearized Navier–Stokes equation defined on an open domain Ωf coupled with an elastic equation defined on another domain Ωs , with boundary conditions matching velocities and normal stresses on the boundary Γs which separates the two open domains Ωf and Ωs . It will be assumed that the solid is subject to small but rapid oscillations (cf. [1]). We consider a boundary control system of this fluid–structure interaction model, with the objective of developing a feedback

∗ Corresponding address: Department of Mathematics, University of Virginia, Kerchof Hall, P O Box 400136, 22903 Charlottesville, VA, United States. Tel.: +1 804 924 8896; fax: +1 804 982 3084. E-mail address: [email protected] (I. Lasiecka). 0167-6911/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2009.02.010

optimal control, acting as a force on the interface between the two media. The construction is based on a solution to an appropriate Riccati equation. It is known that Riccati theory is a very powerful tool for designing and computing feedback controls for finite-dimensional systems. An extension of the Riccati theory has become available for infinite-dimensional systems modeled by PDEs (cf. [8–13] and the references therein). Though actual computations of feedback controllers are performed on finite-dimensional structures, the role of infinite-dimensional Riccati theory need not be defended. In fact, as documented by a large body of literature, rigorous infinitedimensional theory is responsible for stability and consistency of the estimates obtained in finite-dimensional approximations of Riccati equations (cf. [14–20]). The infinite-dimensional Riccati theory was first obtained for control systems generated by strongly continuous semigroups but involving bounded control actions. However, the theory of optimal control attains a higher degree of complexity when treating systems with unbounded control actions which arise in boundary or point control. The analysis of

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I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

such systems goes back to [21,22,8] where the term ‘‘admissible control’’ was used to characterize certain classes of unbounded control actions. More recently, and in the context of Riccati equations, the term ‘‘unbounded coefficients’’ has been used in reference to unbounded control operators (cf. [23,24,46]). These classes of systems have come to attention recently in light of many technological advances in control engineering such as smart materials technologies, the results of which were new types of controls that can only be mathematically captured by unbounded control operators. The mathematical difficulty in extending the Riccati theory available for bounded controls, has to do with the fact that the so-called gain (feedback) operator along with the coefficients in the associated Riccati equation to the system might not be well defined. In fact, a first example of point control system where the gain operator computed from Riccati equation is inconsistent was shown in [25]. Later, [26] shows more general classes of examples exhibiting the same pathology. Therefore, a standard like treatment of an unbounded control system via a Riccati equation and their finite-dimensional approximations ought to be justified with a sound theoretical framework. The framework has indeed been laid out in the case of systems generated by analytic semigroups (cf. [9,23,24,11,27, 13]), where the theory has acquired a reasonable level of maturity and completeness. More recently the analytic approach has been extended to a class of systems referred to as Singular Estimate Control Systems (SECS) which capture partial or approximate analytic dynamics (cf. [28–34]). Coupled PDE systems which do not have analytic generators, but rather combine hyperbolic and parabolic effects are classical prototypes for the SECS systems. The first example of such coupled dynamics was given in [28] where structural acoustic interaction was considered in the framework of SECS systems. Other examples arising in structural or thermal interactions are treated in [31,35,30]. A rather rich theory pertaining to SECS systems has been developed over the last decade or so. Of particular interest to this work is the Bolza Riccati SECS theory which involves penalization of the terminal state (cf. [34,48]). It is known that Bolza problems, even in the case of analytic dynamics, do lead to singular behavior of optimal controls and of the Riccati feedback operators. The general formulation of the SECS (Singular Estimate Control Systems) class is as follows: Consider the dynamics yt = Ay + B g ∈ [D (A? )]0

(1)

with a state space H and a control space U while A is a generator of a strongly continuous semigroup eAt on H , and B is an unbounded control operator such that B ∈ L(U → [D(A∗ )]0 ). The additional singular estimate (SE) condition is

|O eAt B g |Z ≤

C

|g |U , γ

(2)

t where 0 ≤ γ < 1 and where O denotes selected observations of the system defined via bounded operators from the state space H into the observed space Z. The control problem considered is to minimize any functional of the general form J (y, g ) =

Z

2. The control model Let Ω ∈ R3 be a bounded domain with an interior region Ωs and an exterior region Ωf . The boundary Γf is the outer boundary of the domain Ω while Γs is the boundary of the region Ωs which also borders the exterior region Ωf and where the interaction of the two systems takes place. Let u be a function defined on Ωf representing the velocity of the fluid while the scalar function p represents the pressure. Additionally, let w and wt be the displacement and the velocity functions of the solid Ωs . We also denote by ν the unit outward normal vector with respect to the domain Ωs . The boundary-interface control is represented by g ∈ L2 ([0, T ]; L2 (Γs )) and is active on the boundary Γs . We work under the assumption of small but rapid oscillations of the solid body, hence the interface Γs is assumed static (cf. [1,3,6] for more modeling details). Given control g ∈ L2 ([0, T ]; L2 (Γs )), the functions (u, w, wt , p) satisfy the system

 ut − div (u) + Lu + ∇ p = 0    div u = 0     w − div σ (w) = 0   tt u(0, ·) = u0 wt (0, ·) = w1 w(0, ·) = w0 ,    wt = u     u = 0 σ (w) · ν = (u) · ν − pν − g

Qf ≡ Ωf × [0, T ] Qf ≡ Ωf × [0, T ] Qs ≡ Ωs × [0, T ]

Ωf Ωs Σs ≡ Γ s × [ 0 , T ] Σf ≡ Γf × [0, T ] Σs ≡ Γs × [0, T ].

(3)

0

over a set of boundary controls g ∈ L2 ([0, T ]; U ) where G ∈ L(H , W ) is a bounded operator on suitable Hilbert spaces. In the context of the control problem (3), the relevant observation operator in (2) is O = G. Remark 1.1. Note that the singular estimate (2) is automatically satisfied for analytic semigroups and unbounded control operators B which are relatively bounded with respect to the generator A from U → [D ((A? )γ )]0 . Thus, SECS systems provide a proper generalization of control systems governed by analytic semigroups.

(4)

The elastic stress tensor σ and the strain tensor  , respectively, are given by

σij (u) = λ

k=3 X

kk (u)δij + 2µij (u) and

k=1

ij (u) =

1



2

∂ ui ∂ uj + ∂ xj ∂ xi



.

where λ > 0 and µ > 0 are the Lamé constants and ∆u = div (u) which can be easily verified using the divergence free property. The term Lu is a linearization of the convective term in Navier–Stokes (u · ∇)u and is defined as Lu = (∇v) · u + (v · ∇)u

T


|g |2U ds + |Gy(T )|2W

Our aim in this work is to show that the control system under consideration falls into the class of singular estimate control systems for which a satisfactory Riccati theory is available. In particular, we will show that the system satisfies the singular estimate condition (2) with γ = 1/4 +  and the conditions laid out in [34] allow for an application of the results on existence and regularity of the optimal control and most importantly a feedback characterization of the control via solutions to a Riccati equation. The optimal control along with the feedback gain operator while well defined and bounded for transient times, become singular at the terminal time. This singularity is quantified by the algebraic blow up rate of the order (T − t )−1/4− .

