The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002058

THE DYNAMICAL LAME SYSTEM: REGULARITY OF SOLUTIONS, BOUNDARY CONTROLLABILITY AND BOUNDARY DATA CONTINUATION

M.I. Belishev

and I. Lasiecka

Abstract. The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input→state” map inL2-norms is established. A structure of the reachable sets for arbitraryT >0 is studied. In general case, only the first componentu(·, T) of the complete state{u(·, T), ut(·, T)}may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation. If T0 exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input →output” map) R^2T0 uniquely determines R^T for any T >0. A procedure recoveringR^∞viaR^2T0 is also described.

Mathematics Subject Classification. 93C20, 74B05, 35B65, 34K35.

Received December 31, 2001.

1. Introduction 1.1. About the paper

This paper deals with the issue of boundary approximate controllability and related unique continuation property for a system of dynamic elasticity governed by the Lame model.

Our goal is to provide a description of sets which are approximately reachable by the actuation on an arbitrary (possibly small) portion of the boundary within an arbitrary (possibly short ) time – in the system which has variable (in space) coefficients. As such, this problem is very different from a large body of papers dealing with exact controllability for constant coefficient Lame models with seizable portion of the boundary accessible to control action for a sufficiently long time (see,e.g.[2]).

In 1993 Tataru extended the classical Holmgren–John theorem on uniqueness of the continuation across non- characteristic surfaces to solutions of PDE’s with nonanalytic coefficients [20]. In particular, for the case of time independent coefficients the required smoothness of the coefficients in [20] is rather minimal (C¹). One of the corollaries of this remarkable result is that the boundary controllability of the dynamical system governed by the scalar wave equation: on any finite time interval such the system turns out to be approximately controllable

Keywords and phrases:Isotropic elasticity, dynamical Lame system, regularity of solutions, structure of sets reachable from the boundary in a short time, boundary controllability.

1Saint-Petersburg Deptartment of the Steklov Mathematical Institute (POMI), Fontanka 27, St. Petersburg 191011, Russia;

e-mail: belishev@pdmi.ras.ru

∗Supported by the RFBR grant No. 02-01-00260.

2Department of Mathematics, University of Virginia, Charlottesville, VA 22901, USA; e-mail: il2v@weyl.math.virginia.edu

†Supported by the grant DMS 0104305.

c EDP Sciences, SMAI 2002

in the subdomain filled by waves. This last property plays the key role in the BC-method that is an approach to inverse problems based upon their relations with the boundary control theory [4].

In 1998 the Holmgren–John–Tataru theorem was generalized to a class of hyperbolic systems which are principally weakly coupled [9]. This class includes Lame systems withC³coefficients which are space dependent.

Our paper draws on further consequences of this generalization in the context of boundary controllability of the Lame system. Indeed, the unique continuation result in [9] is one of the main tools used in providing characterization of reachable sets. These results should lead to a variant of the BC-method for inverse problems formulated in the context of dynamic elasticity theory.

We dedicate this paper to memory of J.-L. Lions who’s contributions, impact and influence on the field has been and will be everlasting.

1.2. The Lame system

In a bounded domain Ω⊂R³with the smooth enough (sayC³) boundary Γ consider the dynamical system

u_tt−Lu= 0 in Q^T; (1.1)

u|t=0=u_t|t=0= 0 in Ω; (1.2)

u=f on Σ^T, (1.3)

where Q^T := Ω×(0, T); Σ^T := Γ×[0, T];Lis an operator acting onR³-valued functions defined by:

(Lu)i:=ρ⁻¹ X3 j,k,l=1

∂jcijkl∂luk i= 1,2,3

(∂j := _∂x^∂j);cijkl is the elasticity tensor of the Lame model:

cijkl=λδijδkl+µ(δikδjl+δilδjk);

ρ, λ, µ are smooth functions depending on spatial variable only and satisfying the usual ellipticity condition:

ρ >0, µ >0,3λ+ 2µ >0 in ¯Ω. Vector functionf ={f_i(γ, t)}³_i=1 is the Dirichlet boundary control given on Σ^T; u=u^f(x, t) ={u^f_i(x, t)}³i=1 is the solution (wave).

1.3. Metrics and domains of influence

The hyperbolic system (1.1–1.3) has two families of the characteristicsϕ(x, t) = const inQ^T determined by the equations

∂ϕ

∂t ₂

−c²_α|∇ϕ|²= 0, α=p, s, (1.4)

where

cp:=

2µ+λ ρ

¹₂

, cs:=

µ ρ

¹₂

are the velocities of pressure and shear waves,p – characteristics being ordinary whereas s – characteristics being of multiplicity 2.

