Well-posedness of a Structural Acoustics

Control Model with Point Observation of the Pressure

George AvalosDepartment of Mathematics,

Texas Tech University,Lubbock, Texas, 79409

Irena LasieckaDepartment of Applied Mathematics,

Thornton Hall,University of Virginia,

Charlottesville, VA 22903

Richard RebarberDepartment of Mathematics and Statistics,

University of Nebraska-Lincoln,Lincoln, NE 68588-0323, U.S.A.

February 29, 2000

Abstract

We consider a controlled and observed partial differential equation (PDE) which describes astructural acoustics interaction. Physically, this PDE describes an acoustic chamber with a flexiblechamber wall. The control is applied to this flexible wall, and the class of controls under considerationincludes those generated by piezoceramic patches. The observation we consider is point measurementsof acoustic pressure inside the cavity. Mathematically, the model consists of a wave equation coupled,through boundary trace terms, to a structurally damped plate (or beam) equation, and the pointcontrols and observations for this system are modeled by highly unbounded operators. We analyzethe map from the control to the observation, since the properties of this map are central to any controldesign which is based upon this observation. We also show there exists an appropriate state spaceX , so that if the initial state is in X and the control is in L2, then the state evolves continuously inX and the observation is in L2. The analysis of this system entails a microlocal analysis of the wavecomponent of the system, and the use of pseudodifferential machinery.

1 Introduction

1.1 Motivation

In this paper we consider a controlled and observed PDE associated with the mathematical modelingof certain structural acoustic interactions. The spatial domain for the PDE system is a bounded regionΩ ⊂ Rn, n = 2 or 3, with boundary Γ. We will refer throughout to a subset Γ0 of Γ as the “active” partof this boundary. The motivating physical example is sound waves in a cavity Ω which has a flexible wallat Γ0. The control is applied to Γ0 and observations are taken of acoustic pressure inside the cavity. Inthe motivating example the control is implemented via piezoceramic patches on Γ0.

Mathematically, the system is comprised in part of a wave equation satisfied by the function z(t, x)at time t and position x ∈ Ω. The acoustic pressure inside of the cavity Ω is proportional to zt(t, x).Moreover, the wave equation is coupled through boundary “trace” terms to a parabolic–like equation

1

which is satisfied on Γ0. Specifically, when n = 2 this parabolic–like equation is a damped beam equationof Kelvin–Voigt type, and when n = 3 the parabolic component is a plate equation, likewise with Kelvin–Voigt damping. The displacement of this beam or plate is given by the function v(t, ξ), at time t andposition ξ ∈ Γ0. We consider general control input terms of the form Bu(t) in the parabolic component ofthe system for some operator B. In the motivating example, the patches generate bending moments on thebeam or plate, and the control u(t) ∈ L2(0, T ; Rk) contains in its ith component (i = 1, ...., k) the amountof voltage to be conducted through the ith patch at time t. The PDE model will be explicitly given inthe next subsection; it is well-known and has been the subject of past and present laboratory/numericalinvestigations by engineers at NASA and other facilities (for more details, including the physical derivationof the model, see [5, 6, 7, 8] and the references therein).

We are interested in the following physically motivated observations y(t) for this system:

(A) The acoustic pressure at points inside the cavity Ω;

(B) The displacement of the beam (or plate) at points on Γ0;

(C) The velocity of the beam (or plate) at points on Γ0.

In order to suppress noise in the cavity, a control u(t) is typically synthesized by a feedback (dynamicor static) of the observation y(t); see [5, 7] for examples of this practice. Not surprisingly then, a properunderstanding of the properties of the input-output map u → y (with zero initial data) is crucial forthe purposes of feedback control design and analysis. For instance, it is the boundedness properties ofthis map (or lack thereof) which determine whether or not a given feedback stabilization is robust withrespect to a large class of perturbations, see [17, 30]. To make precise this notion of boundedness, weintroduce the following:

Definition 1.0.1 Given a control space U and an observation space Y , an input–output map LT , definedon L2(0, T ;U) by LT u = y, is said to be well–posed if

LT ∈ L(L2(0, T ;U), L2(0, T ;Y )

).

In Avalos, Lasiecka, Rebarber [3], it is shown that when Ω is a rectangle in R2, then LT is a well-posedinput–output map for all three of the observations listed above. If Ω is a general region with a smoothboundary, it is not hard to show that LT is also well-posed if the observation is limited to (B) and (C)only; this is carried out in [3] for n = 2, and for n = 3 it is a much simplified version of the work inthis paper. Our main aim in this paper is to identify a functional analytic framework for this problem,which will give rise to well–posed dynamics. To this end, it becomes necessary to make sense of theinput–output map for general domain Ω. In addition to establishing regularity of the input–output (or“control to observation”) map, we will identify a suitable state space X from which to take initial data,this being necessarily a strict subspace of the natural energy space H (say), such that the solution of thegoverning PDE, corresponding to control from U and initial data from X , evolves continuously in X andmoreover allows for the physically relevant observations. In this way, we will have that for all t, the chainof mappings

u(t) ∈ U → x(t) ∈ X → y(t) ∈ Y,

is well defined.

However, the difficulty in establishing this well definition is that the inherent control and observa-tion operators are strictly unbounded on the natural finite energy space H. Indeed, for our particularapplication, the controls (derivatives of delta functions which model point impulses) and observations(pointwise–in space–evaluations of the velocity of the wave equation, which would correspond to the

2

measurement of pressure within an acoustic chamber) will not be represented by bounded operators onthe natural finite energy space. In addition, the overall dynamics does not have any built-in smoothingmechanism, owing to the contribution from the hyperbolic component.

Therefore, besides validating the input–output map above, we must identify a state space X for thecontrolled and observed structural acoustics PDE, such that:

I. There is a semigroup on X , necessarily a “narrower” space than H, which is associated with theuncontrolled dynamics.

II. The mapping taking the control u(t) from U to the state x(t) in X is bounded.

III. The physically relevant observation y(t) of the state–given by an ill–defined operator on X–whenacting upon a given flow (solution), is also bounded a mapping.

¿From a PDE point of view, the search for a “good” space X on which to set the semigroup is achallenging task. There must be a resort to a complex PDE machinery which can extract the optimalregularity (trace and interior) out of solutions to second order hyperbolic equations under the influenceof given Neumann boundary data. This technical machinery is quite necessary, inasmuch as the waveequation under Neumann boundary conditions will not satisfy the Lopatinski condition. In addition,after this technical work is accomplished for the hyperbolic component, we must labor still further so asto justify that this optimal wave regularity “propagates” onto the plate component via the interface.

By way of discerning the delicacy in choosing X , we look over the objectives I–III above. Note thatwhile objective II seems to call for a “large enough” state space, condition III seems to require that statespace X be “small enough”. Thus, it is not a priori obvious that such a state space should exist at all.And should an appropriate X indeed exist, the conflicting needs of II and III suggest that there will notbe much margin for error in its specification. To complicate matters still further, we are requiring thatthe uncontrolled flow should also satisfy a semigroup property on our choice of X . By appealing to thegeneral theory in Salamon [34], we know that once a system is shown to have a well-posed input-outputmap, there exists a state space X which has yields the properties I–III. However, a construction of thisspace, by means of the prescription in [34], does not result in a physically motivated (or benign–looking)Sobolev space. On the other hand, the X we ultimately choose below has one-half (spatial) derivativemore in the wave component than that for the basic space of well-posedness H; so its characterizationhere is relatively simple.

Once we prove that the mappings described in I and II are “regular”, to use the vernacular of SystemsTheory (see e.g., [39]), all manner of control design issues can be addressed for the model, not only thatof robustness analysis, which we consider in [3], but also those concerning tracking, adaptive control,dynamic stabilization; etc. (see e.g., [25, 26, 30, 31, 32, 40]). Moreover, with this state space X in hand,a rigorous numerical analysis of an observer-based, control design scheme can now proceed, with the cor-responding numerical approximations taking into account the special choice of initial data. But withoutan initial understanding of the “control to state” and “control to observation” maps, an understandingwhich is the objective of the present paper, these subsequent issues cannot be addressed.

1.2 The Model and the Main Results

In this subsection we describe in detail the controlled, observed PDE model describing structural acousticinteractions. This model originates from [9, 28] and more recently has been analyzed in papers by severalauthors, including [5, 6, 8, 10, 11, 12, 13, 24]. We shall use here a slightly abstracted version of thismodel, which retains all basic characteristics of the original structural acoustic problem.

Let Ω be a region in R2 or R3 with a smooth boundary Γ. Furthermore, let Γ0 be a smooth segmentof Γ, with its boundary denoted by ∂Γ0. Let z = z(t, x) for t ∈ [0, T ] and x ∈ Ω, and let v = v(t, ξ)

3

for t ∈ [0, T ] and ξ ∈ Γ0. Denote the outward normal derivative to Γ by ∂/∂ν, and the outward normalderivative to ∂Γ0 by ∂/∂n. Let the control space be U = Rk, and suppose

B ∈ L(U,H− 5

3 (Γ0))

. (1.1)

With this notation, we will refer to the following system as the structural acoustics model:

ztt = ∆z on (0, T )× Ω;

∂z

∂ν=

vt on (0, T )× Γ0;

0 on (0, T )× Γ \ Γ0;

vtt = −∆2v −∆2vt − zt + Bu on (0, T )× Γ0;

v|∂Γ0 =∂v

∂n|∂Γ0 = 0 on (0, T )× Γ0;

[z(t = 0), zt(t = 0), v(t = 0), vt(t = 0)] = X0 := [z0, z1, v0, v1] .

(1.2)

The particular structure of the control operator B is not at all important for the subsequent analysis; itsspecified degree of unboundedness in (1.1) comes from our wish to consider those input operators B whichare associated with control by smart materials. In the motivating example, in which B represents theaction of piezoceramic patches bonded to the flexible wall Γ0, B will take the form of a linear combinationof derivatives of delta functions, and as such is an element of L(U,H− 5

3 (Γ0)) for n = 2. To see this, let ξ0

be a given point in Γ0. By the continuity of the Sobolev embeddings we have H53 (Γ0) ⊂ H3/2+ε(Γ0) ⊂

C1(Γ0), from which we easily deduce that δ′(ξ0) ∈ H− 53 (Γ0).