(5)

where v is a time independent smooth vector function ∈ [C ∞ (Ωf )]n with the property div v = 0. The model considered is in fact a generalization of the model in [48]. The control problem to be considered is of the Bolza type. In particular, we wish to minimize the functional J ( u, g ) =

T

Z 0


|g (t )|2L2 (Γs ) ds + |u(T , ·)|2L2 (Ωf ) ,

(6)

over all g ∈ L2 ([0, T ]; Γs ). Throughout the paper, we denote the energy space for the system by

H ≡ H × H 1 (Ωs ) × L2 (Ωs )

I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

where H ≡ {u ∈ L2 (Ωf ) : div u = 0, u · ν|Γf = 0}. With the above notation, the observation operators introduced before become R ≡ (R, 0, 0) and G(u, w, v) ≡ (I , 0, 0), hence O = G. Notation. Note that all Sobolev spaces H s and L2 spaces pertaining to u and w are in fact (H s )n , (L2 )n , n = 2, 3 but we omit the exponent n for the sake of simplicity. In addition, we define V ≡ {v ∈ H 1 (Ωf ) : div v = 0, u|Γf = 0} and we shall use the following notation for the L2 inner product on the boundary and the interior of the domain

(u, v) =

Z Ω

u · v dx,

hu, vi =

Z Γs

u · v dx.

The space V is topologized with respect to the inner product given by

(u, v)1,f ≡

Z Ωf

(u) · (v)dx.

We also denote the induced norm by | · |1,Ωf which is equivalent to the usual H 1 (Ωf ) norm via Korn’s inequality and Poincaré’s inequality

!1/2

Z |u|1,Ωf =

Ωf

|(u)|2 dx

.

The space H 1 (Ωs ) is topologized with respect to the inner product given by

(w, z )1,s ≡

Z Ωs

w · zdx +

Z Ωs

σ (w) · (z )dx.

We denote by |·|1,Ωs , the norm induced by the inner product above

|w|21,Ωs =

Z Ωs

(c) The Singular Estimate Control condition: There exists γ < 1 and a constant C > 0 such that C |ReAt B u|Z ≤ γ |u|U t and C |GeAt B u|W ≤ γ |u|U t for all 0 < t ≤ 1. (d) R ∈ L(Y , Z ) and the operator G ∈ L(Y , W ) is such that the composition operator GLT : L2 ([0, T ]; U ) → W is closeable where LT is the control to state map at time T . Then for any initial state ys ∈ Y there exists a unique optimal control g 0 (t , s, ys ) ∈ L2 ([s, T ]; U ) and optimal trajectory y0 (t , s, ys ) ∈ C ([0, T ], [D(A∗ )]0 ) with Ry0 (t , s, ys ) ∈ L2 ([0, T ], Z ) such that J (g 0 , y0 , s, ys ) = minu∈L2 ([s,T ],U ) J (g , y(g ), s, ys ). Moreover, there exists a self-adjoint positive operator P (t ) ∈ L(Y ), where t ∈ [0, T ) such that (P (t )x, x)Y = J (g 0 , y0 , t , x). In addition, the following properties hold: (i) The optimal control g 0 (t ) is continuous on [s, T ) but has a singularity of order gamma at the terminal time. In particular, we have

|g 0 (t , s, ys )|U ≤

This is again equivalent to the usual H 1 (Ωs ) norm by Korn’s inequality.

(T − t )γ

|ys |Y ,

s ≤ t < T.

(9)

C

|ys |Y , s ≤ t < T . (10) (T − t )2γ −1+ (iii) P (t ) is continuous on [0, T ] and P (t ) ∈ L(Y , L∞ ([0, T ]; Y )). (iv) B ∗ P (t ) exhibits the following singularity |Ry0 (t , s, ys )|Z ≤

C |x|Y , (T − t )γ

0 ≤ t < T.

(11)

(v) The optimal control satisfies the estimate g 0 (t , s, ys ) = −B ∗ P (t )y0 (t , s, ys ),

s ≤ t < T.

(12)

(vi) P (t ) satisfies the Riccati Differential equation for t < T and x, y ∈ D (A)

3. Main results We first recall an abstract result from [34,47], which provides the Riccati theory pertinent to SECS control systems.

hPt x, yiY + hA? P (t )x, yiY + hP (t )Ax, yiY = hB ∗ P (t )x, B ∗ P (t )yiU lim P (t )x = G Gx ∗

Theorem 3.1. Let U, Y , Z and W be given Hilbert spaces. Spaces U and Y denote, respectively, control and state spaces while Z and W are observation spaces. We consider the dynamics governed by the state equation on [D (A? )]0 ;

yt = Ay + B g ;

y(s) = ys ∈ Y ,

(7)

with a state y(t ) ∈ Y and control g (t ) ∈ U. The control problem is to minimize the functional J (g , y, s, ys ) =

C

(ii) The optimal output y0 (t ) is continuous on [s, T ] when γ < 1/2 with values in the observation space W , but has a singularity of order 2γ − 1 at the terminal time when γ ≥ 1/2. We also have

|B ∗ P (t )x|U ≤

σ (w) · (w)dx + |w|20,Ωs .

501

T

Z


|Ry(t )|2Z + |g (t )|2U dt + |Gy(T )|2W

(8)

s

subject to the state (4) over all g ∈ L2 ([s, T ]; U ) and under the assumptions (a) A is a generator of a strongly continuous semigroup denoted by eAt on the Hilbert space Y . (b) The control operator B is a linear operator from U → [D (A? )]0 , satisfying the condition R(λ, A)B ∈ L(U , Y ), for some λ ∈ ρ(A) where R(λ, A) is the resolvent of A and ρ(A) is the resolvent set.

t →T

∀x ∈ Y .

(13) (14)

(vii) When γ < 1/2, the solution of the Riccati equation above is unique within the class of positive and self-adjoint operators satisfying (11). The main result of this paper is the following theorem pertaining to the model in (4) with the functional cost given by (6). Theorem 3.2. In reference to the model in (4) and the control problem in (6), and for every initial condition y0 = [u0 , w0 , w1 ] ∈ H , there exists a unique optimal control g 0 (t , ·) ∈ L2 ([0, T ]; Γs ) and a corresponding optimal state y0 (t , ·) = (u0 (t , ·), w 0 (t , ·), wt0 (t , ·))

∈ C ([0, T ]; H × H 1 (Ωs ) × L2 (Ωs ))

(15)

such that J (g , y ) = ming ∈L2 ([0,T ];Γs ) J (g , y). Moreover, 0

0

(i) The optimal control g 0 satisfies singular estimate given by (9) with blow up rate γ = 1/4 +  , where  is positive and can be taken arbitrarily small.

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I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

(ii) There exists a positive self-adjoint P (t ) ∈ L(H ) such that J (g 0 , y0 ) = (P (0)y0 , y0 )H . (iii) For B defined in (22), the feedback gain operator B ∗ P (t ) ∈ L(H → L2 (Γs )) for all 0 ≤ t < T and at the terminal point t = T blows up with the rate given by

|B ∗ P (t )y|L2 (Γs ) ≤

C |y|H . (T − t )1/4+

Remark 4.1. Note that the definition of weak solutions postulates the trace regularity σ (w) · ν ∈ L2 ([0, T ]; H −1/2 (Γs )) which does not follow from the interior regularity of solutions. This, in fact, a ‘‘hidden regularity’’ result on the boundary. Thus, the proof of existence of weak solutions depends on information regarding boundary regularity of the normal stresses. To obtain this information, methods of microlocal analysis were used (cf. [6]).

for t < T and x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ D (AL ) along with the terminal condition

Remark 4.2. We note that the variational formulation of weak solutions provided above is amenable to ‘‘friendly’’ numerical implementations. In contrast with other approaches (cf. [5,1, 4]), the test functions considered are not required to satisfy compatibility conditions on the interface. This is an important feature when designing FEM approximations for the corresponding control problem (cf. [38,14,18,19]). It is our hope that the above formulation will provide a first step towards a finite-dimensional approximation of the feedback control.