The velocities determine two Riemannian metrics in ¯Ω:

ds²= |dx|²

c²_α , α=p, s;

we denote distαthe corresponding distances.

For an open subsetσ⊆Γ define the domains of influence ofσ

Ω^σ,T_α :={x∈Ω|distα(x, σ)< T}, T >0 and the times of filling

T_α^σ:= inf{T >0|Ω^σ,T_α ⊃Ω}·

The relationcs< cp implies Ω^σ,T_s ⊂Ω^σ,T_p , T_s^σ> T_p^σ.

Introduce the space H:=L_2,ρ(Ω;R³) (with measureρdx) and its subspaces H^σ,Tα :={y∈ H |suppy⊂Ω¯^σ,T_α }, α=p, s;

with the following obvious relationH^σ,Ts ⊆ Hp^σ,T.

1.4. The results

Our first result concerns the regularity of the “input→trajectory” map.

Theorem 1. The map f →u^f is continuous fromL₂(Σ^T;R³)intoC([0, T];L₂(Ω;R³)).

We shall discuss next reachability properties. To this end we introduce: Σ^σ,T := ¯σ×[0, T]. The sets of waves U^σ,T :={u^f(·, T)|f ∈L₂(Σ^T;R³); suppf ⊆Σ^σ,T};

U₀^σ,T :={u^f(·, T)|f ∈C^∞(Σ^T; R³); suppf ⊂σ×(0, T)}

play the role of reachable sets. Due to hyperbolicity of the system (1.1–1.3) and the above regularity result one has

U^σ,T ⊂ H^σ,Tp . (1.5)

In the spaceHintroduce the projections:

X_α^σ,Ty:=

(y in Ω^σ,T_α ; 0 in Ω\Ω^σ,T_α ;

so thatX_α^σ,TH=H^σ,Tα , α=p, s.The following result clarifies the character of the embedding (1.5).

Theorem 2. For any T >0 the equality

clos_HX_s^σ,TU^σ,T =H^σ,Ts (1.6)

holds. In particular, whenT > T_s^σ,clos_HU^σ,T =H.

This equality is interpreted as an approximate controllability of the system (1.1–1.3) in the subdomain Ω^σ,T_s filled bys–waves at the momentt=T. Simple examples (for small enough T) demonstrate that in the wider subdomain Ω^σ,T_p such the controllability does not occur. Notice, that the theorem deals with controllability

with respect to only the first component of the complete state{u, ut}. This is natural: as we show, for times T < T_p^σ u^f(·, T) determinesu^f_t(·, T).

The next result concerns a completeness of waves in the Sobolev classes for large enoughT connected with geometry of Ω.

Theorem 3. If T > T_s^σ the relation

closH² U₀^σ,T =H²(Ω;R³)\

H₀¹(Ω;R³) (1.7)

holds.

Our last result concerns to the response operator (“input → output” map) R^σ,T : L₂(Σ^σ,T;R³) → L₂(Σ^σ,T;R³); DomR^σ,T =C₀^∞(Σ^σ,T;R³),

(R^σ,Tf)i:=

X3 j,k,l=1

νjcijkl∂lu^f˜k on Σ^σ,T,

where{νj}³j=1are the components of the outward normal, the control ˜fin the right hand side is the continuation of f from Σ^σ,T to Σ^T by zero.

Theorem 4. The operator R^σ,2T given for a finite fixed T > T_s^σ determines the operators R^σ,T0 for all T⁰

∈(0,∞).

In other words, dynamical boundary measurements (data) given on Σ^σ,2T with a fixed T > T_s^σ may be uniquely continued onto Σ^σ,∞ without resorting to the evolution equation(1.1). We also describe at the end of the paper an effective procedure for the continuation.

Remark. Just for simplicity it is assumed in this paper that the coefficients of the system are C^∞. A more detailed analysis shows that C³-smoothness of Γ, λ, µ, ρ is enough to preserve all of the results. The last requirement is motivated by applicability of unique continuation results in [9]. On the other hand, other arguments in the paper do not require that much regularity (C¹suffices). Thus, if any further progress is made in relaxing regularity for unique continuation results, our results will apply as well.

2. The Lame system 2.1. Spaces and subspaces

Unless otherwises stated, we assume Γ, λ, µ ρ to beC^∞–smooth. Everywhere in the paperσ⊆Γ is a fixed open subset of the boundary.

The space of controls

F^σ,T := {f ∈L₂(Σ^T;R³)|suppf ∈Σ^σ,T} plays the role of external spaceof the system (1.1–1.3); we denote

M^σ,T :={f ∈ F^σ,T\

C^∞(Σ^T;R³)|suppf ⊂σ×(0, T)}, so that eachf ∈ M^σ,T vanishes near Γ× {t= 0}and Γ× {t=T}.