As indicated in the previous subsection, there are three natural observations for the PDE model (1.2),and in this paper we are interested in the most “unbounded” of these, the observation of acoustic pressureat a given point in the cavity, since this is the observation for which well-posedness of the control-to-observation map is difficult to prove. Let x0 be a specified point in Ω. If [z, v] is a solution of (1.2), wethen define

y(t) = zt(t,x0), (1.3)

so that in this paper the observation space is Y = R. This observation is proportional to the acousticpressure at x0, a physical quantity which in practice could be measured by means of a microphone.Initially, (1.3) is only a formal expression, inasmuch as there is no guarantee that zt(t, ·) can be evaluatedpointwise in Ω, see Remark 1.5 below. More generally, one can consider any finite collection of pointsxim

i=0 ∈ Ω, and with it take our observation y(t) to be a linear combination of zt(t,xi)mi=0. This

would clearly not affect the analysis below.

One may represent (1.2), (1.3) formally by a triple (A,B, Cx0) (defined below) on the “natural” statespace H, where

H = H1(Ω)× L2(Ω)×H20 (Γ0)× L2(Γ0). (1.4)

For X0 = [z0, z1, v0, v1]T ∈ H, we define A : D(A) ⊂ H → H by

AX0 =

z1

∆z0

v1

−∆2v0 −∆2v1 − z1

,

with D(A) =

[z0, z1, v0, v1]T ∈

[H1(Ω)

]2 × [H20 (Γ0)

]2: ∆z0 ∈ L2(Ω),

∂z0

∂ν= v1 on Γ0,

∂z0

∂ν= 0 on Γ \ Γ0; ∆2v0 + ∆2v1 ∈ L2(Γ0)

. (1.5)

4

One can readily show thateAt

t≥0generates a C0–semigroup on H; see for instance [2]. Letting

X(t) = [z(t), zt(t), v(t), vt(t)]T and B = [0, 0, 0, B]T , (1.2) is equivalent to

d

dtX(t) = AX(t) + Bu, X(0) = X0.

Therefore the solution of (1.2) may be given by the variation of parameters formula

X(t) = eAtX0 + +LT u(t),

where

LT u(t) ≡∫ t

0

eA(t−s)Bu(s)ds, for all t ∈ [0, T ].

It is shown in [2] that if u ∈ L2(0, T ;U), then X(t) ∈ H for every X0 ∈ H. In systems theoretic languagethen, B is an “admissible control operator” for eAt in H, see Weiss [39].

For arbitrary X0 = [z0, z1, v0, v1]T ∈ H, we formally define the observation map Cx0 by

Cx0X0 = z1(x0), where x0 ∈ Ω, (1.6)

so the observation map y(t) may be formally expressed as

y(t) = Cx0eAtX0 + Cx0LT u(t). (1.7)

The input–output map LT may then be formally represented by LT u(t) = Cx0LT u(t) for u ∈ L2(0, T ;U).The fact that both B and Cx0 are unbounded operators makes the analysis of this map especially chal-lenging. Our main results, Theorem 1.1 and 1.4, give a proper meaning to the observed state (1.7).

Theorem 1.1 For every 0 < T < ∞ and all x0 ∈ Ω, LT is well–posed (in the sense of Definition 1.0.1,with U = Rk and Y = R therein), with the norm bound of LT generally depending on T .

Remark 1.2 In the special case that Ω is a rectangle, this was proved in Section 3 of [3] by a Fourier/Harmonicanalysis. These techniques are wholly inapplicable in the present case that Ω is an arbitrary domain withsmooth boundary.

Note that it is not true that Cx0eAtX0 ∈ L2(0, T ) for all X0 ∈ H—that is to say, in the language of

systems theory, Cx0 is not an “admissible” observation operator for eAt on the space H, see [39]. Wewish to identify a state space X such that B is an admissible control operator and Cx0 an admissibleobservation operator. To this end, we define the elliptic operator AN : L2(Ω)

R ⊃ D(AN ) → L2(Ω)R by

AN = −∆ with

D(AN ) =

f ∈ H2(Ω)R

:∂f

∂ν= 0 on Γ

. (1.8)

AN is positive definite, self–adjoint on L2(Ω)R , and so by the characterization of the fractional powers in

[15], we have

D(AαN ) =

H2α(Ω)R

, 0 ≤ α <34;

D(A34N ) =

f ∈ H

32 (Ω)R

3 ∇f ∈ L2− 1

2(Ω)

, (1.9)

5

where, as in [15], L2− 1

2(Ω) denotes the space of functions h(x) such that %(x)−

12 h(x) ∈ L2(Ω), with %(x)

being the distance from x to boundary Γ. Furthermore, we define the Neumann map N : L2(Γ) → L2(Ω)R

by Ng = h if

∆h = 0 on Ω ; (1.10)∂h

∂ν= g on Γ.

By standard elliptic theory (see e.g., [20]), we have that for all real s,

N ∈ L

(Hs(Γ),

Hs+ 32 (Ω)

R

). (1.11)

With these operator definitions, we now present our choice of state space X which is principally motivatedby the following “trace” result of R. Triggiani, which gives meaning to pointwise spatial evaluations ofsolutions of the wave equation.

Lemma 1.3 For n = 2, let z(t, x) be the solution of the wave equationztt = ∆z on (0, T )× Ω;∂z∂ν = 0 on (0, T )× Γ;

z(0, ·) = z0 ∈ D(A34N ), zt(0, ·) = z1 ∈ H

12 (Ω)R ;

(1.12)

Then for any x0 ∈Int(Ω), there exists CT > 0 such that

‖zt(·,x0)‖L2(0,T ) ≤ CT

∥∥∥[z0, z1]T∥∥∥

D(A34N )×H

12 (Ω)R

.

Proof: Note that the wave equation above is well posed with [z, zt]T ∈ C([0, T ];D(A

34N )×H

12 (Ω)/R).

Thus for all 0 ≤ t ≤ T , ztt(t) = ANz(t) ∈[H

12 (Ω)/R

]′, after using the characterization in (1.9). If we

make the change of variable p = zt, then p solvesptt = ∆p on (0, T )× Ω;∂p∂ν = 0 on (0, T )× Γ;

p(0, ·) = z1 ∈ H12 (Ω)R , pt(0, ·) = ANz0 ∈

[H

12 (Ω)R

]′.

By Theorem 3.2 of [38], we then have for arbitrary x0 in the interior of Ω,

‖p(·,x0)‖L2(0,T ) ≤ CT

∥∥∥[z1, ANz0]T∥∥∥

H12 (Ω)/R×[H

12 (Ω)/R]′

≤ CT

∥∥∥[z0, z1]T∥∥∥

D(A34N )×H

12 (Ω)R

. (1.13)

Transforming back to variable zt now gives the result. 2

With this result in mind, we define our prospective state space to be functions in the following productof Hilbert spaces, which moreover satisfy a certain compatibility condition:

X :=

[z0, z1, v0, v1] ∈

H32 (Ω)R

× H12 (Ω)R

×H20 (Γ0)× L2(Γ0),

such that z0 −Nv1 ∈ D(A34N )

. (1.14)

6

As given, then X has the same regularity as H in the beam component, but has one-half more derivativeregularity in the wave component. X is easily seen to be a Hilbert space with the norm

‖[z0, z1, v0, v1]‖X

=(‖z0‖2

H32 (Ω)

+ ‖z1‖2H

12 (Ω)

+ ‖v0‖2H20 (Γ0)

+ ‖v1‖2L2(Γ0)+∥∥∥A 3

4N (z0 −Nv1)

∥∥∥2

L2(Ω)

) 12

. (1.15)

We are now in a position to state our second main result, which shows that the given X is a statespace for the structural acoustics model (1.2), and moreover allows pointwise observations of the acousticpressure.

Theorem 1.4 (i) Let n = 2, 3. Then A generates a strongly continuous semigroup on X .

(ii) Let n = 2, 3. Then LT ∈ L(L2(0, T ;U), C([0, T ];X )).

(iii) Let n = 2. Then for x0 ∈Int(Ω), Cx0eA(·) ∈ L

(X , L2(0, T )

).

In (ii) and (iii), the respective norm bounds will generally depend upon T .

Remark 1.5 For X0 ∈ X and u ∈ L2(0, T ;U), the known interior regularity for this problem (see [2])is zt ∈ C([0, T ];L2(Ω)). This is not sufficient to allow a well-defined pointwise evaluation (in Ω) of zt

via the Sobolev Embedding Theorem. Thus, Theorems 1.1 and 1.4 provide an additional trace regularityfor the hyperbolic component of (1.2). In systems theoretic language, the triple (A,B, Cx0) is a well-posedsystem (see [39]); using the techniques in Theorem 4.8 of [3] we can see that (A,B, Cx0) is in fact aregular system with “feedthrough” 0. We should also note that for the admissibility of the observationoperator Cx0 (part (iii) in Theorem 1.4), we need to restrict ourselves to n = 2 and x0 ∈ Ω, rather thanthe more general situation considered in parts (i) and (ii); this is owing to our critical use of Lemma 1.3in proving (iii).

2 Proof of Theorem 1.1

2.1 Analysis of the wave component

For the remainder of this paper the letter C will denote a generic constant, which will vary according tocontext. If C depends on a variable, we put that variable as a subscript on C; e.g., CT .

We begin by listing some properties of the structural acoustic system which were established in [1, 2],and which will be used throughout this paper. In addition to the regularity property for zt noted inRemark 1.5, note that we also have the improved regularity of the velocity of the displacement vt.

Proposition 2.1 (See [1],[2]). For all u ∈ L2(0, T ;U) and X0 ∈ H, the solution of (1.2) satisfies thefollowing estimate:∥∥∥[z, zt, v, vt]

T∥∥∥

C([0,T ];H)+∥∥∥[zt|Γ0

, vt

]T∥∥∥L2(0,T ;H− 1

3 (Γ0)×H20 (Γ0))

≤ CT

[‖u‖L2(0,T ;U) + ‖X0‖H

].

Remark 2.2 In [1, 2] this result was only established for n = 2, but the same proof holds for n = 3.It should be noted that the proof of Proposition 2.1 relies in part upon estimates derived from a rathertechnical microlocal analysis of the wave component of (1.2). In particular, the “sharp” regularity oftraces of solutions to the Neumann problem, which was established recently in [1, 21, 22], plays a criticalrole.