lim (P (t )x)1 = x1

In order to be able to apply the result of Theorem 3.1 we represent the solution to (4) as an abstract equation of the form

Moreover, the optimal feedback synthesis (12) holds. (iv) The operator P (t ) is a unique solution of the Riccati Differential Equation

hPt x, yiH + hA?L P (t )x, yiH + hP (t )AL x, yiH = hB ∗ P (t )x, B ∗ P (t )yiΓs

t →T

(16)

in H ∀x ∈ H

(17)

yt = AL y + B g ,

where the operators AL and B are defined in (22). The main mathematical difficulty of the problem studied is due to a mismatch of parabolic and hyperbolic regularity occurring at the interface. The strategy pursued in this work is to transport via the interface, the maximal (abstract) parabolic regularity (cf. [36, 11]) resulting from the fluid component onto the boundary traces of hyperbolic (wave) solutions. The so-called ‘‘Hidden’’ regularity of the wave boundary traces (cf. [37]) plays a pivotal role in this transfer. The remainder of this paper is devoted to the proof of Theorem 3.2. The proof is based on the following two main technical ingredients: (i) sharp hyperbolic-like regularity theory for traces of solutions to fluid–structure interaction, and (ii) abstract maximal parabolic regularity (cf. [36,11]) applied to the boundary value problem driven by the fluid component. 4. Weak solutions and c0 semigroup solutions We consider a weak solution to the system (4) defined to be

y0 ∈ H

(21)

where

AL =

A−L 0 0

AN σ () · ν 0 div σ

0 I , 0

!

B=

AN 0 0

! .

(22)

Here A : V → V 0 is defined by

(Au, φ) = −((u), (φ)),

∀φ ∈ V

(23)

while the Neumann map N : L2 (Γs ) → H is defined by Ng = h ⇔ {((h), (φ)) = hg , φi, ∀φ ∈ V }.

(24)

It follows immediately from the Lax–Milgram Theorem that the map A ∈ L(V → V 0 ) and the map N enjoys the property N ∈ L(H −1/2 (Γs ) → V ⊂ H 1 (Ωf )).

(25)

This allows us to consider the operator A (denoted by the same symbol) as acting on H with the domain D(A) ≡ {u ∈ V ; |((u), (φ))| ≤ C |φ|H }. A is self-adjoint, negative and generates an analytic semigroup eAt on H. Fractional powers of −A, denoted by Aα are then well defined (cf. [39]). In particular

(u, w, wt ) ∈ C ([0, T ]; H × H 1 (Ωs ) × L2 (Ωs )) = C ([0, T ]; H ) and

|Aα eAt |L(H ) ≤ Ct −α ,

such that

In addition, the perturbation A − L still generates an analytic semigroup since L is compact from D (A) to H (cf. [40]). Let the space X be the trace space corresponding to V and X 0 its dual (with respect to L2 inner product). Elements of X are defined as X ≡ {z = φ|Γs , φ ∈ V }. As a consequence, elements of X are in the 1/2 Rspace H (Γs ) and they satisfy the boundary compatibility relation g · ν = 0. Γs It was shown in [6] that the operator AL given by (22) and defined on D (AL ) ⊂ H → H with

(a) (u0 , w0 , w1 ) ∈ H × H 1 (Ωs ) × L2 (Ωs ) (b) u ∈ L2 ([0, T ]; V ) (c) σ (w) · ν ∈ L2 ([0, T ]; H −1/2 (Γs )) and wt |Γs = u|Γs ∈ L2 ([0, T ]; H 1/2 (Γs )) (d) The functions (u, w, wt ) satisfy the variational equations

(ut , φ)Ωf + ((u), (φ))Ωf + (Lu, φ)Ωf + hσ (w) · ν + g , φi = 0,

(18)

and

(wtt , ψ)Ωs + (σ (w), (ψ))Ωs − hσ (w) · ν, ψi = 0.

(19)

a.e. t ∈ (0, T ) for all test functions φ ∈ V and ψ ∈ H 1 (Ωs ). −1/2

It has been shown (cf. [6]) that for any g ∈ L2 ([0, T ], H (Γs )) and (u0 , w0 , w1 ) ∈ H there exists a unique weak solution (u, w, wt ) ∈ C ([0, T ], H ) such that

|u(t )|20,Ωf + |w(t )|21,Ωs + |wt (t )|20,Ωs Z t
+ |u(s)|21,Ωf + |σ (w) · ν|2−1/2,Γs ds 0

≤ C eωt |u(0)|20,Ωf + |w(0)|21,Ωs + |wt (0)|20,Ωs
+ |g |2L ([0,T ];H −1/2 (Γs )) . 2

(20)

0 < t ≤ 1.

(26)

D (AL ) = {y ∈ H : u ∈ V , A(u + N σ (w) · ν) − Lu ∈ H ; z ∈ H 1 (Ωs ), div σ (w) ∈ L2 (Ωs ); z |Γ s = u|Γs } indeed generates a strongly continuous semigroup eAL t ∈ L(H ). Remark 4.3. There is another approach available in recent literature that leads to semigroup solutions of fluid–structure interaction models. This has been pursued in (cf. [41,42] also the references therein) where the ‘‘generator’’ is explicitly constructed via non-local Green maps. The main difference between these two approaches is that our framework has explicit boundary conditions involving stresses on the boundary, which are defined for weak solutions (see definition (18)). In contrast, the approach taken in (cf. [28,41]) does not exhibit boundary traces at the level of weak solutions. These become apparent only for strong solutions which are characterized by a membership in the domain of the generator. Instead, the analysis in this paper critically relies on the notion of boundary traces defined for finite energy solutions.

I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

4.1. Properties of the Neumann map In what follows we shall establish additional properties of the map N defined in (24) and a useful PDE interpretation of solutions to (18). To this end we define for each (u, p) ∈ V × L2 (Ωf ) the fluid tensor T (u, p) given by

T (u, p) ≡ (u) − pI . Proposition 4.1. Let g ∈ H −1/2 (Γs ). There exists a function p ∈ L2 (Ωf ) such that h ≡ Ng satisfies distributionally div (T (h, p)) = 0

in Ωf ,

(27)

div h = 0

in Ωf ,

(28)

T (h, p) · ν = g

on Γs ,

(29)

h=0

in Γf .

(30)

Proof. The form ((h), (φ)) is V -elliptic and continuous on V × V and hence by the Lax–Milgram theorem there exists h ∈ V for a given g ∈ H −1/2 (Γs ) such that ((h), (φ))f = hg , φiΓs for every φ ∈ V . Hence, the map N is well defined as a bounded linear operator from H −1/2 (Γs ) → H 1 (Ωf ) ∩ H = V . Let Ng = h, where g ∈ H −1/2 (Γs ) then

((h), (φ))f = hg , φi for all φ ∈ V . Consider all φ ∈ H01 (Ωf )

Proposition 4.2. Let g ∈ H 1/2 (Γs ) and T h = Ng then the map N acts continuously from H 1/2 (Γs ) → H 2 (Ωf ) V . Moreover, the function p associated with the PDE problem (27)–(30) satisfied by h belongs to the space H 1 (Ωf ). Proof. By Proposition 4.1, h ∈ H 1 (Ωf ) and satisfies the boundary value problem (27)–(30) for some function p ∈ L2 (Ωf ). To prove this higher regularity result, we follow the strategy of Agmon–Douglis–Nirenberg where it suffices to consider (27)–(30) in the neighborhood of the boundary Γs (Interior regularity is straightforward) which is accomplished via a partition of unity. We then differentiate in the tangential direction by introducing the tangential differential operator S with respect to the boundary Γs to obtain the local problem in aP collar neighborhood of the n boundary Γs . The operator S = i=1 bi (x)Di is a first order operator (time independent) with b = {b1 , b2 , b3 } smooth in Ω , such that S is tangent to Γs (i.e. b|Γs · ν = 0). Denoting by [D, S ] the commutator of S with an operator D and letting S h = hˆ and S p = pˆ , we apply S to (27)–(30) in order to obtain div (hˆ ) − ∇ pˆ = [div , S ]h − [∇, S ]p in Ωf ,

(31)

(hˆ ) · ν = gˆ + pˆ ν + pνˆ + [ · ν, S ]h on Γs ,

(32)

3

div hˆ =

X

Di bj Dj hi

in Ωf .