Theinner space isH:=L₂,ρ(Ω;R³); as above, we select its subspaces H^σ,Tα :={y∈ H |suppy⊂Ω¯^σ,T_α }, α=p, s.

Everywhere below, simplifying the notations, we omitσin the case ofσ= Γ: F^Γ,T :=F^T; M^Γ,T :=M^T etc.

2.2. Operator representation of Lame system

We will find convenient to provide operator– in fact spectral– representation of solutions to Lame system:

u_tt−Lu= 0 inQ^T; (2.1)

u|t=0=u₀, ut|t=0=u₁ in Ω; (2.2)

u=f on Σ^T. (2.3)

To accomplish this we introduce the following spaces and operators. H² := H²(Ω;R³),H₀¹ := H₀¹(Ω;R³).

H⁻¹:= (H¹₀)⁰. The operatorL₀;H → H,DomL₀=H²∩ H₀¹, L₀y:=Lyis self-adjoint, negatively defined with Dom (−L₀)¹² =H₀¹. Therefore, it generates sine and cosine operators [19] with the properties:

S(·)∈ L(H →C([0, T];H¹₀)); C(t) := d

dtS(t)∈ L(H → H) L₀S(t)u= d

dtC(t)u, u∈ H¹₀; d²

dt²C(t)u=L₀C(t)u; u∈DomL₀. (2.4) It is well known that S(t), C(t) commute with L₀ and they obey the following trigonometric relations

S(t−s) =S(t)C(s)−C(t)S(s); C(t−s) =C(t)C(s) +L₀S(t)S(s). (2.5) Since L₀ has discrete spectrum, we can write down spectral representation for sine and cosine operators: Let {λ_k}^∞_k=1 : 0 > λ₁ ≥ λ₂ ≥ ... and {ϕ_k}^∞_k=1 denote eigenvalues and eigenfunctions of L₀: L₀ϕ_k

= λ_kϕ_k; (ϕ_k, ϕ_l)_H = δ_kl. The existence and continuity of L⁻¹₀ follows from the fact that λ_k 6= 0. With the above notation we have

S(t)y= X∞ k=1

sin√

−λ_kt

√−λ_k (y, ϕ_k)_Hϕ_k, 0≤t≤T; (2.6)

C(t)y= X∞ k=1

cosp

−λkt (y, ϕk)_Hϕk, 0≤t≤T. (2.7) We also introduce the so called Green’s mapGgiven byGψ:= v iff Lv= 0 in Ω;v=ψon Γ. It is well known from standard elliptic theory that Gis bounded fromL²(Γ,R³) toH.

With the help of Green’s map we introduce the operator W :F^T →L₂((0, T);H⁻¹) (“input →trajectory”

map) defined by

(W f)(t) := L₀ Z t

0

S(t−s)Gf(s)ds. (2.8)

From (2.4) we clearly have: W ∈ L(F^T →C([0, T];H⁻¹)). Moreover,W ∈ L(H₀¹(Σ^T;R³) → C([0, T];H)), where this last assertion follows from the representation of W which is valid for with any f ∈ H₀¹(Σ^T;R³) (W f)(t) =−Gf(t) +Rt

0C(t−s)Gf_s(s)ds.

With the above notation, the solution u(t) to Lame system (2.1–2.3) with f ∈H₀²(ΣT,R³) can be written as (see Sect. 3 in [13])

u(t) =C(t)u₀+S(t)u₁−(W f)(t).

For f ∈ F^T,u₀ ∈ H, u₁ ∈ H⁻¹ the formula above provides a definition for “ultra-weak” solution to (2.1–2.3), which resides inC([0, T];H⁻¹).

Our first goal is to show that this “ultra–weak” solution is, in fact more regular by one spatial derivative.

This is the regularity statement which amounts to proving that the operatorW : F^T →C([0, T];H) is bounded.

2.3. Regularity of solutions

Iff ∈C^∞(Σ^T;R³) vanishes near Γ× {t= 0}then the problem (1.1–1.3) (subject to appropriate regularity of the coefficients) has the unique classical solutionu^f ∈C^∞( ¯Q^T;R³), the relations

u^ftt =u^f_tt=Lu^f, (2.9)

and

suppu^f(·, t)⊂Ω¯^σ,t_p , 0≤t≤T, (2.10) being valid due to time-invariant coefficients and hyperbolicity. In what follows we shall not need so much regularity and we shall show thatu^f ∈C([0, T];H) forf ∈ F^T, which in turn allows to define functionu^f a.e.

in Ω. Indeed, the above assertion follows from

Theorem 1. The map W is continuous from F^T into C([0, T];H). Moreover, the map _dt^dW is continuous from F^T intoC([0, T];H⁻¹).