7

The extra regularity for vt cited in Proposition 2.1 leads us to a study of the following wave equation:

ztt = ∆z in (0, T )× Ω;∂z

∂ν= g on (0, T )× Γ;

z(t = 0) = zt(t = 0) = 0; in Ω, (2.1)

where the boundary data g is taken to be better than L2 in space. Indeed, the bulk of our effort in thissection will be devoted to establishing the following Lemma:

Lemma 2.3 Let z solve (2.1) with g ∈ L2(0, T ;H2(Γ)) and fixed x0 ∈ Ω ⊂ Rn, n ≤ 3. Then,

‖zt(·,x0)‖L2(0,T ) ≤ CT ‖g‖L2(0,T ;H2(Γ)) . (2.2)

Assuming for the time being the validity of Lemma 2.3, the proof of Theorem 1.1 is now straightforward.In fact, let

g :=

vt on Γ0

0 on ΓΓ0;(2.3)

so by Proposition 2.1 and the fact that vt|∂Γ0=

∂vt

∂n

∣∣∣∣∂Γ0

= 0, we see that g ∈ L2(0, T ;H2(Γ)). Theorem

1.1 follows immediately then from Lemma 2.3.

Remark 2.4 We note that the result stated in Lemma 2.3 does not follow from the previously knownregularity for the Neumann problem. Indeed, the classical hyperbolic results in [20, 29], and even the“sharp” hyperbolic results in [21] require much more regularity on g in the time variable, a regularitywhich is not available in our present problem.

Our proof of Lemma 2.3 is to be done in the forthcoming sections. In Section 2.2 we first prove theLemma for the case where Ω is a half-space. Our reason for singling out this canonical geometry isthat the corresponding computations are much simpler than in the general case, yet an analysis on thehalf space reveals key features of the problem, and illuminates the proof in the general case. The proofof Lemma 2.3 for a smooth, bounded domain will require the use of microlocal analysis, and will beundertaken in Section 2.3.

2.2 The Proof of Lemma 2.3 for the Half-Space Problem

In this subsection we assume that

Ω = (x,y); x > 0, y ∈ Rn−1; Γ = 0 × Rn−1,

where again, n ≤ 3. With this geometry we consider the equation

ztt(t, x,y) = zxx(t, x,y) + ∆yz(t, x,y) on (0, T )× Ω;zx(t, 0,y) = g(t,y) on (0, T )× Γ; (2.4)z(0, x,y) = zt(0, x,y) = 0,

where 0 < T < ∞, and ∆y =n−1∑i=1

∂2

∂y2i.

In this special case, we are in a position to prove a somewhat stronger result than the one in Lemma2.3. In fact, we shall show the following:

8

Lemma 2.5 Let x0 = (x0,y0) ∈ Ω, and ε > 0 be arbitrary. Then if z solves (2.4) with g ∈ L2(0, T ;H3/2+ε(Γ)),we have the estimate:

‖zt(·, x0,y0)‖L2(0,T ) ≤ CT ‖g‖L2(0,T ;H3/2+ε(Γ)) .

Proof: The key component in the proof of this Lemma is to show the following estimate for fixedx0 ∈ R: ∫ T

0

‖zt(t, x0, ·)‖2H1+ε(Γ) dt ≤ CT ‖g‖2L2(0,T ;H3/2+ε(Γ)) . (2.5)

To this end, We extend g(t, ·) to all of R by zero outside the interval t ∈ [0, T ] (given that the problem(2.4) has zero initial data) and apply the Fourier–Laplace transform. For this purpose we will denotethe Fourier variable corresponding to y by η ∈ Rn−1, and the Laplace variable corresponding to t bys = γ + iσ ∈ C, where γ > 0 is fixed.

With z denoting the Fourier–Laplace Transform of z, we obtain, after using the fact that (2.4) haszero initial data,

s2z(s, x, η) = zxx(s, x, η)− |η|2z(s, x, η)zx(s, 0, η) = g(s, η). (2.6)

Solving this initial value problem explicitly gives the formulae

z(s, x, η) = − g(s, η)√s2 + |η|2

e−x√

s2+|η|2 ;

zt(s, x, η) = − sg(s, η)√s2 + |η|2

e−x√

s2+|η|2 . (2.7)

The behaviour of the symbol s2 + |η|2 is critical here. Since

s2 + |η|2 = |η|2 + γ2 − σ2 + 2iγσ,

we have asymptotically as |η| → ∞ and |σ| → ∞,∣∣s2 + |η|2∣∣ ∼ (||η|2 + γ2 − σ2|+ |σ|

). (2.8)

We shall now localize the Fourier variable according to the following partition of Rn: let

R0 ≡ (η, σ) : |σ|2 + |η|2 ≤ 1; R1 ≡ (η, σ) : |σ| ≥ 2|η|; R2 ≡ Rn\R1.

Step 1: Analysis in R1 \R0.

In R1 \R0 we have from (2.8) that asymptotically

|s2 + |η|2| ≥ Cσ2. (2.9)

Hence in R1 \R0, we have ∣∣∣∣∣s g(η, s)√s2 + |η|2

∣∣∣∣∣ ≤ C|sg(η, s)||σ|

≤ C|g(η, s)|. (2.10)

This estimate and (2.7) thus yield

|zt(x0, η, s)||η|1+ε ≤ C|η|1+ε||g(η, s)|. (2.11)

Step 2: Analysis in R2 \R0.

9

In this region |σ| < 2|η|, and so as∣∣s2 + |η|2

∣∣ ≥ C |σ|, we then have∣∣∣∣∣s g(η, s)√s2 + |η|2

∣∣∣∣∣ ≤ C|σ|1/2|g(η, s)| ≤ C|η|1/2|g(η, s)|. (2.12)

Therefore (2.7) and (2.12) show that in R2 \R0

|zt(x0, η, s)||η|1+ε ≤ C|η|3/2+ε|g(η, s)|. (2.13)

Step 3 : Analysis in Rn.

Combining (2.11) and (2.13), we obtain

|zt(x0, η, s)||η|1+ε ≤ Cγ |η|3/2+ε|g(η, s)| for all (η, σ) ∈ Rn\R0. (2.14)

On the other hand, since R0 is bounded then easily we deduce from (2.7) and (2.14) that for all (η, σ) ∈ Rn

|zt(s, x0, η)||η|1+ε ≤ C[|η|3/2+ε + 1]|g(η, s)|.

Using this inequality and the generalized Parseval’s relation (see p. 212 of [14]) we obtain

2πe−2γT

∫ T

0

‖zt(t, x0, ·)‖2H1+ε(Γ) dt ≤ 2π

∫ ∞

0

e−2γt ‖zt(t, x0, ·)‖2H1+ε(Γ) dt

=∫ ∞

−∞

∫Rn−1

|zt(γ + iσ, x0, η)|2(1 + |η|2

)1+ε

dηdσ

= ≤ C

∫ ∞

−∞

∫Rn−1

[(|η|3/2+ε + 1)2|g(η, s)|2

]dηdσ

≤ 2πC

∫ T

0

e−2γt ‖g(t)‖2H

32 +ε(Γ)

dt,

from which the estimate (2.5) follows.

Finally, by the Sobolev embedding theorem H1+ε(Γ) ⊂ C(Γ) (recall that the dimension of Γ ≤ 2), andthis combined with (2.5) yields,

‖zt(·, x0,y0)‖L2(0,T ) ≤ CT ‖zt(·, x0, ·)‖L2(0,T ;H1+ε(Γ)) ≤ CT ‖g‖L2(0,T ;H3/2+ε(Γ)) ,

which is the desired result of Lemma 2.5. 2

Remark 2.6 The above computations reveal that the trace regularity of zt is better in the sector R1, wherethe Lopatinski condition holds true. The deterioration occurs in R2, which contains the characteristicsector |η| = |σ|. This structure is typically seen in hyperbolic problems. In fact, we shall observe the samephenomenon in the general case of smooth domains. The difference, however, will be that establishing theregularity of traces for the general case will require more tangential differentiability of g than when Ω isa half plane. This is due to the appearance of commutator terms.

Remark 2.7 Note that if dim (Ω) ≤ 2, then the result of Lemma 2.5 holds with g ∈ L2(0, T ;H1+ε(Ω))only.

10

2.3 Proof of Lemma 2.3 (The General Case)

2.3.1 Space Localization

Since the initial conditions are zero, we can extend the boundary data g by zero for t < 0 and considerthe system (2.1) for all t ∈ R. To make things easier we also multiply g by e−γt with γ > 0. This doesnot influence the regularity over a finite time horizon, but it does allow the application of the transformover an infinite interval. We adopt the following notation:

Qt := Ω× (0, t); Q := Q∞;

Σt := Γ× (0, t); Σ := Σ∞;

|u|s,D := ‖u‖Hs(D) ,

for any s ∈ R and any appropriate smooth domain D. Letting, as usual, OPSs(Ω), OPSs(Γ), OPSs(Q),and OPSs(Σ) denote the spaces of pseudodifferential operators (ΨDO’s) with homogenous symbol oforder s (see [36]), we now proceed to prove a trace regularity result for wave equations from whichLemma 2.3 immediately follows. Namely, our main result in this section is the following:

Lemma 2.8 With Ω ⊂ Rn, n ≤ 3, let z solve the wave equation

ztt = ∆z + Lz in (0, T )× Ω;∂z

∂ν= g + Mz on (0, T )× Γ;

z(t = 0) = zt(t = 0) = 0; in Ω, (2.15)

whereg ∈ L2(0, T ;H2(Γ)), and ΨDO’s L ∈ OPS1(Ω); M ∈ OPS0(Γ). (2.16)

Then for all fixed x0 ∈ Ω, we have the estimate

‖zt(·,x0)‖L2(0,T ) ≤ CT ‖g‖L2(0,T ;H2(Γ)) . (2.17)

Remark 2.9 In the case that dim (Ω) = 2, then Lemma 2.8 holds true with g ∈ L2(0, T ;H32−ε(Γ)) only.