(33)

i,j=1

T

V . Therefore,

(div (h), φ)f = 0. This implies by De Rham’s theorem that there exists a function p ∈ L2 (Ωf ) such that div T (h, p) = 0 in the sense of distributions. Since T (h, p) ∈ L2 (Ωf ), the distributional derivative div T (h, p) coincides with a function equal to zero. Hence, we also have div T (h, p) = 0,

503

a.e

which relation reconstructs equation (27). This also shows (by the Divergence Theorem) that for all φ ∈ V

hT (h, p) · ν, φiΓs = ((h) − pI , (φ))f
≤ |φ|V |h|V + |p|L2 (Ωf ) . Consequently, we have T (h, p) · ν ∈ X 0 . On the other hand, the definition of the map N implies for every φ ∈ V that

hg , φiΓs = ((h), (φ))f = ((h), (φ))f − (pI , (φ))f = hT (h, p) · ν, φiΓs . Here we used the fact that (pI , (φ))f = (p, div φ)f = 0. Hence

hg − T (h, p) · ν, φiΓs = 0, for every φ ∈ V and by the definition of X

hg − T (h, p) · ν, z i = 0, for every z ∈ X . This implies g − T (h, p) · ν belongs to the normal cone in X which can be identified with {λ = kν, k ∈ R}. Redefining the pressure p by adding a suitable constant yields the boundary conditions and the PDE form of the map N asserted in the Proposition.  Now, we turn to higher regularity of the Neumann map N. The analogous result is known in the case of Dirichlet boundary conditions (cf. [38]). However, in the case of Neumann boundary conditions, the issue is more subtle due to the presence of the pressure term on the boundary. More specifically, we show that

Taking the L2 inner product of (31) with hˆ and integrating by parts we have

( hˆ ,  hˆ ) + hˆg , hˆ i + hpνˆ , hˆ i + h[ · ν, S ]h, hˆ i − (ˆp, div hˆ ) + ([div , S ]h, hˆ ) + ([∇, S ]p, hˆ ) = 0.

(34)

We next note that since p ∈ L2 (Ωf ) is harmonic (which can be seen from applying the divergence operator to (27)), we have p|Γs ∈ H −1/2 (Γs ) ⊂ X 0 (cf. [43]). Recalling that T (p, h) · ν ∈ X 0 from (29), we conclude that (h) · ν ∈ X 0 and we have the estimate

|∇ h|Γs |X 0 ≤ C |∇ h · ν|X 0 + C |hˆ |H −1/2 (Γs ) ≤ C |g |H −1/2 (Γs ) + C |h|H 1/2 (Γs ) ≤ C |g |H −1/2 (Γs ) .

(35)

Moreover, the commutator [div , S ] is a second order differential operator and both [∇, S ] and [ · ν, S ] are first order differential operators. Hence, we estimate the term ([div , S ]h, hˆ ) after we integrate by parts as

|([div , S ]h, hˆ )f | ≤ C |(∇ h, ∇ hˆ )f | + |(∇ h, hˆ )f | + C |h∇ h · ν, hˆ i|. Similarly, we estimate the term ([∇, S ]p, hˆ )f to obtain

|([∇, S ]p, hˆ )f | ≤ C |(p, ∇ hˆ )f | + |(p, hˆ )f | + C |hpν, hˆ i|. This allows us to estimate the V norm of hˆ via Eq. (34) obtaining


|(hˆ )|20,Ωf ≤ |ˆg |−1/2,Γs + |p|X 0 + |∇ h|Γs |X 0 |hˆ |X + |p|0,Ωf |S div hˆ |0,Ωf + |∇ h|0,Ωf |hˆ |1,Ωf + |p|0,Ωf |hˆ |1,Ωf .

(36)

Now, since div hˆ = Σi,j Di bj Dj hi which comes from the fact that h ˆ Therefore, this is divergence free, we can estimate S div hˆ by ∇ h. yields

|(hˆ )|20,Ωf ≤

C

(|g |21/2,Γs + |p|2−1/2,Γs + |∇ h|Γs |2−1/2,Γs δ + |p|20,Ωf + |h|21,Ωf ) + δ|hˆ |21,Ωf .

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I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

Finally, choosing δ = 1/2 and using the fact that the H 1 norm of h and the L2 norm of p are continuously dependent on the H −1/2 (Γs ) norm of g by Proposition 4.1 in addition to the estimate in (35) we finally have

|hˆ |21,Ωf ≤ K |g |21/2,Γs . Hence, hˆ ∈ H 1 (Ωf ), and from (31), it follows that pˆ ∈ L2 (Ωf ). Let ν = (n1 , n2 , n3 ) the unit normal vector to the boundary Γs while τ and κ two linearly independent orthogonal tangential unit vectors to the boundary Γs . Consider the C 1 extension of τ , κ and ν to the whole domain Ω which preserves their orthogonality. Let v = Dτ h ≡ ∇ h · τ then v ∈ H 1 (Ωf ) and the derivatives Dτ v and Dν v belong to L2 (Ωf ), since the above estimates could be reproduced with any tangential operator S . Similarly, Dτ p and Dκ p also belong to L2 (Ωf ). To show h ∈ H 2 (Ωf ), it remains to establish that D2ν,ν h ∈ L2 (Ωf ). Let z ≡ Dν h, and notice that div z ∈ L2 (Ωf ) which follows from h being divergence free. Rewriting z in terms of τ , κ , and ν we have

Indeed, taking the dot product of the above equation with τ , we obtain the expression for Dνν h · τ Dνν h · τ = ∆h · τ + ∇ p · τ − Dκκ h · κ − Dτ τ h · τ

− (div ν Dν h + div τ Dτ h + div κ Dκ h) · τ + Dτ p ∈ L2 (Ωf ). Hence, Dν z · τ ∈ L2 (Ωf ) and similarly Dν z · κ ∈ L2 (Ωf ) and we have n3

q

n23 + n21

−n 1 n 2 q Dν z1 + n23 + n21

+ Dτ (z3 n3 ) + Dκ (z1 κ1 ) + Dκ (z2 κ2 ) + Dκ (z3 κ3 ) + (z · ν)div ν + (z · τ )div τ + (z · κ)div κ. 1 n23 + n21

Dν z3 ∈ L2 (Ωf ),

(38)

q

n23 + n21 Dν z2 − q

n3 n2 n23 + n21

Dν z3 ∈ L2 (Ωf ). (39)

n1 n3

 q   n23 + n21 M =  − q n1 n2 n23 + n21

n2 0

n3 n1

−q

n23

q

n23 + n21

−q

+

 n21

n3 n2

   .  