Proof.

Step 1: Preliminaries

We begin with some notation. An R³- valued function y = {y_j(x)}³j=1, x ∈ Ω is said to be a¯ field; the term “function” is reserved for scalar functions. We use the summation over repeating indexes and denote y·v :=yivi, hα, βi:=αijβij, the scalar products of vectors and matrices; |y|:= (y·y)¹²; ||α|| :=hα, αi¹². The product α y is the vector αijyj. If y is a field we denote ∇y the matrix (∇y)ij := ∂iyj (∂i := _∂x^∂

i) and set (divα)_i := ∂_jα_ji. Finally, τ is the matrix conjugation: (α^τ)ij = α_ji. The elastic moduli tensor C ={c_ijkl}³_i,j,k,l=1 is considered as a “matrix →matrix” map: (Cα)_ij =c_ijklα_kl, the moduli c_ijkl are smooth functions in ¯Ω satisfying the symmetry relations

c_ijkl=c_jikl=c_ijlk =c_klij. (2.11)

The Lame model corresponds to

Cα=µ(α+α^τ) +λtrα I (2.12)

whereIis the unit matrix, trα:=αkk. Simple calculations with regard to 3λ+ 2µ >0, µ >0 allow to establish positivity ofC: for anyα=α^τ one has

hCα, αi ≥c₀||α||² (2.13)

with a constantc₀>0. Introducing the strain tensor ε(y) := 1

2[∇y+ (∇y)^τ] the basic operatorLmay be written in the form

L=ρ⁻¹divCε(·) (2.14)

or, by components, (Ly)i=ρ⁻¹∂j[2µεij(y) +λεkk(y)δij], where the densityρis a positive function in ¯Ω. Denote H:=L₂,ρ(Ω;R³) (with measureρdx),G:=L₂(Γ;R³) and recall Green’s formula

(Lu, v)_H−(u, Lv)_H= (Nu, Dv)_G−(Du, Nv)_G (2.15) where Dand N are the trace operators:

Du:=u|_Γ; Nu:=Cε(u)ν|_Γ,

ν ={νj}³j=1 is the outward normal, so that (Nu)i=cijklεkl(u)νj =cijkl∂lukνj. By using formula (2.15) we easily establish the following representation

G^∗L₀u=Nu , u∈DomL₀. (2.16)

Indeed, it suffices to apply formula (2.15) withu∈DomL₀, v=Gψ, ψ∈ G, which leads to (G^∗L₀u, ψ)_G= (L₀u, Gψ)_H = (u, LGψ)_H+ (Nu, DGψ)_G

−(Du, NGψ)_G= (Nu, DGψ)_G = (Nu, ψ)_G (2.17) implying the conclusion in (2.16).

Step 2: The bound for G^∗L₀S(·)

We shall consider composition operators G^∗L₀S(·), and G^∗(−L₀)¹²C(·) as (potentially) unbounded but densely defined operators : H → F^T given by:

(G^∗L₀S(·)u)(t) :=G^∗L₀S(t)u; u∈DomL₀ (G^∗(−L₀)¹²C(·)u)(t) :=G^∗(−L₀)¹²C(t)u u∈DomL₀.

Similarly, W^∗ which is originally defined as an element of L(L2((0, T);H¹₀) → F^T), can be considered as an unbounded operator (denoted by the same symbol): L₂((0, T); H)→ F^T.

The lemma stated below shows that these operators are in fact bounded.

Lemma 1. The operators

G^∗L₀S(·) :H → F^T; G^∗(−L₀)¹²C(·) :H → F^T; W^∗:L₂((0, T);H)→ F^T are bounded.

Proof. To prove the lemma we shall follow the same strategy as in [13] with support of computations performed in [14] for the von Karman system.

The main task in proving the lemma is establishing the appropriate bound for elements in the domains of respective operators. By virtue of (2.16) the result of the Lemma 1 follows from the following proposition:

Proposition 1. Let a, b∈DomL₀, g∈L₂((0, T);H¹₀)and letw(t) :=C(t)a+S(t)b+Rt

0S(t−s)g(s) ds. Then the following estimate holds

Z

Σ^T

dΓ dt||ε(w)||²≤ch

||a||²_H1

0+||b||²_H+||g||²L₁((0,T);H)

i. (2.18)

Proof. Follows the same strategy as in [13].