Proof of Lemma 2.8: As before, we extend g by zero for T < t < 0, so that this extension of g is inL2(R;H2(Γ)). We shall first consider the case when the point x0 lies on the boundary. The general caseof x0 ∈ Ω can readily be reduced to the former (see Remark 2.15 below). For each ξ0 ∈ Γ, we choose aneighborhood Nξ0 of ξ0. Subsequently, we introduce, in Nξ0 , a local coordinate coordinate system (ν, τ),where ν is the normal vector to Γ at ξ0 and τ the tangent vector to Γ at ξ0. This can be done due to thesmoothness requirements imposed upon the boundary Γ. Dν will denote the directional derivative in thedirection of ν. It is known (see e.g., [16, 27]) that for Nξ0 sufficiently small,

∆ = D2ν + R

(ν(x), τ(x),

∂

∂τ

), (2.18)

where ν(x) and τ(x) are the components of x in the directions of ν and τ ; and where in Nξ0 , R(ν(x), τ(x), ∂∂τ )

is a second order, strongly elliptic operator in the tangential direction τ . In the sequel, we shall frequentlydenote this local normal and tangential representation of the Laplacian as simply ∆.

Let [A,B] denote the commutator between operators A and B. Furthermore, let φ denote a C∞(Ω)–function whose support is contained in Nξ0 ; we will also denote the associated (multiplication) ΨDO byφ. We will use this φ to localize the wave equation (2.1); to wit, the function φz satisfies

(φz)tt = ∆(φz)− [∆, φ]z + φLz on Q; (2.19)Dν(φz) = φg + (∇φ · ν)z + φMz on Σ.

11

In addition, we introduce the Fourier transform variables η ∈ Rn−1 (corresponding to tangential τ) andσ − iγ (corresponding to t), where σ ∈ R1 and γ > 0 is fixed. Given the spatial (time) dual variable, welet Ds

τ (Dsσ) denote the ΨDO with symbol

√|η|2 + 1

s(resp.,

√|σ|2 + 1

s, see e.g. [36]). With this ΨDO,

we make the change of variablew := Dτ

3/2(φz), (2.20)

where z solves (2.15). Applying D3/2τ to the PDE (2.19), we have that w, supported in a neighborhood

of the boundary Γ, satisfies the following equation:

wtt = ∆w + [Dτ3/2,∆]φz −Dτ

3/2[∆, φ]z + Dτ3/2 (φLz) on R× Ω;

Dνw = D3/2τ (φg) + Dτ

3/2(∇φ · ν)z + Dτ3/2 (φMz) on R× Γ. (2.21)

We now evaluate the commutator terms which appear in (2.21). We begin with the commutators inthe PDE. First, note that by the algebra for ΨDO’s we have

[Dτ3/2,∆]φz = [Dτ

3/2,∆]Dτ−3/2w;

Dτ3/2[∆, φ]z = Dτ

3/2[∆, φ]Dτ−3/2Dτ

3/2z. (2.22)

By (2.18) and the commutator and product rules (see e.g., Theorems 4.3 and 4.4 of [36]),

[Dτ3/2, ∆]Dτ

−3/2 ∈ OPS1(Ω);[∆, φ] ∈ OPS1(Ω). (2.23)

As for the commutators in the boundary functions of (2.21), we similarily have

Dτ3/2 (∇φ · ν) z = Dτ

3/2(∇φ · ν)Dτ−3/2

Dτ3/2z. (2.24)

Setting

f0(z) := [Dτ3/2,∆]φz −Dτ

3/2[∆, φ]z + D32τ (φLz) ;

g0(z) := Dτ3/2(∇φ · ν)z + D

32τ (φMz) ,

we rewrite (2.21) as

wtt = ∆w + f0(z) in Q

Dνw = g0(z) + D3/2τ (φg) on Σ. (2.25)

Using (2.22)–(2.24), (2.16), and the Sobolev continuity of ΨDO operators (see [16, 36]), we obtain thefollowing (pointwise in time) estimate:

|f0(z)(t)|0,Ω ≤ C[|w(t)|1,Ω + |Dτ

3/2z(t)|1,Ω

];

|g0(z)(t)| 12 ,Γ ≤ C|Dτ

3/2z(t)| 12 ,Γ. (2.26)

Now applying Theorem 3 in [29] (with k = 0 therein) to equation (2.25), the regularity estimatesin (2.26), classical trace theory (see [20]), and the fact that w has zero initial data, we obtain for all0 < t ≤ T :

|w|21,Qt+ |w(t)|21,Ω + |Dτ

−1/2w|Γ|21,Σt≤ CT

[|f0(z)|20,Qt

+∥∥∥g0(z) + D3/2

τ φg∥∥∥2

L2(0,t;H1/2(Γ))

]≤ CT

[‖w‖2L2(0,t;H1(Ω)) +

∥∥∥Dτ3/2z

∥∥∥2

L2(0,t;H1(Ω))+∥∥∥Dτ

3/2φg∥∥∥2

L2(0,t;H1/2(Γ))

]. (2.27)

12

To refine the right hand side, we have

w = D32τ φz = φD

32τ z +

[D

32τ , φ

]D− 3

2τ D

32τ z,

and with this and the Sobolev continuity of ΨDO’s, the above inequality becomes

|w|21,Qt+ |w(t)|21,Ω + |Dτ

−1/2w|Γ|21,Σt≤ CT

[∥∥∥Dτ3/2z

∥∥∥2

L2(0,t;H1(Ω))+ ‖g‖2L2(0,t;H2(Γ)

]. (2.28)

It should be noted that in (2.28), the constant CT depends upon the choice of Nξ0 .

Let us now say that φ = φi is the ith component of the resolution of the identity over a relativelycompact set containing Ω, corresponding to a partition of unity, and with the support of each φi beingsmall enough so that in the normal and tangential coordinates, ∆|supp(φi)

has the general elliptic rep-

resentation in (2.18). Using w = Dτ3/2(φz), we can apply the same argument to each φi (noting that

analysis for those φi supported in the interior of Ω is much simpler) and sum up the estimates in (2.28)with respect to i. This gives for all 0 < t ≤ T :

|Dτ3/2z|21,Qt

+ |Dτ3/2z(t)|21,Ω + |Dτz|Γ|21,Σt

≤ CT

[∥∥∥Dτ3/2z

∥∥∥2

L2(0,t;H1(Ω))+ ‖g‖2L2(0,T ;H2(Γ))

], (2.29)

(where here, we are implicitly using z =∑

φiz). Applying Gronwall’s inequality to (2.29), we obtain forall 0 < t ≤ T ,

|Dτ3/2z(t)|21,Ω ≤ CT ‖g‖2L2(0,T ;H2(Γ)) . (2.30)

In turn, this inequality applied to (2.29) and (2.28) gives

|Dτ3/2z|21,Q + |Dτz|Γ|21,Σ + |w|21,Q + |Dτ

−1/2w|Γ|21,Σ ≤ CT ‖g‖2L2(R;H2(Γ)) . (2.31)

(Alternatively, we could have obtained (2.31) by taking the parameter γ “large enough” in the formulationof Theorem 1 in [29].) Applying this estimate to (2.26) yields

|f0(z)|0,Q + ‖g0(z)‖L2(R;H1/2(Γ)) ≤ CT ‖g‖L2(R;H2(Γ)) . (2.32)

Using the notationf1 := f0(z); g1 := Dτ

3/2 (φg) + g0(z), (2.33)

we rewrite equation (2.25) as

wtt = ∆w + f1 on Q,Dνw = g1 on Σ, (2.34)

whereby (2.32), the “forcing term” f1 and boundary term g1 satisfy

|f1|0,Q + ‖g1‖L2(R;H1/2(Γ)) ≤ CT ‖g‖L2(R;H2(Γ)) . (2.35)

It is the equation (2.34) which now constitutes a basis for further analysis. In this analysis, we willfrequently invoke the following estimate, which follows immediately from (2.31):

|w|1,Q +∣∣∣Dτ

−1/2wt|Γ∣∣∣0,Σ

≤ CT ‖g‖L2(0,T ;H2(Γ)) . (2.36)

We first analyze (2.34) with a generic choice of φ (recall that w is associated with φ by (2.20)), and afterobtaining the appropriate bounds for w, we will again use the particular partition of unity φi chosenabove. Our next step is microlocalization.

13

2.3.2 Microlocalization (Continuation of the Proof of Lemma 2.8)

Having introduced the transform variables η, σ, we shall microlocalize the problem to a conic neighborhood(τ, t, η, σ) ∈ T ∗(Σ). Let c > 1 and

R1 := (η, σ) : |σ| > 2c|η|; R3 := (η, σ) : |σ| < c|η|; R2 = R \ R1 ∪R3.

The microlocalization is done with a help of a “localizer”, by which we mean here the zero order ΨDOΛ ∈ OPS0(Σ) (see [16]), with associated symbol λ ∈ S0(Σ) given by

λ(η, σ) :=

1 in R1

a C∞ function with range [0, 1] in R2

0 in R3.(2.37)

We shall study the behaviour of the solution w to (2.34), particularly on Γ, in each sector Ri. To thisend we consider the decomposition

w = Λw + (I − Λ)w.

The idea here is that in the sector R1 the Lopatinski condition is satisfied, so the trace of w willaccordingly have “better” regularity in R1. Lacking the Lopatinski condition in the characteristic sectorR2 ∪R3, we instead take advantage of extra spatial regularity prescribed on Γ (that is, the regularity ofg1), and the fact that in this sector “space dominates time”, so as to compensate for the loss of ellipticity.

We formalize this idea in the proofs of the following Propositions:

Proposition 2.10 If w is the solution of the wave equationwtt = ∆w + f1 on Q

∂w

∂ν= g1 on Σ,

(2.38)

where arbitrary f1 ∈ L2(Q) and g1 ∈ L2(0, T ;H12 (Γ)), then

|Λ wt|Γ|0,Σ ≤ C[|w|1,Q + |f1|0,Q

].

Proposition 2.11 If w is the solution of (2.38), then

|(I − Λ) w|Γ| 35 ,Σ ≤ C[|w|1,Q + |f1|0,Q + ‖g1‖

L2(0,T ;H12 (Γ))

].

The proofs of both Propositions are given in the next subsection. Taking for granted the validity ofPropositions 2.10 and 2.11, we shall continue with the proof of Lemma 2.8 (and thus Lemma 2.3).