n23 + n21

The coefficients of the matrix are continuous and its determinant is 1, which gives the desired result of Dν z ∈ L2 (Ωf ). T Therefore, h ∈ H 2 (Ωf ) V . It also follows that p ∈ H 1 (Ωf ) and (h) · ν ∈ H 1/2 (Γs ). In addition, the following estimate is implied:

We now set

τ= q

n23 + n21

These three Eqs. (37), (38) and (39) produce a system MDν z = b ∈ [L2 (Ωf )]3 where



div z = Dν (z1 n1 ) + Dν (z2 n2 ) + Dν (z3 n3 ) + Dτ (z1 τ1 ) + Dτ (z2 τ2 )

n1

and

z = (z · ν)ν + (z · τ )τ + (z · κ)κ. Thus the divergence of z can be expressed as

Dν z1 − q

(n3 , 0, −n1 )

|h|H 2 (Ωf ) ≤ C |h|L2 (Ωf ) + |Dτ h|H 1 (Ωf ) + |Dκ h|H 1 (Ωf ) + |Dν h|H 1 (Ωf )

and

κ= q

1

n23 + n21

(−

,

n1 n2 n23

+

n21

≤ C |g |H 1/2 (Γs ) .

, −n3 n2 ).

We finally conclude that N is bounded from H 1/2 (Γs ) → H 2 (Ωf ). By interpolation, N ∈ L(H s (Γs ) ∩ X 0 → H s+3/2 (Ωf ) ∩ V ), for −1/2 ≤ s ≤ 1/2. The above regularity can be extended to a full range of s, but this will not be needed in this paper. 

Therefore, div z = Dν (z1 n1 + z2 n2 + z3 n3 ) + (z · ν)div ν + (z · τ )div τ





+ (z · κ)div κ + Dτ z1 q  − Dκ z1 q

n23 + n21

 − Dτ z3 q

Some further properties of N are given below.



 n1 n23 + n21

Proposition 4.3. 1. The map given u ∈ V , we have that N ∗ Au = −u|Γs where the adjoint is computed with respect to L2 topology. 2. The map N ∈ L(L2 (Γs ) → D (A3/4− )) ∩ L(H −1/2 (Γs ) → D (A1/2 )) for all  > 0.



 n1 n2 n23 + n21

 q   + Dκ z2 n23 + n21

Proof. 1. Let u ∈ V , g ∈ H −1/2 (Γs ) then

(ANg , u)f = (Ng , Au)f = −((Ng ), (u))f .



 − Dκ z3 q

n3

1 n23 + n21




n3 n2  .

By the definition of the map N, we then have

Recalling that div z, Dτ z, and Dκ z all belong to L2 (Ωf ) while τ and κ are C 1 functions, we have

−hg , ui = hg , N ? Aui. Hence, N ? Au = −u|Γs . 2. By part (1) and trace theory we have

Dν (z1 n1 ) + Dν (z2 n2 ) + Dν (z3 n3 ) ∈ L2 (Ωf ).

|N ? Ah|L2 (Γs ) = |N ? A3/4− A1/4+ h|L2 (Γs ) ≤ |A1/4+ h|L2 (Ωf ) .

With ν ∈ C 1 and z ∈ L2 (Ωf ) we then have

The second membership in (2) follows from (25).

n1 Dν z1 + n2 Dν z2 + n3 Dν z3 ∈ L2 (Ωf ).

(37)

On the other hand, since we know the tangential derivatives of h belong to H 1 (Ωf ), we can deduce the regularity of Dν z · τ by rewriting (27) as 0 = ∆h + ∇ p

= Dνν h + Dκκ h + Dτ τ h + div ν Dν h + div τ Dτ h + div κ Dκ h + Dτ pτ + Dν pν + Dκ pκ.



Remark 4.4. We note that the method of the proof of Proposition 4.2 also leads to higher regularity of the map A−1 . Indeed, one obtains A−1 : H → H 2 (Ωf ) ∩ V , (cf. [44]). The above regularity, along with interpolation, allow us to identify domains of fractional powers of A as D(Aθ ) ∼ H 2θ (Ωf ), θ

2θ

D(A ) ⊂ H (Ωf ),

0 ≤ θ < 3/4

θ ∈ [0, 1].

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I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

505

4.2. The generator and the control operator

5. Singular estimate property

The operator AL . It was shown (cf. [6]), that AL generates a c0 semigroup if L : V → V 0 is locally lipschitz and satisfies

In order to prove Theorem 3.2, the critical step is to prove that the Singular Estimate (SE) is valid for the model (4). This step is necessary in order to be able to apply Theorem 3.1 to the abstract system (21). To this end, we define a scale of Hilbert spaces parameterized by the parameter α ≥ 0: H−α ≡ H × H 1−α (Ωs ) × H −α (Ωs ). Note that with the above notation H = H0 .

|(Lu, u)| ≤ δ|u|2V + Cδ |u|2H

(41)

for any δ > 0. The linear operator Lu = (∇v)u +(v·∇)u is bounded when acting from V → V 0 since v is smooth, and indeed satisfies the condition above. This is sufficient (cf. [6]) for the establishment of c0 semigroup solutions to the system, which are generated by the operator AL whose action is given by (22) and whose domain is given by

Theorem 5.1. The semigroup eAL t and the control operator B satisfy the Singular Estimate (SE) C

D (AL ) = {y ∈ H : u ∈ V , (A − L)u + AN σ (w) · ν ∈ H ; z ∈ H 1 (Ωs ); div σ (w) ∈ L2 (Ωs ); z |Γ s = u|Γs }.

|eAL t B g |H−α ≤

Remark 4.5. We note that the domain D (AL ) is not compact in H . This has been noticed in [41] and in the context of studying strong stability of the uncontrolled model.

5.1. Preliminary results

> 0 such that R(λ, AL )B ∈

Proof. Writing (AL − λ)Y = f = (f1 , f2 , f3 ) leads to the system Au + AN σ (w) · ν + Lu − λu = f1 z − λw = f2 z |Γs = u|Γs ,

(42)

Lemma 5.2. Let (w0 , w1 ) ∈ H α+1 (Ωs ) × H α (Ωs ) and let f ∈ L2 ([0, T ]; H 1/2 (Γs )) where 0 ≤ α ≤ 1/4 and let (w, wt ) be the solution to the wave equation

( wtt − div σ (w) = 0 w(0, ·) = w0 , wt (0, ·) = w1 wt = f

Qs ≡ Ωs × [0, T ]

Ωs Σs ≡ Γs × [0, T ].

(46)

Then, w can be decomposed into w1 + w2 such that σ (w1 ) · ν ∈ C ([0, T ]; H −1/2 (Γs )) and σ (w2 )·ν ∈ L2 (Σs ) = L2 ([0, T ]× Γs ). If we further have that f ∈ H α (Σs ), then σ (w2 ) · ν ∈ H α (Σs ). Moreover, we have the estimates

≤ K (|w0 |21,Ωs + |w1 |20,Ωs + |f |2L ([0,T ];H 1/2 (Γs )) )

(47)

2

u|Γf = 0. Since AL generates a c0 semigroup, there exists ω > 0 such that AL − I is injective for all λ > ω. Setting f = B g = (ANg , 0, 0), we have

−((u), (φ))f −hσ (w) · ν, φi−(Lu, φ)f −λ(u, φ)f = −hg , φi (43) form all φ ∈ V and

− (σ (w), (ψ))s + hσ (w) · ν, ψi − λ(w, ψ)s = 0,

(44)

for all ψ ∈ H (Ωs ). Setting φ = u and ψ = λw and using the fact u|Γs = z |Γs = λw|Γs we add the two equations to get 1

|(u)|20,Ωf + λ|u|20,Ωf + λ(σ (w), (w))s + λ2 |w|20,Ωs + (Lu, u)f = hg , ui.