Step 1. Integrating by parts the equationwtt−Lw=gsatisfied bywand accounting for boundary conditions, leads to the following variational equality

Z

Q^T

dxdt[ρwtt·η−ρg·η+hCε(w), ε(η)i]− Z

Σ^T

dΓdtCε(w)ν·η= 0 (2.19)

for any test field η∈H¹(Q^T;R³). In what follows we choose and fixη= (∇w)^τh, so thatηi=∂jwihj, with a smooth field h=h(x).

Step 2. Introduce the tensorse(w) andm(w):

e_lk(w) = 1

2[(∂_iw_l)∂_kh_i+ (∂_iw_k)∂_lh_i];

mlk(w) = 1

2[(∂_ik²wl)hi+ (∂_il²wk)hi]

where ∂_ik² :=∂i∂k. Using the symmetry of tensorC (2.11) it is straightforward to show that

ε((∇w)^Th) =e(w) +m(w), (2.20)

and

div[hCε(w), ε(w)ih] =hCε(w), ε(w)idivh+ 2hCε(w), m(w)i+h(∂kC)ε(w), ε(w)ihk. (2.21) Step 3. In order to simplify notations we omit differentials in integrals. Inserting chosenη in (2.19) yields:

0 = Z

Q^T

{ρ(wtt−g)·(∇w)^τh+hCε(w), ε((∇w)^τh)i}

− Z

Σ^T

Cε(w)ν·(∇w)^τh=hsee (2.20)i

= Z

Ω

ρwt·(∇w)^τh |^t=Tt=0 − 1 2

Z

Q^T

∇|wt|²·ρh− Z

Q_T

g·(∇w)^τh

+ Z

Q^T

{hCε(w), e(w)i+hCε(w), m(w)i} − Z

Σ^T

Cε(w)ν·(∇w)^τh. (2.22)

Applying the divergence theorem and taking into accountw|Γ = 0 one has Z

Q^T

∇|wt|²·ρh= Z

Q^T

|wt|²divρh;

using (2.21) one obtains Z

Q^T

hCε(w), m(w)i = 1 2

Z

Q^T

{div[hCε(w), ε(w)ih]− hCε(w), ε(w)idivh− h(∂kC)ε(w), ε(w)ih_k}

= 1 2

Z

Σ^T

hCε(w), ε(w)ih·ν−1 2

Z

Q^T

{hCε(w), ε(w)idivh+h(∂kC)ε(w), ε(w)ihk}·

Inserting in (2.22) gives 0 =

Z

Ω

ρw_t·(∇w)^τh |^tt⁼=0^T +1 2

Z

Q^T

|w_t|²divρh− Z

Q_T

g·(∇w)^τh

+ Z

Q^T

hCε(w), e(w)i − 1

2 [hCε(w), ε(w)idivh+h(∂kC)ε(w), ε(w)ihk]

+ Z

Σ^T

1

2hCε(w), ε(w)ih·ν− Cε(w)ν·(∇w)^τh

· (2.23)

Step 4. By exploiting the fact thatwvanishes on Γ, we shall show that

Cε(w)ν·(∇w)^τh=hCε(w), ε(w)iν·h on Γ. (2.24) Indeed,w|Γ = 0 implies∂_jw_i=∂_νw_iν_j on Γ, (∂_ν:= _∂ν^∂ ); therefore, denotingCε(w) =:αone has

Cε(w)ν·(∇w)^τh=α_ikν_k(∂_jw_i)h_j=α_ikν_k(∂_νw_i)ν_jh_j. (2.25) On the other side, due to symmetry ofα

hCε(w), ε(w)i=αij1

2[∂iwj+∂jwi] =αij∂jwi=αij(∂νwi)νj. (2.26) Comparing (2.25) with (2.26) leads to (2.24).

Step 5. Takinghparallel to ν on Γ, and using (2.24) one transforms (2.23) to Z

Σ^T

hCε(w), ε(w)i = Z

Ω

ρw_t·(∇w)^τh|^t=T_t=0 − Z

Q_T

g·(∇w)^τh

+ Z

Q^T

1

2|w_t|²divρh+hCε(w), e(w)−1

2(divh)ε(w)i −1

2h(∂kC)ε(w), ε(w)ih_k

·

Since only the 1^st-order derivatives ofwenter the right hand side we easily get Z

Σ^T

hCε(w), ε(w)i ≤c sup_t∈[0,T]

h||w(·, t)||²_H1

0+||wt(·, t)||²_H+||g||²L₁((0,T);H)

i. (2.27)

On the other hand, by (2.4)

||w(·, t)||²_H1

0+||wt(·, t)||²_H ≤ c h

||a||²_H1

0+||b||²_H+||g||²L₁((0,T);H)

i,

whereas positivity (2.3) allows to transform (2.20) into (2.12). The Proposition is proved.