The estimate in Proposition 2.10, in combination with (2.35) and (2.36), yields

|Λwt|Γ|0,Σ ≤ CT ‖g‖L2(R;H2(Γ)) . (2.39)

With this estimate in mind, we recall the definition of w to write

Dτ3/2Λ(φzt|Γ) = Dτ

3/2ΛDτ−3/2Dτ

3/2(φzt|Γ)

= Dτ3/2[Λ, Dτ

−3/2]wt|Γ + Λwt|Γ

=(Dτ

3/2[Λ, Dτ

−3/2]Dτ

1/2)

Dτ−1/2(wt|Γ) + Λwt|Γ.

14

Norming and majorizing both sides of this expression with the commutator rule, (2.36) and (2.39), weobtain

‖Λφzt|Γ‖L2(R;H3/2(Γ)) ≤ CT ‖g‖L2(R;H2(Γ)) .

Since the dimension of Γ is 1 or 2, the Sobolev Embedding theorem gives H3/2(Γ) ⊂ C(Γ), and so

‖Λφzt|Γ(x0)‖L2(R) ≤ CT ‖g‖L2(R;H2(Γ)) for fixed x0 ∈ Γ. (2.40)

To handle the quantity (I − Λ)φzt, we first write out the equality

D35τ (I − Λ)D

12τ φ zt|Γ = D

35τ (I − Λ)

d

dtD

12τ φ z|Γ

= D35τ (I − Λ)

d

dtD

12τ D

− 32

τ D32τ φ z|Γ (2.41)

=[D

35τ (I − Λ),

d

dtD−1

τ

]w|Γ +

d

dtD−1

τ D35τ (I − Λ) w|Γ .

To majorize this last expression, we note initially that Proposition 2.11, (2.35) and (2.36) give collectively

|(I − Λ)w|Γ|3/5,Σ ≤ CT ‖g‖L2(R;H2(Γ)) . (2.42)

We next notice that on the supp(1− λ) we have the dominance of the tangential transform variable ηover that of time; i.e.,

|σ| ≤ 2c|η| on supp (I − λ) .

We thus conclude thatd

dtDτ

−1 is bounded on supp (I − Λ) . (2.43)

Hence, taking the L2-norm in (2.41), and majorizing the resulting expression by means of (2.43), (2.42),trace theory, (2.36), and the Sobolev continuity of ΨDO’s, we obtain∣∣∣D3/5

τ (I − Λ)Dτ1/2φzt|Γ

∣∣∣0,Σ

≤ CT ‖g‖L2(R;H2(Γ)) . (2.44)

Moreover,D3/5

τ

[D1/2

τ , (I − Λ)]φzt|Γ = D3/5

τ

[D1/2

τ , (I − Λ)]D−1

τ

(D−1/2

τ wt|Γ)

.

Using (2.36) and the commutator rule, we then have∣∣∣D3/5τ [D1/2

τ , (I − Λ)]φzt|Γ∣∣∣0,Σ

≤ CT ‖g‖L2(0,T ;H2(Γ)). (2.45)

Combining (2.44) and (2.45) with another application of the commutator rule now gives

|(I − Λ)φzt|Γ|L2(R;H11/10(Γ)) ≤ CT ‖g‖L2(R;H2(Γ)) . (2.46)

By the Sobolev’s Embedding H11/10(Γ) ⊂ C(Γ) (recall dim Γ ≤ 2), so we conclude that

‖(I − Λ)φzt|Γ(x0)‖L2(R) ≤ CT ‖g‖L2(R;H2(Γ)) for all x0 ∈ Γ. (2.47)

Combining (2.40) and (2.47) gives the pointwise (in space) estimate

‖φzt|Γ(x0)‖L2(R) ≤ ‖Λφzt|Γ(x0)‖L2(R) + ‖(I − Λ)φzt|Γ(x0)‖L2(R) ≤ C ‖g‖L2(R;H2(Γ)) .

This inequality holds with φ = φi, for every φi in our partition of unity. Summing over i leads to thefinal estimate

‖zt|Γ(x0)‖L2(0,T ) ≤ C ‖g‖L2(R;H2(Γ)) for all x0 ∈ Γ.

This will complete the proof of Lemma 2.8 (and thus Lemma 2.3), as soon as we prove Propositions 2.10and 2.11. 2.

15

2.3.3 The Proof of Proposition 2.10

We start by applying the localizing operator Λ to both sides of the first equation in (2.38). This leads tothe PDE

Λwtt = ∆Λw + Λf1 + [Λ,∆]w (2.48)

By the commutator rule and the Sobolev regularity of ΨDO’s we have

|[Λ,∆]w|0,Q ≤ C|w|1,Q.

Therefore, denotingf := Λf1 + [Λ,∆]w,

we have (since Λ ∈ L(Hs(Q)))|f |0,Q ≤ C[|w|1,Q + |f1|0,Q]. (2.49)

With this new forcing function f , we rewrite (2.48) as

Λwtt = ∆Λw + f . (2.50)

To this equation, we next apply a multiplier method in very much the same way as in [18, 21, 22].We merely sketch the method here; for a full proof, see Appendix A of Triggiani [37]. We set the normalvector ν = [ν1, . . . νn]T , and moreover, for x ∈ Rn let H be the n× n matrix defined by

H(x) =[∂νi(x)∂xj

]i,,j

.

We multiply both sides of the equation (2.50) by ν · ∇Λw = DνΛw and subsequently integrate by parts.Manipulations of this equation which involve the use of Green’s Formula and the Divergence Theoremeventually give:

12

∫Σ

[|Λwt|2 + |DνΛw|2 −

∣∣∣∣ ∂

∂τΛw

∣∣∣∣2]

dΣ =∫

Q

H∇Λw · ∇ΛwdQ

+12

∫Q

(|Λwt|2 − |∇Λw|2

)div(ν)dQ−

∫Q

fDνΛwdQ

(here we have also used the fact that w has zero initial data).

Estimating the right hand side of this expression, using Cauchy-Schwartz, the estimate (2.49), and factthat Λ ∈ L(Hs(Q)), we obtain

12

∫Σ

[|Λwt|2 + |DνΛw|2 −

∣∣∣∣ ∂

∂τΛw

∣∣∣∣2]

dΣ ≤ C[|w|21,Q + |f1|20,Q

].

Rewriting the inequality above in terms of the transform variables (σ, η) gives∫Rn

[σ2 − |η|2||λ(σ, η)w(σ, η)|2]dηdσ + |DνΛw|20,Σ ≤ C[|w|21,Q + |f1|20,Q

], (2.51)

where w is the Fourier (in the tangential and time variable) transform of w|Γ. Note that by (2.37), thesymbol σ2 − |η|2 is elliptic in σ on supp(λ). Thus (2.51) yields∫

Rn

[σ2|λ(σ, η)w(σ, η)|2]dηdσ ≤ C[|w|21,Q + |f1|20,Q

],

or ∫Σ

|Λwt|2dΣ ≤ C[|w|21,Q + |f1|20,Q

].

This completes Proposition 2.10. 2.

16

Remark 2.12 We notice that the proof of Proposition 2.10 did not use any boundary conditions satisfiedby Λw. Accordingly, the same argument will apply if we replace Σ in Proposition 2.10 by any othernoncharacteristic time-like surface.

2.3.4 Proof of Proposition 2.11

The proof of Proposition 2.11 is based on sharp trace regularity results established for solutions to theNeumann problem in [21, 22, 23] . These results improve the classical results [20, 29] by “1/6 derivative”.As we shall see, this improvement is critical to the analysis below.

To begin, we apply the operator I − Λ to both sides of the first equation in (2.38), leading to

(I − Λ)wtt = ∆(I − Λ)w + (I − Λ)f1 + [I − Λ,∆]w. (2.52)

Since the commutator term [I − Λ,∆] ∈ OPS1(Q), we then have, by the Sobolev continuity for ΨDO’s,

|[I − Λ,∆]w|0,Q ≤ C|w|1,Q.

Therefore, denotingf := (I − Λ)f1 + [I − Λ,∆]w,

we have|f |0,Q ≤ C[|w|1,Q + |f1|0,Q]. (2.53)

We thus have from (2.38) the wave equation

(I − Λ)wtt = ∆(I − Λ)w + f

Dν(I − Λ)w = (I − Λ)g1. (2.54)

To analyze (2.54), we recall the sharp regularity results for the Neumann-to-Dirichlet map defined onsmooth domains:

Theorem 2.13 (see [23]) Let A(x, ∂) be an arbitrary second order elliptic operator with smooth coeffi-cients, and let ∂

∂νA denote the associated conormal derivative. For the system

ptt +A(x, ∂)p = f in Q;∂p

∂νA= h on Σ;

p(t = 0) = pt(t = 0) = 0 on Ω, (2.55)

the following regularity holds:

(i) The mapping [f, h = 0] → [p, p|Σ] is bounded from L2(Q) into[H1(Q),H

35 (Σ)

].

(ii) The mapping [f = 0, h] → [p, p|Σ] is bounded from L2(Σ) into[H

35−ε(Q),H

15−ε(Σ)

].

(iii) The mapping [f = 0, h] → p|Σ is bounded from H1/2(Σ) into H710−ε(Σ).

Parts (i)–(ii) of this theorem is a direct statement from Theorems 3.3 and 3.1 in [23]. The result (iii)comes from interpolating between Theorems 3.1 and 3.2(b).

17

Remark 2.14 We note that in comparison to the Dirichlet problem, for which the Lopatinski conditionholds, the traces of the solutions lose some measure of differentiability. Indeed, for the Dirichlet problem,a boundary trace statement equivalent to that for (2.55) would be that p|Σ ∈ H1(Σ). Thus, the Neumannsolutions lose “3/10+ ε” derivative. The fact that this is unavoidable (when the dimension is higher thanone) follows from elementary computations performed on special domains like spheres or parallelepipeds(see [23]).

We now apply these trace results directly to the analysis of (2.54). Since on the support of I − Λ thespace tangential variable dominates that of time (i.e., |σ| ≤ 2c|η| on supp(1− λ)),

g1 ∈ L2(R;H1/2(Γ)) ⇒ (I − Λ)g1 ∈ H1/2(Σ).