(45)

Using inequality (5) we then obtain 1

|u|2V + |w|21,Ωs − δ|u|2V + (λ − Cδ )|u|2H ≤ K |g |2H −1/2 (Γs ) + |u|2V . 2

We now choose δ so that 1/2 − δ > 0 and choose λ > 0 so that λ − Cδ > 0. Hence

|u|2V + |w|21,Ωs ≤ C |g |2H −1/2 (Γs ) . Since z = λw we also obtain the estimate

|z |

We begin with the following regularity result established for boundary traces of the dynamic wave equation.

|σ (w1 ) · ν|2C ([0,T ];H −1/2 (Γs ))

div σ (w) − λz = f3

2 1,Ωs

|g |L2 (Γs )

for every g ∈ L2 (Γs ), t ≤ T0 , and α > 0.

Control operator B . The operator B , defined in (22), is unbounded when acting from L2 (Γs ) to H since AN is an unbounded operator from L2 (Γs ) → H though bounded from L2 (Γs ) → V 0 . Keeping in mind the aim of applying Theorem 3.1 to the model described above, we proceed with the verification of the conditions required by that theorem. We start with the resolvent condition (b). Proposition 4.4. There exists ω L(L2 (Γs ) → H ), where λ > ω.

t 1/4+

≤ C λ |g | 2

2 H −1/2 (Γs )

and

|σ (w2 ) · ν|2H α (Σs ) ≤ K [|w0 |21+α,Ωs + |w1 |2α,Ωs + |f |2H α (Σs ) ].

Proof. The result of Lemma 5.2 with α = 0 was proved in [6]. The extension of the estimate (48) to α ∈ [0, 1/4) follows via tangential lifting from the same arguments as given in [6]. For the reader’s convenience, we include the details of this extension in the appendix section.  We next prove the improved boundary regularity for the velocity field u. Lemma 5.3. Consider the uncontrolled system (18) with g = 0. If in addition the initial condition (u0 , w0 , w1 ) ∈ L2 (Ωf ) × H 1+α (Ωs ) × H α (Ωs ) for 0 < α < 1/4, then u|Γs ∈ H α (Σs ) and it satisfies the estimate

|u|2H α (Σs ) ≤ C (|u0 |20,Ωf + |w0 |21+α,Ωs + |w1 |2α,Ωs ).

(49)

Proof. The proof relies on the decomposition given by Lemma 5.2, Proposition 4.3 and abstract parabolic maximal regularity methods (cf. [36]). From (21), we use the variation of parameters formula to express the solution u as u(t , ·) = eAt u0 +

,

and the desired result follows from the estimates.

(48)

t

Z

eA(t −s) Lu(s, ·)ds

0 t

Z 

eA(t −s) AN σ (w1 )(s, ·) · ν + σ (w2 )(s, ·) · ν ds.




+ 0

506

I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

Here, we used Lemma 5.2 to decompose σ (w) · ν into σ (w1 ) · ν and σ (w2 ) · ν . Now, by Proposition 4.1, the map N ? A acts as the restriction on the boundary Γs . Therefore, we express u on Γs via this map and estimate each of these terms separately by U1 = N ? AeAt u0 , t

Z U2 = U3 =

t

Moreover, (58) and (59) together imply that

N ? AeA(t −s) AN σ (w2 )(s, ·) · ν ds,

(52)

N ? AeA(t −s) Lu(s, ·) ds.

(53)

Step 1: Estimating U1 —Note that U1 is the restriction to Γs of the solution generated by the analytic semigroup d ≡ eAt u0 , which fact implies (cf. [36,11]) (note that A is self-adjoint) that for u0 ∈ H, we have A

d ∈ L2 ([0, T ]; H ).

(54)

By Remark 4.4, d ∈ L2 ([0, T ]; H (Ωf )), and the restriction to the boundary yields 1

U1 ∈ L2 ([0, T ]; H 1/2 (Γs ))

(55)

d = eAt u0 ∈ H 1 ([0, T ]; [D(A1/2 )]0 ) which combined with (54) via interpolation implies

while the restriction to the boundary yields that U1 ∈ H α ([0, T ]; H 1/2−2α (Γs )) whenever α < 1/4. Combining the estimates, we have U1 ∈ L2 ([0, T ]; H


|U2 |H 1/2,α (Σs ) ≤ C |u0 |0,Ωf + |w0 |1,Ωs + |w1 |0,Ωs . Hence


|U2 |2H α (Σs ) ≤ KT |u0 |20,Ωf + |w0 |21,Ωs + |w1 |20,Ωs . Step 3: Estimating U3 —We first observe that U3 is the restriction on Γs of the solution h to the ‘‘abstract’’ parabolic problem d h = Ah + AN σ (w2 ) · ν dt h(0, ·) = 0.

α

(Γs )) ∩ H ([0, T ]; L2 (Γs )) ⊂ H (Σs )

|U3 |2H α (Σs ) ≤ |U3 |2H α+1−,α/2+1/2−/2 (Σs ) ≤ K |h|2H α+3/2−,α/2+3/4−/2 (Ω

f

≤ K |σ (w2 ) · ν|

Step 2: Estimating U2 —Note that U2 is the restriction of the solution h to the parabolic problem h = Ah + AN σ (w1 ) · ν

for an initial condition h(0, ·) = 0 and where σ (w1 ) · ν ∈ C ([0, T ]; H −1/2 (Γs )). We shall use maximal parabolic regularity methods (cf. [36,11]) along with the characterization of fractional powers of the R tStokes operator A (see Remark 4.4). To wit, first note that h(t ) = 0 eA(t −s) A1/2 f (s)ds where f (s) ≡ A1/2 N σ (w1 (s)) · ν ∈ L∞ ([0, T ]; H ).

(56)

Indeed, on the strength of Lemma 5.2, Proposition 4.3, and the characterization of fractional powers of A we obtain that N σ (w1 (s)) · ν ∈ C ([0, T ]; D(A1/2 )), which implies the desired conclusion. By using next maximal parabolic regularity (cf. [11]) along with (56) one obtains for all p ∈ (1, ∞) that h ∈ Lp ([0, T ]; D(A

1/2

))

×[0,T ])

2 H α−,α/2−/2 (Σs )

|U1 |H α (Σs ) ≤ C |u0 |0,Ωf .

d

(61)

where the last inclusion follows since α < 1 and thus α/2 + 1/2 − /2 > α . Therefore,

for α < 1/4 along with the estimate

dt

(60)

U3 = h|Γs ∈ H α+1−,α/2+1/2−/2 (Σs ) ⊂ H α (Σs )

d ∈ H α ([0, T ]; D(A1/2−α )) ⊂ H α ([0, T ]; H 1−2α (Ωf ))

α

U2 = N ∗ Ah ∈ H 1/2,α (Σs )

Following the existing results in parabolic theory (cf. [36,45]), and identifying D(Aθ ) ∼ H 2θ (Ω ), θ < 3/4 we have that if σ (w2 ) · ν ∈ H α− (Σs ) ⊂ H α−,α/2−/2 (Σs ) then h ∈ H α+3/2−,α/2+3/4−/2 (Ωf × [0, T ]) and consequently

for a given u0 ∈ H. On the other hand, we also have

1/2

(59)

for α < 1/4. Moreover, the above regularity is expressed by the estimate

0

1/2

From (57) we also obtain

(51)

A(t −s)

and

Z

(58)

AN σ (w1 )(s, ·) · ν ds,

?