Let g = 0. Since DomL₀ is dense in H and H¹₀, the Proposition 1 yields continuity of the map {a, b}

→ε(w)|_Σ^T fromH₀¹× HintoF^T. Thus, eachH¹(Q^T;R³)-solutionwof Lame system with finite energy initial data possesses the traceε(w)|_Σ^T inL₂sense. The result of Proposition 1 when applied witha= 0, band then a, b= 0 implies,viaprinciple of superposition and (2.16) the result stated in Lemma 1.

Step 3: Completion of the proof of Theorem 1

To complete the proof, we proceed as in [13] Section 3. We define the following operators as elements of L(F^T →L_∞((0, T);H⁻¹)).

(J₀f)(t) :=L₀ Z t

0

S(s)Gf(s) ds; (J₁f)(t) := (−L₀)¹² Z t

0

C(s)Gf(s) ds.

From Lemma 1 and duality it follows that J₀ and J₁ are bounded: F^T → L_∞((0, T);H). Indeed, the above assertion followsviaRiesz Representation theorem from the following estimate valid with an arbitrary test field φ∈L₁((0, T);H₀¹):

Z T 0

((J₀f)(t), φ(t))_H−1,H¹₀dt≤ ||f||_F^T||G^∗L₀S(·)||_L(H→F^T)

Z T 0

||φ(t)||Hdt.

Analogous estimate applies to the second operatorJ₁.

Moreover, by standard density argument one shows that J₀ and J₁ are bounded: F^T → C([0, T];H) see Corollary 3.2 in [13]. To complete the proof it suffices to use trigonometric identities in (2.5) along with properties of sine and cosine operators in (2.4)

(W f)(t) =L₀S(t) Z t

0

C(s)Gf(s)ds−L₀C(t) Z t

0

S(s)Gf(s)ds

=−C(t)L₀ Z t

0

S(s)Gf(s)ds+ (−L₀)¹²S(t)(−L₀)¹² Z t

0

C(s)Gf(s)ds

= (−L₀)¹²S(t)(J₁f)(t)−C(t)(J₀f)(t). (2.28) The requisite boundedness ofW follows now from the continuity ofJ₀, J₁as operators: F^T →C([0, T];H), and continuity of (−L₀)¹²S(t) andC(t) as operators: H →C([0, T];H).The same argument applies to the velocity component:

d dtW f

(t) =−L₀ Z t

0

C(t−s)Gf(s)ds=−(−L₀)¹²[C(t)(J₁f)(t) + (−L₀)¹²S(t)(J₀f)(t)] (2.29) where, again, we have a composition of continuous operators acting on respective spaces such that the product is inC([0, T]; [Dom (−L₀)¹²]⁰)- as desired of the final conclusion. The proof of Theorem 1 is thus completed.

Remark. An alternative way of completing the proof of Theorem 1 is to use the fact that W^∗ a-priori in L(L2((0, T);H¹)→ F^T) is bounded: L₂((0, T);H)→ F^T. (This last assertion follows directly from Lem. 1.) The above implies,viaduality (see Sect. 3 in [13]), the boundedness ofW :F^T →L₂((0, T);H). Final conclusion can be then reached by appealing to “lifting theorem” in [16] which allows to boostL₂time regularity toC for time reversible dynamics.

Remark. TheL₂→L₂regularity of solutions of the Lame system is the same as of the scalar wave equation [13].

This is in contrast to the Maxwell system where the mapf →u^f is not continuous inL₂-norms.

Remark. Extending the mapf →u^fontoF^Tby continuity we consider its images inC([0, T];H) as generalized solutions of Lame system.

Theorem 1 implies the continuity of the control operatorW^T :F^T → H, W^Tf := (W f)(T) =L₀

Z T 0

S(t−s)Gf(s)ds=u^f(·, T).

Its reduction W^σ,T :=W^T |_F^σ,T is understood as an operator fromF^σ,T into H. Also, taking the adjoint of W^T (see Sect. 3 [13]) one obtains (W^T)^∗∈ L(H → F^T) is given by:

W^T_∗

y=G^∗L₀S(T− ·)y. (2.30)

2.4. Boundary control problem

U^σ,T := RanW^σ,T =W^σ,TF^σ,T.

Taking into account the relation (2.10) the following statement of the boundary control problem (BCP) appears relevant: givenT >0 anda∈ H^σ,Tp , findf ∈ F^σ,T such that

u^f(·, T) =a (2.31)

holds. In the lemma below we shall show that, at least for small T, the reachable set is rather poor, and the BCP is not solvable in general.