Hence, using Theorem 2.13(iii) we obtain that

[f = 0, g1] → (I − Λ)w|Σ ∈ H710−ε(Σ). (2.56)

In addition, as f ∈ L2(Q), we can use (2.56) and Theorem 2.13(i) to obtain

|(I − Λ)w|Γ|3/5,Σ ≤ C[|f |0,Q + |(I − Λ)g1|1/2,Σ] ≤ C[|f |0,Q + ‖g1‖L2(R,H1/2(Γ)) . (2.57)

Proposition 2.11 now follows from estimating the right hand side of (2.57) with (2.53) . 2

Remark 2.15 In order to obtain the same result valid for an arbitrary point x0 ∈ Ω (rather than justx0 ∈ Γ), we proceed as follows: For a given x0 we choose a noncharacteristic time-like surface, say Σ1,such that x0 ∈ Σ1. As noted in Remark 2.12, the result of Proposition 2.10 holds for any such surface(i.e. replace Σ with Σ1). The reason for this is that the boundary conditions are not used in the proofthereof. As for proving the counterpart of the result of Proposition 2.11 for x0 ∈ Σ1: instead of invokingTheorem 2.13, one can use Theorem 2 and 3 in [35] which gives the same trace regularity for any H1(Q)solutions, and any time-like noncharacteristic surface without any a priori knowledge of the boundaryconditions. With these modifications, the rest of the argument is the same.

3 Proof of Theorem 1.4

3.1 Two Supporting Results

To begin, we give a regularity result for the velocity of the beam component, which essentially followsfrom Proposition 2.1.

Proposition 3.1 For u ∈ L2(0, T ;U) and X0 = [z0, z1, v0, v1]T ∈ H, the component vt of the solution

[z, zt, v, vt]T to (1.2) satisfies

‖vt‖H

12 (0,T ;L2(Γ0))

≤ CT

[‖u‖L2(0,T ;U) + ‖X0‖H

].

Proof: Taking norms in the third equation of (1.2), we have first

‖vtt‖L2(0,T ;H−2(Γ0))=∥∥−∆2v −∆2vt − zt|Γ0 + Bu

∥∥L2(0,T ;H−2(Γ0))

. (3.1)

To majorize this relation, we note from Proposition 2.1 that[‖[v, vt]‖[L2(0,T ;H2

0 (Γ0))]2+ ‖zt|Γ‖L2(0,T ;H−2(Γ0))

]≤ CT

[‖u‖L2(0,T ;U) + ‖X0‖H

]. (3.2)

18

Majorizing (3.1) with (3.2) and (1.1) we obtain

‖vtt‖L2(0,T ;H−2(Γ0))≤ CT

[‖u‖L2(0,T ;U) + ‖X0‖H

];

and so‖vt‖H1(0,T ;H−2(Γ0))

≤ CT

[‖u‖L2(0,T ;U) + ‖X0‖H

]. (3.3)

Interpolation between (3.3) and (3.2) now gives the asserted result.2

In the sequel we will make frequent use of the parametrized Sobolev norms introduced in [29]. To wit,for positive integer m and function f defined on R× (0,∞)× Rn−1 (time and space), we set

|f |2s,γ :=∑

i+j+k+|α|=m

∫R

∫ ∞

0

∫Rn−1

∣∣∣e−γtγi

Djt D

kxDα

y f(t, x, y)∣∣∣2 dydxdt. (3.4)

We also define the boundary norms 〈f〉s,γ , by having for arbitrary real s and function f defined onR× Rn−1,

〈f〉2s,γ :=∑

i+j+|α|=m

∫R

∫Rn−1

∣∣∣e−γtγi

Djt D

αy f(t, x, y)

∣∣∣2 dydt. (3.5)

For all real s, the norms |·|s,γ and 〈·〉s,γ can subsequently be defined by interpolation.

With these norms, we next give a nontrivial regularity result for the wave equation with prescribedNeumann data and forcing term, which is due to Miyatake.

Lemma 3.2 ([29], Theorem 1) Let z(t, x) satisfy the wave equationztt = ∆z + f on Q∂z

∂ν= g on ⊀

where f ∈ H12 (Q), and g ∈ H

12 (R;H

12 (Γ)) ∩ L2(R;H1(Γ)). Then there exists a γ0 > 0, such that for

γ > γ0, z satisfies the estimate

γ |z|232 ,γ ≤

C

γ

(|f |21

2 ,γ +⟨D

12τ g⟩2

12 ,γ

)(where, above, we are implicitly taking z =

∑i φiz, φi being the aforesaid resolution of the identity).

3.2 Sharp Regularity Results for Wave Equations

In this section, we prove two regularity result for wave equations under the action of boundary data ofprescribed smoothness, in time and space. This result is in support of the proof of Theorem 1.4.

Lemma 3.3 For Ω a smooth, bounded subset of Rn, n ≤ 3, Let z(t, x) satisfy the following wave equationztt = ∆z on (0, T )× Ω∂z

∂ν= g on (0, T )× Γ

z(0, ·) = 0, zt,(0, ·) = 0.

(3.6)

Here the boundary data g = g1 + g2, where g1 ∈ Gη ≡[L2(0, T ;Hη(Γ)) ∩H

12 (Σ) ∩ C([0, T ];L2(Γ))

], and

g2 ∈ Hη(0, T ;L2(Γ)), where (to be specified) η > 12 . Moreover, we assume g(t = 0) = 0.

19

(i) Let dimension n = 2 or 3, and parameter η = 1. Then the corresponding solution [z, zt] ∈

C

([0, T ]; H

32 (Ω)R × H

12 (Ω)R

), with the accompanying estimate

‖[z, zt]‖C([0,T ];H

32 (Ω)×H

12 (Ω))

≤ CT

[‖g1‖G1

+ ‖g2‖H1(0,T ;L2(Γ))

]. (3.7)

(ii) Let dimension n = 2 and parameter η = 32 − ε. Then for fixed x0 ∈ Ω, we have that the velocity

zt satisfies the pointwise (in space) estimate

‖zt(·,x0)‖L2(0,T ) ≤ CT

(‖g1‖G 3

2−ε+ ‖g2‖

H32−ε(0,T ;L2(Γ))

). (3.8)

Remark 3.4 If we define the operator

A1 :=[

0 I−AN 0

]; D(A1) = D(AN )× H1(Ω)

R,

where AN is the elliptic operator given in (1.8), then A1 : D(AN ) ⊂ H1(Ω)R × L2(Ω)

R → H1(Ω)R × L2(Ω)

R

generates a C0 unitary groupeA1t

t≥0

on H1(Ω)R × L2(Ω)

R , and by extension, on D(A34N )× H

12 (Ω)R . With

these dynamics, the solution to (3.6) can be written as[z(t)zt(t)

]= eA1t

[z0

z1

]+∫ t

0

eA1(t−s)

[0

ANNg(s)

]ds. (3.9)

This identification will be used in what follows.

Proof of (i): To start, we again extend the function g for t < 0; and so given the compatibilitycondition g(t = 0) = 0, we have that the Sobolev regularity of g with this extension is retained at themicrolocal level. That is to say, with the partition of unity φi introduced in Subsection 2.3.1, and thespace defined in (3.5) we have that φig = φig1 + φig2, where

φig1 ∈ H 12 ,γ(Σ); D

12τ φig1, Dσφig2 ∈ H0,γ(Σ). (3.10)

We next recall the localizing function λ(η, σ), and its corresponding ΨDO Λ which were introduced inSubsection 2.3.2 (see (2.37)). With this localizer, we will now analyze each component of the decompo-sition

φiz = Λφiz + (I − Λ)φiz. (3.11)

Handling the component Λφiz: Applying the ΨDO Λφi to (3.6) we obtain(Λφiz)tt = ∆ (Λφiz) + [Λφi,∆] z on Q∂

∂ν(Λφiz) = Λφig + Λ (∇φi · ν) z on Σ.

(3.12)

We decompose Λφiz into Λφiz ≡ w(a) + w(b), where w(a)tt = ∆w(a) + [Λφi,∆] z on Q∂

∂νw(a) = Λ (∇φi · ν) z on Σ;

(3.13)

w(b)tt = ∆w(b) on Q∂

∂νw(b) = Λφig on Σ.

(3.14)

20

We note first, by interpolation, that if z ∈ H32 (Q), then z ∈ H

12 (0, T ;H1(Ω)). Coupling this with

trace theory, and using the spaces defined in (3.4) and (3.5), we then have that

z ∈ H32 (Q) ⇒ D

12τ (∇φi · ν) z

∣∣∣Σ∈ H 1

2 ,γ(Σ). (3.15)

To handle the wave equation in (3.13), one can then directly appeal to Lemma 3.2, followed by a use ofthe commutator rule and (3.15) to obtain the following estimate for γ > 0 large enough:

γ∣∣∣w(a)

∣∣∣232 ,γ

≤ C

γ

[|[Λφi,∆] z|21

2 ,γ +⟨D

12τ Λ (∇φi · ν) z

⟩2

12 ,γ

]≤ C

γ|z|23

2 ,γ . (3.16)

To handle the wave equation in (3.14), we note that in supp(λ), “time dominates space”, and so wededuce from from (3.10) that

[g1, g2]T ∈ G1 ×H1(0, T ;L2(Γ)) ⇒ Λφig ∈ H 1

2 ,γ(Σ).

Recalling now from Section 2.3.3 that the wave operator satisfies the Lopatinski condition in supp(λ), wecan apply the results of [21] (see also [33]) to show that w(b) ∈ H 3

2 ,γ(Q), with the estimate∣∣∣w(b)∣∣∣232 ,γ

≤ C0 ‖Λφig‖212 ,γ ≤ Cγ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.17)

Combining (3.16) and (3.17) then gives

γ |Λφiz|232 ,γ ≤

C

γ|z|23

2 ,γ + Cγ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.18)

Handling the component (I − Λ)φiz: Here, we will need a further decomposition. To this end, wedefine the sectors

Re := (η, σ) : |σ| < c0|η|; Rne := (η, σ) : |η| < |σ||; Rtr = R \ Re ∪Rne,

where the constant c0 is chosen small enough so that on Re, the symbol corresponding to ∆ − ∂tt iselliptic in η. Therewith we define the (elliptic) localizer λe(η, σ) by

λe(η, σ) =

1 in Re

a C∞ function with range [0, 1] in Rtr

0 in Rne,(3.19)

with Λe ∈ OPS0(Ω) being its corresponding ΨDO. We now write (I − Λ) φiz =(I − Λ

)φiz + Λeφiz,

whereΛ ≡ Λ + Λe. (3.20)

The symbol λ being so defined, we then have from (2.37) and (3.19)

supp(1− λ

)⊂ (η, σ) : c1|η| ≤ |σ| ≤ c2|η| , (3.21)

for appropriately chosen constants c1 and c2.