0

U4 =

for α < 1/4.

h ∈ Lp ([0, T ]; H 1 (Ωf )) → N ∗ Ah ∈ Lp ([0, T ]; H 1/2 (Γs )).

N Ae t

N ∗ Ah ∈ Wpα ([0, T ]; L2 (Γs )),

(50)

0

Z

and by standard trace theorem

≤ K |σ (w2 ) · ν|2H α− (Σs ) .

(62)

On the other hand, estimate (48) with α replaced by α −  and f replaced by u|Γs implies

|σ (w2 ) · ν|2H α− (Σs ) ≤ K |u|Γs |2H α− (Σs ) +|w0 |21+α−,Ωs + |w1 |2α−,Ωs




≤ K |u|Γs |2L ([0,T ];H 1/2 (Γs )) +|u|Γs |2H α− ([0,T ];L2 (Γs )) 2
+ |w0 |21+α,Ωs + |w1 |2α,Ωs   ≤ K |u|2H α− ((0,T ),L2 (Γs )) + |w0 |21+α,Ωs + |w1 |2α,Ωs + |u0 |20,Ωf where in the last inequality, we used the a priori estimate |u|2L ([0,T ];H 1/2 (Γ )) ≤ CE (0) which comes from energy estimates 2

s

(20) for the system (4). Our next step is to apply the interpolation inequality to the space H α− ([0, T ], L2 (Γs )) to obtain

|u|H α− ([0,T ],L2 (Γs )) ≤ |u|θH α ([0,T ],L2 (Γs )) z |u|1L2−θ ([0,T ],L2 (Γs ))

Thus, for all p ∈ (1, ∞) we have

for a suitable θ ∈ (0, 1), whose exact value depends on  and α . Young’s inequality then implies that for any arbitrary small δ , we have

h ∈ Wpα ([0, T ]; D(A1/2−α )) ⊂ Wpα ([0, T ]; H 1−2α (Ωf ))

|u|H α− ([0,T ],L2 (Γs )) ≤ δ|u|H α ([0,T ],L2 (Γs )) + Cδ |u|L2 ([0,T ],L2 (Γs )) .

ht ∈ Lp ([0, T ]; [D(A

1/2

)]0 ).

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I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

Using the a priori estimate for the term |u|L2 ([0,T ],L2 (Γs )) once more, we get

|σ (w2 ) · ν|2H α− (Σs ) ≤ δ|u|2H α ([0,T ],L2 (Γs )) + Cδ (|w0 |21+α,Ωs + |w1 |2α,Ωs + |u0 |2Ωf ) ≤ δ|u|

2 H α (Σs )

+ Cδ (|w |

2 0 1+α,Ωs

2

u0 20,Ωf

+ |w1 |α,Ωs + | |

(63)

|U3 |2H α (Σs ) ≤ δ|u|2H α (Σs ) + Cδ (|w0 |21+α,Ωs + |w1 |2α,Ωs ),

u0

? A?L t

z ∈ L2 ([0, T ]; D(A)) ∩ H 1 ([0, T ]; H ).

?

= [N A, 0, 0]e

A?L t

u0

!

w0 w1

= N ? Auˆ |Γs = uˆ |Γs .

It is sufficient then to estimate the norm of uˆ (t )|Γs in L2 (Γs ) for uˆ satisfying (66). As in Lemma 5.3, we have uˆ (t )|Γs = U1 (t ) + U2 (t ) + U3 (t ) + U4 (t )

U1 = N ? AeAt u0 ,

z ∈ H α ([0, T ]; D(A1−α )) ⊂ H α (0, T ; H 2−2α (Ωf )).

t

Z

Hence

U2 =

z |Γs ∈ H ([0, T ]; H

(67)

where the terms Ui are defined to be

By interpolation, we have

3/2−2α

!

w0 w1

B e

A(t −s)

Step 4: Estimating U4 —Let z ≡ 0 e Lu(s)ds, and recall from (20) that Lu ∈ L2 (0, T ; H ) for given initial data (u0 , w0 , w1 ) ∈ H . We then obtain from maximal parabolic regularity theorems (cf. [11]) that


|ˆu|21,Ωf + |σ (w) ˆ · ν|2−1/2,Γs ds ≤ C eωt |ˆy0 |2H .

In addition, the results of Lemma 5.2 and of Lemma 5.3 are valid with (u, w) replaced by (ˆu, w) ˆ . In order to establish (65), we ? compute the adjoint B ? eAL t obtaining

(64)

where δ > 0 can be taken arbitrarily small.

α

t

Z 0

Thus

Rt

used to show that AL generates a c0 semigroup since L? : V → V 0 satisfies the same condition as L (cf. [6]). Hence, the same regularity as in (20) holds for the solution yˆ = [ˆu, w, ˆ zˆ ] to the adjoint system.

|ˆy(t )|2H +

).

507

(68)

N ? AeA(t −s) AN σ (w ˆ 1 )(s, ·) · ν ds,

(69)

N ? AeA(t −s) AN σ (w ˆ 2 )(s, ·) · ν ds,

(70)

N ? AeA(t −s) L∗ uˆ (·, s)ds.

(71)

0

(Γs ))

t

Z

as long as α < 3/4. As a consequence, we also obtain

U3 =

U4 ∈ H 3/4− ([0, T ]; L2 (Γs )) ∩ L2 ([0, T ]; H 3/2 (Γs ))

and

0

⊂ H 3/4−,3/2 (Σs ) ⊂ H α (Σs )

t

Z

for α < 3/4. Moreover, the above inclusion is controlled by the estimate

U4 =

|U4 |2H α (Σs ) ≤ C |Lu|2L2 ([0,T ];H ) ≤ C |u|2L2 ([0,T ];V )

Estimate of U1 . The term U1 is precisely the source of the singular estimate and it is estimated on the strength of Proposition 4.3 and (26) so that we have


≤ C |u0 |2H + |w0 |21,Ωs + |w1 |20,Ωs . Step 5. Collecting the estimates for U1 , U2 , U3 and U4 , and recalling (20), we obtain 2 H α (Σs )

|u|

u0 20,Ωf

≤ Kδ | |

2 0 1+α,Ωs

+ |w |

+ |w1 |α,Ωs + δ|u| 2




2 H α (Σs )

C t 1/4+

|y0 |H .

Estimate of U2 . To estimate U2 and U3 , we utilize the properties of A and of the map N in addition to the estimates from Lemmas 5.2 and 5.3 to obtain t

Z

C

|A1/2 N σ (w ˆ 1 )(s, ·) · ν|L2 (Ωf ) ds (t − s)3/4+ ≤ Ct 1/4− |σ (w ˆ 1 ) · ν|C ([0,T ];H −1/2 (Γs )) ≤ CT |y0 |H , 0

Proof. It is equivalent to prove the following estimate for every y0 = [u0 , w0 , w1 ] ∈ H × H 1+α (Ωs ) × H α (Ωs ) = Hα :

|y0 |Hα .