Lemma 2. Let T < T_p^σ and nonzero a ∈ H^σ,Tp are such that the set Ω\suppa is connected and contains a neighborhood of σin Ω. Then a6∈ U^σ,T.

Proof. We are going to show that the conjecturea∈ U^σ,T leads to a contradiction.

Step 1: Letf ∈ F^σ,T be such thatu^f(·, T) =a. As one can check, the field

u(x, t) :=















0 in Ω¯ ×(−∞,0];

u^f(x, t) in Ω¯ ×(0, T];

−u^f(x,2T−t) in Ω¯ ×(T,2T];

0 in Ω¯ ×(2T,+∞)

belongs to the classC(R\ {t=T};H) and satisfies (in the sense of distributionsD⁰(Ω×R) ) the equation utt−Lu=−2δ⁰Tu^f(·, T) =−2δ⁰Ta in Ω×R.

Here δ⁰_T(t) := _dt^dδ(t−T) is the dipole supported att=T. This easily implies that, for eachkthe field

˜u(x, k) := 1

√2π Z∞

−∞

e^−iktu(x, t) dt, (x, k)∈Ω×R

satisfies the equation

(L+k²)˜u(·, k) = 2ike^−ikTa in Ω as aD⁰(Ω))–distribution.

Since suppu^f(·, t)⊂Ω¯^σ,T_p for all t ∈[0, T], one has suppu(·, t)⊂Ω¯^σ,T_p for all t ∈ R. The latter obviously implies that its Fourier transform satisfies

u(·˜ , k) = 0 ∈ Ω\Ω¯^σ,T_p . (2.32)

Step 2: Thus, the field ˜u satisfies the homogeneous equation (L+k²)˜u(·, k) = 0 in the open set Ω\suppa and vanishes on its open subset (2.32). By the well known unique continuation principle for the Lame operator L[21] (see also [17]) one has ˜u(·, k) = 0 everywhere in Ω\suppa, and, in particular, in a neighbourhoodω^σ⊂Ω of σ.

Step 3: The last fact implies

u^f(x, t) = 1

√2π Z∞

−∞

e^iktu(x, k) dk˜ = 0, (x, t)∈ω^σ×[0, T];

hencef =u^f|σ×[0,T]= 0. Consequently,a=u^f(·, T) = 0, which fact contradictsa6= 0.

Remark. Taking a = 0 in (2.31), and repeating all the steps of the proof we can easily obtain f = 0 that is equivalent to KerW^σ,T = {0}. Therefore, in the case of T < T_p^σ the BCP can have at most one solution.

The injectivity of W^T means that on the time interval (0, T_p^σ) velocity is determined by displacement. Indeed, consider the set of pairs {{u^f(·, T), u^f_t(·, T)} |f ∈ F^T}; ifu^f(·, T) =W^σ,Tf = 0 then f = 0 by injectivity of W^σ,T, that yieldsu^f_t(·, T) = 0. Thus, the set is the graph of a linear operator expressingu^f_t(·, T) troughu^f(·, T).

This means that only one component of the complete state{u^f;u^f_t} may be controlled at timest < T_p^σ. This phenomenon should be contrasted with the case when the time is large enough (seee.g.[2]).

2.5. Comments

• An analog of the result stated in Theorem 1 in the case of the scalar wave equation with constant coefficients was first shown in [15] and later generalized to various topological levels in [13]. Thus the L₂-regularity of “control→ state” map is extended from the scalar case to the case of systems of dynamic elasticity.

We note, that as in the case of the wave equation, the regularity result has no analogue for the Neumann problem. This is to say thatL₂ tractions prescribed on the boundary do not produceH¹ solutions.

• By using the result in Theorem 1 along with semigroup methods one can prove higher (or lower) level optimal regularity of solutions with respect to various levels of regularity of controls. In fact, this can be done in the same way as in [13] where scalar wave equations are treated.

• The proof of Lemma 2 follows the scheme of the paper [1] (see also [7]).

3. Controllability in subdomain Ω

– proof of Theorem 2 3.1. The dual system

We recall from (2.30) and (2.16) that withy∈ Hwe have W^T_∗

y=G^∗L₀S(T− ·)y=NS(T − ·)y∈ F^T

where we recall (Nu)i =νjcijkl∂luk |Γ; PDE interpretation of this is that for a solution v=v^y(x, t), y∈ H of the dynamical system

vtt−Lv= 0 in Q^T; (3.1)

v|t=T = 0, vt|t=T =y in Ω; (3.2)

v= 0 on Σ^T; (3.3)

we have the following representation

W^σ,T∗y=Nv^y|_Σ^σ,T . (3.4)

The rest of Section 3 is devoted to the proof of Theorem 2.