(a) Handling the subcomponent(I − Λ

)φiz: Applying (I− Λ)φi to (3.6), we see that (I− Λ)φiz solves

the PDE (I − Λ)φiztt = ∆(I − Λ)φiz +[(I − Λ)φi,∆

]z on Q

∂

∂ν

((I − Λ)φiz

)= (I − Λ)φig + (I − Λ) (∇φi · ν) z on Σ.

(3.22)

21

¿From (3.21), we see that on supp(1− λ), the tangential spatial variable “is comparable” to that of time.In consequence,

[g1, g2]T ∈ G1 ×H1(0, T ;L2(Γ)) ⇒ D

12τ (I − Λ)φig ∈ H 1

2 ,γ(Σ). (3.23)

Using this information and (3.15), we can again appeal to Lemma 3.2 to have for large enough γ > 0,

γ∣∣∣(I − Λ)φiz

∣∣∣232 ,γ

≤ C

γ

(∣∣∣[(I − Λ)φi,∆]z∣∣∣212 ,γ

+⟨D

12τ

[(I − Λ) (∇φi · ν) z + (I − Λ)φig

]⟩2

12 ,γ

)≤ C

γ|z|23

2 ,γ + Cγ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.24)

(b) Handling the component Λeφiz: Applying Λeφi to (3.6), we have the PDE(Λeφiz)tt = ∆ (Λeφiz) + [Λeφi,∆] z on Q∂

∂ν(Λeφiz) = Λeφig + Λe (∇φi · ν) z on Σ.

Here we use the fact that on supp(Λe), the operator ∆ − ∂tt is elliptic, and so by classical elliptictheory we obtain the estimate

|Λeφiz|232 ,γ ≤ C

[〈Λeφig + Λe (∇φi · ν) z 〉20,γ + |[Λeφi,∆] z |2− 1

2 ,γ

]≤ C

[|g|20,Σ + |z |21

2 ,γ

]≤ C |g|20,Σ ≤ C

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

), (3.25)

where above, we have also used the a priori regularity given in Theorem 2.13(ii). Combining the decom-position (3.20) with (3.24) and (3.25) thus yields

γ |(I − Λ)φiz|232 ,γ ≤ γ

∣∣∣(I − Λ)φiz∣∣∣232 ,γ

+ γ |Λeφiz|232 ,γ

≤ C

γ|z|23

2 ,γ + Cγ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.26)

With (3.18) and (3.26), we hence get the local estimate

γ |φiz|232 ,γ ≤

C

γ|z|23

2 ,γ + Cγ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.27)

Now using the decomposition z =∑

φiz and taking γ large enough in this inequality, we obtain

|z|232 ,Q ≤ CT,γ

(‖g1‖2G1

+ ‖g2‖2H1(0,T ;L2(Γ))

). (3.28)

Interpolation with this inequality further gives that [z, zt]T ∈ L2(0, T ;H

32 (Ω)×H

12 (Ω)), with continuous

dependence on the data. The improvement to continuity in time now follows from the “lifting” argumentinvoked in [19]. This completes the proof of (i).

Proof of (ii): Again, owing to the compatibility condition g(t = 0) = 0, we extend g = g1 + g2 byzero to all of time and space via the partition of unity φi, thereby preserving the Sobolev regularity ofthis data. Now invoking the localizer Λe of (3.19), we apply the decomposition φiz = Λeφiz +(I−Λe)φizto (3.6) to obtain

(Λeφiz)tt = ∆ (Λeφiz) + [Λeφi,∆] z on Q∂

∂ν(Λeφiz) = Λeφig + Λe (∇φi · ν) z on Σ;

(3.29)

(I − Λe)φiztt = ∆(I − Λe)φiz + [(I − Λe)φi,∆] z on Q∂

∂ν((I − Λe)φiz) = (I − Λe)φig + (I − Λe) (∇φi · ν) z on Σ.

(3.30)

22

Handling the wave equation (3.29): With p ≡ (Λeφiz)t, then p solves the wave equationptt = ∆p + [Λeφi,∆] zt on Q∂p

∂ν= Λeφigt + Λe (∇φi · ν) zt on Σ.

(3.31)

In analyzing the data of this equation, we have by the regularity posted in part (i) of the theorem that

[g1, g2]T ∈ G1 ×H1(0, T ;L2(Γ)) ⇒ z ∈ H32 (Q). (3.32)

This, together with the algebra for ΨDO’s gives

|[Λeφi,∆] zt|H− 12 ,γ

(Q) ≤ CT

(‖g1‖G1

+ ‖g2‖H1(0,T ;L2(Γ))

). (3.33)

Moreover, by (3.32), the fact that space dominates time on supp(Λe) (see (3.19)) and trace theory, wehave

[g1, g2]T ∈ G1 ×H1(0, T ;L2(Γ))⇒ Λeφigt + Λe (∇φi · νzt) ∈ H0,γ(Σ). (3.34)

To solve the problem (3.31), we can now use the fact that supp(Λe) was chosen so that ∆− ∂tt is elliptictherein; so as to have by classical elliptic theory that (Λeφiz)t = p ∈ H 3

2 ,γ(Q). This, combined with(3.33) and (3.34), gives

|Λeφizt|H 32 ,γ

(Q) ≤ CT

(‖g1‖G1

+ ‖g2‖H1(0,T ;L2(Γ))

). (3.35)

Combining the Sobolev Embedding Theorem (for dim (Ω) ≤ 2) with the estimate above now gives forarbitrary x0 ∈ Ω,

‖Λeφizt(x0)‖L2(0,T ) ≤ CT

(‖g1‖G1

+ ‖g2‖H1(0,T ;L2(Γ))

). (3.36)

Handling the wave equation (3.30): Here we use the fact that on supp(1− λe), time dominates space,so that

[g1, g2]T ∈ G 32−ε ×H

32−ε(0, T ;L2(Γ))

⇒ (I − Λe)φig ∈ L2(0, T ;H32−ε(Γ)).

Moreover, [(I − Λe)φi,∆] ∈ OPS1(Ω), and (I − Λe) (∇φi · ν) z ∈ OPS0(Γ). Appealing to Lemma 2.8and Remark 2.9 thus yields

‖(I − Λe)φizt(·,x0)‖L2(0,T ) ≤ CT ‖(I − Λe)φig‖L2(0,T ;H

32−ε(Γ))

≤ CT

(‖g1‖G 3

2−ε+ ‖g2‖

H32−ε(0,T ;L2(Γ))

). (3.37)

Finally, combining (3.36) and (3.37) with the partition of unity φi gives the asserted point evaluation,thereby completing the proof of (ii).

3.3 Proof of Theorem 1.4(i)-(ii)

The proof of Theorem 1.4(i)-(ii) will readily follow as a deduction from the following Lemma.

23

Lemma 3.5 For initial data X0 ∈ X and control u ∈ L2(0, T ;L2(Γ0)), one has that the correspondingsolution [z, zt, v, vt] of (1.2) is an element of C([0, T ];X ), with the estimate

‖[z, zt, v, vt]‖C([0,T ];X ) ≤ C(‖X0‖X + ‖u‖L2(0,T ;U)

). (3.38)

Proof: By virtue of the definition of X in (1.14), and the estimate in Propostion 2.1, we can restrictour attention to the wave component of (1.2).

Step 1 (Demonstration of L2–time regularity).

(1.A). Taking initial data X0 = [z0, z1, v0, v1] from X and u ∈ L2(0, T ;U), we can write the wavecomponent [z, zt] via the representation (see (3.9))[

z(t)zt(t)

]= eA1t

[z0

z1

]+∫ t

0

eA1(t−s)

[0

ANNvt(s)

]ds. (3.39)

Integrating by parts, we have then[z(t)zt(t)

]= eA1t

[z0

z1

]+∫ t

0

eA1(t−s)A1A−11

[0

ANNv1

]ds +

∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds

= eA1t

[z0

z1

]−∫ t

0

d

dseA1(t−s)A−1

1

[0

ANNv1

]ds +

∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds

= eA1t

[z0

z1

]+∫ t

0

d

dseA1(t−s)

[Nv1

0

]ds +

∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds

= eA1t

[z0 −Nv1

z1

]+[

Nv1

0

]+∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds

=[

z(0)(t)z(0)t (t)

]+[

z(1)

z(1)t

]+[

z(2)(t)z(2)t (t)

], (3.40)

where [z(0)(t)z(0)t (t)

]= eA1t

[z0 −Nv1

z1

];

[z(1)

z(1)t

]=[

Nv1

0

];[

z(2)(t)z(2)t (t)

]=

∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds. (3.41)

To estimate[z(0), z

(0)t

]: As the initial data was taken from X , then using the invariance of the group

eA1t

t≥0, we have∥∥∥∥[ z(0)

z(0)t

]∥∥∥∥C([0,T ];D(A

34N )×D(A

14N ))

≤∥∥∥∥[ z0 −Nv1

z1

]∥∥∥∥D(A

34N )×D(A

14N )

≤ ‖X0‖X . (3.42)

To estimate[z(1), z

(1)t

]: By (1.11), we have∥∥∥∥[ z(1)

z(1)t

]∥∥∥∥H

32 (Ω)×H

12 (Ω)

≤ C ‖v1‖L2(Γ0)≤ C ‖X0‖H . (3.43)

24

To estimate[z(2), z

(2)t

]: We first note that vt ∈ L2(0, T ;H1(Γ0))∩H

12 (0, T ;L2(Γ0))∩C([0, T ];L2(Γ0)),

with continuous dependence on the data and control, by Propositions 2.1 and 3.1. Moreover, initial datumv1 ∈ L2(Γ0). Given then that

[z(2), z

(2)t

], as it appears in (3.41), solves the wave equation with Neumann

data vt − v1, with vt(t = 0)− v1 = 0, we can use Lemma 3.3 (with g = vt − v1 therein) to obtain∥∥∥∥[ z(2)

z(2)t

]∥∥∥∥C([0,T ];H

32 (Ω)×H

12 (Ω))

≤ C ‖[vt, v1]‖[L2(0,T ;H1(Γ0))∩H

12 (0,T ;L2(Γ0))∩C([0,T ];L2(Γ0))

]×[H1(0,T ;L2(Γ0))]

≤ C(‖X0‖H + ‖u‖L2(0,T ;U)

). (3.44)

Together, the estimates (3.42)–(3.44) give∥∥∥∥[ zzt

]∥∥∥∥C([0,T ];H

32 (Ω)×H

12 (Ω))

≤ C(‖X0‖X + ‖u‖L2(0,T ;U)

). (3.45)

(1.B) We have by means of the representation (3.40),

z(t)−Nvt(t) = z(0)(t) + z(1)(t) + z(2)(t)−Nvt(t). (3.46)

In analyzing the right hand side of this expression, we note by (3.41) that

∂

∂ν

(z(1)(t) + z(2)(t)−Nvt(t)

)∣∣∣∣Γ0

= 0;

and this, combined with (1.11), (3.43), (3.44), the characterization (1.9) and interpolation gives thededuction that ∥∥∥z(1) + z(2) −Nvt

∥∥∥C([0,T ];D(A

34N ))

≤ C(‖X0‖H + ‖u‖L2(0,T ;U)

). (3.47)

Combining (3.46), (3.42), (3.47) and Proposition 2.1 yields then

‖[z, zt, v, vt]‖C([0,T ];X ) ≤ C(‖X0‖X + ‖u‖L2(0,T ;U)

), (3.48)

the desired result.