(65)

This term represents the solution (ˆu, w, ˆ w ˆ t ) to the adjoint system of (18), when the initial condition is u0 , w0 , w1 ∈ H × H 1+α (Ωs ) × ? H α (Ωs ). Here, the semigroup eAL t generates the solution to the ? equation yˆ t = AL yˆ , where yˆ = (ˆu, w, ˆ w ˆ t ) satisfies the system

( ) (ˆut , φ)f = −((ˆu), (φ))f − (L? uˆ , φ)f + hσ w ˆ · ν, φi (w ˆ tt , ψ)s = (div σ (w), ˆ ψ)s w ˆ t |Γs = −ˆu|Γs

where we have used Proposition 4.3 and the estimate

|N ∗ AeA(t ) A1/2 |H →L2 (Γs ) ≤

! u0 C ? A?L t w0 ≤ 1/4+ |y0 |H ×H 1+α (Ωs )×H α (Ωs ) B e t w1 H t 1/4+

≤

|U2 (t )|L2 (Γs ) ≤

5.2. Proof of Theorem 5.1

=

|U1 (t )|L2 (Γs ) = |N ? AeAt u0 |L2 (Γs ) ≤ |N ? A3/4− eAt A1/4+ u0 |L2 (Γs )

.

We now choose δ small enough (this is possible since α −  < α ) and absorb the last term into the left-hand side of the inequality to obtain the desired result. 

C

0

(66)

for all φ ∈ V and ψ ∈ H 1 (Ωs ). The system above is regularity-wise equivalent to the system in (18) with g = 0. Moreover, A?L also generates a c0 semigroup on H using the same argument as that

C t 3/4+

.

Estimate of U3 . Note that H α+1,α/2+1/2 (Γs × [0, T ]) ⊂ C ([0, T ]; L2 (Γs )) by Sobolev embedding theorems in one dimension. On the other hand U3 is the restriction on the boundary Γs of h which solves problem (61). By (62), U3 satisfies the estimate

|U3 (t )|L2 (Γs ) ≤ |U3 |H α+1,α/2+1/2 (Σs )×[0,T ] ≤ K |σ (w ˆ 2 ) · ν|H α,α/2 (Σs ) ≤ K |σ (w2 ) · ν|H α (Σs ) . We next apply the estimate (48) from Lemma 5.2 and the estimate in Lemma 5.3 to obtain


|U3 (t )|L2 (Γs ) ≤ K |ˆu|H α (Σs ) + |y0 |H ×H 1+α (Ωs )×H α (Ωs ) ≤ KT |y0 |H ×H 1+α (Ωs )×H α (Ωs ) = KT |y0 |Hα , for any α > 0.

508

I. Lasiecka, A. Tuffaha / Systems & Control Letters 58 (2009) 499–509

Estimate of U4 . In estimating U4 (t ) we again invoke Proposition 4.3 and (67) to obtain

|U4 (t )|L2 (Γs ) ≤

t

Z

|N ? AeA(t −s) L? uˆ (s, ·)|L2 (Γs ) ds

0 t

Z ≤K

|A1/4+ eA(t −s) L? u(s, ·)|L2 (Ωf ) ds

0 t

Z ≤K

C

(t − s)1/4+ ≤ C |y0 |H .

|u|V ds ≤ CT |u|L2 ([0,T ];V )

0

Collecting the estimates of U1 , U2 , U3 , and U4 leads to the final estimate C

?

|B ? eAL t y0 |L2 (Γs ) = |ˆu(t )|L2 (Γs ) ≤

t 1/4+

|y0 |Hα .

By duality, the above implies

|eAL t B g |H−α ≤ Ct −1/4− |g |L2 (Γs ) . 

6. Completion of the Proof of Theorem 3.2 The conclusion of Theorem 3.2 follows from Theorem 3.1, Theorem 5.1 and the regularity estimate in (20). Assumptions (a) and (b) of Theorem 3.1 are satisfied on the strength of the results presented in Section 4, including Proposition 4.4. Assumption (d) follows from the fact that R(λ, AL )B ∈ L(H ) and from the structure of the operator G (projection). The most critical assumption is the singular estimate assumption (c) and its validity for system (4) follows from Theorem 5.1. To see this we first note that Theorem 5.1 implies that with G = (I , 0, 0) we have

|GeAL t B g |W = |GeAL t B g |0,Ωf ≤ |eAL t B g |H−α ≤

C t 1/4+

|g |L2 (Γs ) .

Similarly for all α < 1

|ReAL t B g |H ≤ |eAL t B g |H−α ≤

C

t

|g |L2 (Γs ) . 1/4+

Thus, condition (c) in Theorem 3.1 is satisfied with γ = 1/4 +  . Finally, the regularity of the trajectories is stronger than that postulated in Theorem 3.1, but this follows from the estimate in (20). The proof of Theorem 3.2 is then complete. Remark 6.1. One could generalize the result in this paper by considering more general functionals incorporating the state of the solid body. This could be done in a straightforward manner provided that the penalization is controlled by H −α norms. Whether one can take α = 0 remains an open question. Acknowledgment The research of first author has been supported by NSF Grant DMS 0606682. Appendix. Proof of Lemma 5.2 The proof of Lemma 5.2 is essentially contained in [6] where trace regularity for the model (46) has been established by the methods of microlocal analysis. In fact the result in (48) with α = 0 is proved there. Generalization for α ∈ [0, 1/4) stated in (48)

follows by a simple tangential rescaling argument. The inequality in (47) follows by tracing the steps of the proof in [6], which requires boosting L2 time regularity shown in [6] to continuous-intime regularity, as stated in (47). However, due to the fact that w1 corresponds to microlocalization in the elliptic sector, the behavior in time domain is nice (smooth) and dominated by tangential boundary derivatives. This fact allows for tangential (time and space) boosting. After these preliminary remarks, we will provide more details. Inequality in (48). We proceed with a microlocal analysis argument following Section 5 in [6]. By moving into half space through a partition of unity and a change of coordinates, we obtain a second order hyperbolic equation in a new variable wc . We next use pseudo-differential operators dividing the domain into a hyperbolic and an elliptic sector hence decomposing wc into w1 and w2 . The boundary tangential derivative of w2 obtained from the hyperbolic sector is dominated by the time derivative wt = f ∈ H α (Σs ) implying that w2 ∈ H α+1 (Σs ). This allows for an application of the sharp trace regularity result for wave equation (cf. [37]) concluding that σ (w2 ) · ν ∈ H α (Σs ). Inequality in (47). w1 satisfies an elliptic problem in the elliptic sector where |σ |  |η|, with σ denoting the dual variable corresponding to time t and η is a dual variable corresponding to the tangential τ derivative. Thus, regularity of w1 is determined from the following relations: ∆x,t w1 ∈ C ∞ (Qs ) and w1t = f |Γs ∈ L2 (H 1/2 ), where ∆x,t denotes the second order elliptic operator in space and time variables and Qs denotes Ωs × [0, T ]. Since w1t is supported in an elliptic sector so that its time derivative is dominated by its tangential derivative, we automatically obtain w1,t ∈ H 1/2 (Σs ) (hence space AND time 1/2 derivative). By elliptic theory (applied to w1,t ) we obtain w1t ∈ H 1 (Qs ). Since the support considered is in the elliptic sector L2 ([0, T ]; H 1 (Ωs )) regularity, automatically translates into H 1 (Qs ). Thus, applying Green’s formula to z ≡ w1t we obtain with any test function φ ∈ L2 (H 1 (Ωs ))

hσ (z ) · ν, φiΣs = (div σ (z ), φ)Qs − (σ (z ), (φ))Qs = −(zt + lot (z ), φt )Qs − (∇ z , ∇φ)Qs where lot (z ) denote lower order terms. Thus,

|hσ (z ) · ν, φiΣs | ≤ C |φ|H 1 (Qs ) |z |H 1 (Qs )





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σ (z ) · ν ∈ L2 ((0, T ); H −1/2 (Γs )). Hence

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