3.4 Proof of Theorem 1.4(iii)

Again we consider here the solution [z, zt, v, vt]T of (1.2) with initial data X0 := [z0, z1, v0, v1]

T ∈ X andcontrol u = 0. As we have done before (see (3.40)), the wave component [z, zt]

T of the solution of can bewritten explicitly as[

z(t)zt(t)

]= eA1t

[z0

z1

]+∫ t

0

eA1(t−s)

[0

ANNvt(s)

]ds

= eA1t

[z0 −Nv1

z1

]+[

Nv1

0

]+∫ t

0

eA1(t−s)

[0

ANN (vt(s)− v1)

]ds. (3.49)

For x0 ∈ Int(Ω), let Πx0 : D(Πx0) ⊂ H1(Ω)× L2(Ω) be defined by

Πx0 [z0, z1]T = z1(x0).

25

Note that from Propositions 2.1 and 3.1 (with u = 0), we have

‖vt‖L2(0,T ;H2(Γ))∩H1/2(Σ)∩C([0,T ];L2(Γ)) ≤ ‖X0‖H .

We can hence invoke Lemma 3.3(ii) to have∥∥∥∥∥Πx0

∫ (·)

0

eA1(·−s)

[0

ANN (vt(s)− v1)

]ds

∥∥∥∥∥L2(0,T )

≤ CT

(‖vt‖L2(0,T ;H2(Γ))∩H1/2(Σ)∩C([0,T ];L2(Γ)) + ‖v1‖L2(Γ)

)≤ CT ‖X0‖H . (3.50)

Moreover, as X0 ∈ X , then [z0 −Nv1, z1]T ∈ D(A

34N ) × H1(Ω)

R , and so we have from Triggiani’s Lemma1.3, ∥∥∥∥Πx0e

A1(·)[

z0 −Nv1

z1

]∥∥∥∥L2(0,T )

≤ CT ‖[z0 −Nv1, z1]‖D(A

34N )×H1(Ω)

R

≤ CT ‖X0‖X . (3.51)

Combining (3.50) and (3.51) with the expression in (3.49) and the definition of the observation in (1.6)now leads to the desired result.

References

[1] G. Avalos. Sharp regularity estimates for solutions to wave equations and their traces with prescribedNeumann data. Applied Mathematics and Optimization, 35:203–221, 1997.

[2] G. Avalos and I. Lasiecka. Differential Riccati equation for the active control of a problem instructural acoustics. J. Optimization Theory and Applications, 91:695–728, 1996.

[3] G. Avalos, I. Lasiecka, and R. Rebarber. Lack of time robustness for dynamic stabilization of astructural acoustic model. to appear in SIAM J. Control and Optimization.

[4] J.M. Ball. Strongly continuous semigroups, weak solutions, and the variation of constants formula.Proc. Amer. Math. Soc., 63: 370-373, 1977.

[5] H. T. Banks and R. Smith. Active control of acoustic pressure fields using smart material technologyin flow control. In M. Gunzburger, editor, IMA, Volume 68. Springer Verlag, 1995.

[6] H. T. Banks and R. C. Smith. Well-posedness of a model for structural acoustic coupling in a cavityenclosed by a thin cylindrical shell. Journal of Mathematical Analysis and Applications, 191:1–25,1995.

[7] H. T. Banks, M. A. Demitriou and R. C. Smith. An H∞ minimax periodic control in a 2–D structuralacoustics model with piezoceramic actuators. IEEE Trans. on Automatic Control 41:943–959, 1996.

[8] H. T. Banks, R. C. Smith, and Y. Wang. The modeling of piezoceramic patch interactions withshells, plates and beams. Quarterly of Applied Mathematics, 53(2):353–381, 1995.

[9] J. Beale. Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J., 9:895–917,1976.

[10] E. F. Crawley and E. H. Anderson. Detailed models of piezoceramic actuation of beams. In Proc.of AIAA Conference, 1989.

[11] E. F. Crawley and J. de Luis. Use of piezoelectric actuators as elements of intelligent structures.AIAA Journal, 25:1373–1385, 1987.

26

[12] E. F. Crawley, J. de Luis, N. W. Hagood, and E. H. Anderson. Development of piezoelectrictechnology for applications in control of intelligent structures. In Proc. of Applications in Controlof Intelligent Structures, American Controls Conference, pages 1890–1896, Atlanta, June 1988.

[13] E. K. Dimitriadis, C. R. Fuller, and C. A. Rogers. Piezoelectric actuators for distributed noise andvibration excitation of thin plates. Journal of Vibration and Acoustics, 13:100–107, 1991.

[14] G. Doetsch, Introduction to the theory and application of the Laplace transform. Springer–Verlag,New York, 1994.

[15] P. Grisvard. Caracterization de quelques espaces d’interpolation, Arch. Rational Mech. Anal., 25(1967), pp. 40–63.

[16] L. Hormander. The Analysis of Linear Partial Differential Operators I-IV. Springer Verlag, NewYork, 1984.

[17] T. Georgiou and M.C. Smith. Graphs, causality and stabilizability: linear, shift-invariant systemson L2[0,∞). Mathematics of Control, Signals, and Systems 6:195-223, 1993.

[18] I. Lasiecka, J.L. Lions, and R. Triggiani. Non-homogenous boundary value problems for second orderhyperbolic operator. J. Math. Pure et Applic., 65:149–192, 1986.

[19] I. Lasiecka and R. Triggiani, A lifting theorem for the time regularity of solutions to abstractequations with unbounded operators and applications to hyperbolic equations. Proceedings of theAMS, 104(3): 745–755, 1988.

[20] J.L. Lions and E. Magenes. Non-homogenous Boundary Value Problems and Applications. SpringerVerlag, 1972.

[21] I. Lasiecka and R. Triggiani. Sharp regularity results for mixed second order hyperbolic equations ofNeumann type. Part I: the L2 boundary data. Annali di Mat. Pura et Applicata, IV CLVII:285–367,1990.

[22] I. Lasiecka and R. Triggiani. Sharp regularity results for mixed second order hyperbolic equationsof Neumann type. Part II: general boundary data. J. Differential Equations, 94:112–164, 1991.

[23] I. Lasiecka and R. Triggiani. Recent advances in regularity of second-order hyperbolic mixed prob-lems, and applications. In Dynamics Reported–Expositions in Dynamical Systems, Volume 3, pp.25–104. 1994.

[24] H. C. Lester and C. R. Fuller. Active control of propeller induced noise fields inside a flexible cylinder.In Proc. of AIAA Tenth Aeroacoustics Conference, Seattle, WA, 1986.

[25] H. Logemann, R. Rebarber and G. Weiss. Conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays in the feedback loop. SIAM Journal onControl and Optimization, 34:572-600, 1996.

[26] H. Logemann and S. Townley. Low-gain control of uncertain regular linear systems. SIAM J. Controland Optimization, 35:pp. 78-116, 1997.

[27] R. Melrose and J. Sjostrand. Singularities of boundary value problems. Comm. Pure Applied Math.,31:593–617, 1978.

[28] P.M. Morse and K.U. Ingard. Theoretical Acoustics. McGraw-Hill, New York, 1968.

[29] S. Miyatake. Mixed problems for hyperbolic equations. J. Math. Kyoto Univ, 130:435–487, 1973.

[30] A. J. Pritchard and S. Townley. A real stability radius for infinite dimensional systems. Proc. of the1989 MTNS, Birkhauser, Boston, pp. 635–646, 1990.

27

[31] R. Rebarber. Conditions for the equivalence of internal and external stability for distributed param-eter systems. IEEE Trans. Aut. Contr., 38(6):994–998, 1993.

[32] R. Rebarber. Exponential stability of coupled beams with dissipative joints: a frequency domainapproach. SIAM J. Control and Optimization, 33:1–28, 1995.

[33] R. Sakamoto, Mixed problems for hyperbolic equations, I, II, J. Math. Kyoto Univ., 10, no. 2 (1970),pp. 343–373 and 10, no. 3 (1970), pp. 403–417.

[34] D. Salamon. Realization Theory in Hilbert Spaces. Math. Systems Theory, 21:147-164, 1989.

[35] D. Tataru. On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup.Pisa Cl. Sci. (4) 26 (1998), no. 1, pp. 185–206.

[36] M. Taylor. Pseudodifferential Operators. Princeton University Press, 1981.

[37] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach,J. Math. Anal. Appl., 137:438–461, 1989.

[38] R. Triggiani, Interior and boundary regularity of the wave equation with interior point control,Differential and Integral Equations, 6(1):111–129, 1993.

[39] G. Weiss. Transfer functions of regular linear systems, part I: characterization of regularity. Trans.Amer. Math. Soc. 342:827–854, 1994.

[40] G. Weiss and R. F. Curtain. Dynamic stabilization of regular linear systems. IEEE Trans. Aut.Control, to appear.

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