3.2. Odd continuation and Cauchy data

We are going to show that existence ofh∈ Hsatisfying:

(α) h⊥ U^σ,T; (β) supph∩Ω^σ,T_s 6={∅}

which statements will lead to a contradiction.

Letv^h be the solution of (3.1–3.3) withy=h; in accordance with (α) one has 0 = (W^Tf, h)_H=hsee (3.4)i= (f, Nv^h)_FT

for anyf ∈ F^T,σ; hence

Nv^h= 0 on σ×[0, T]. (3.5)

Fix γ ∈ Γ and choose the coordinates so that ν = ν(γ) = {0,0,−1}. From (3.3) we have ∂jv^h_i = 0 for j = 1,2i= 1,2,3, whereas equality (3.5) takes the form of the system

ci3k3∂₃v^h_k = 0, i= 1,2,3

with diagonal matrixci3k3=µδik+ (λ+µ)δi3δ₃k.Solving this system, one gets∂₃v_k^h= 0 and, finally, v^h= 0, ∂_νv^h= 0 onσ×[0, T],

so that the solution v^h has zero Cauchy data on Σ^σ,T. As is easy to verify, due to v|t=T = 0 the odd continuation

w(x, t) :=

v^h(x, t) Ω×(0, T);

−v^h(x,2T−t) Ω×[T,2T) turns out to be an H¹–solution of the problem

w_tt−Lw= 0 in Ω×(0,2T); (3.6)

w= 0, ∂νw= 0 onσ×[0,2T]. (3.7)

3.3. The extension Ω

Let Ω⁰be a domain with smooth boundary such that Ω⁰⊃Ω. Continue the Lame parameters in Ω¯ ⁰preserving smoothness and conditions ρ >0, µ >0,3λ+ 2µ >0 in ¯Ω⁰.

Continue the solutionwin Ω⁰×(0,2T) puttingw= 0 in [Ω⁰\Ω]×(0,2T). Due to (3.6, 3.7) this continuation turns out to be an H¹– solution of the equation

wtt−Lw= 0 in Ω⁰×(0,2T) (3.8)

vanishing near one side of the smooth surface σ×(0,2T). Since the surface is noncharacteristic with respect to both speeds s, p (time-like), by virtue of the uniqueness theorem [9] one has w= 0 in an open set K₀^σ,2T

⊂Ω×(0,2T) such that∂K₀^σ,2T ⊃σ×[0,2T].

Let us show that the solutionwis continued by zero fromK₀^σ,2T onto the domain K^σ,2T :={(x, t)∈Ω×(0,2T)| |t−T|< T−dists(x, σ)}

bounded by characteristics of equation (3.6).

3.4. Paraboloids

Fix a pointx₀ ∈Ω^σ,T_s ; choose a point γ₀ ∈σ. Let l ⊂Ω be a smooth curve connectingγ₀ with x₀, of the s-length less than T, transversal toσ. Such a choice is possible due to dist_s(x₀, σ)< T.

Continuel smoothly across beyondσin Ω⁰. On this continuation fix a pointaclose enough toσso that the s-length of the segment [a, x₀] is less thanT. Parametrize this interval putting s(m) equal to the s-length of [a, m], so thats(x₀) =:T₀< T.

Consider a (small) tube neighborhood ω of the segment [a, x₀]⊂l in Ω⁰ determined by the condition: each x ∈ω is connected withl by unique shortests-geodesic orthogonal to l. Letmx be the point onl s-nearest to x∈ω.

In ω introduce the “cylinder coordinates” r(x) := dists(x, l), s(x) := s(mx); notice the inequality s(x)

≥dists(x, σ), and the relations

∇s· ∇r= 0 inω; |∇s|²= 1

c²_s onl (3.9)

easily following from the definitions.

Takingε >0 small enough one can ensure in the “tube”ωε:={x∈ω|r(x)< ε} the inequality T₀+T

2T₀ 2

c²_s(x)|∇s(x)|²>1, x∈ωε (3.10) which is valid due toT₀< T and the second of the relations in (3.9).

Choose r₀>0 so small that the subdomain (paraboloid) π_r^x0,γ0,l

0,κ :=

(

x∈ω_ε∩Ω|κ−s(x) T₀ −

r(x) r₀

₂

>0 )

for anyκ∈(0,1] is contained in Ω^σ,T_s whereas the components ∂π_r^x₀⁰_,κ^,γ0,l∩Ω and ∂π_r^x₀⁰_,κ^,γ0,l∩σ of its boundary are smooth surfaces. We call these paraboloids admissible; notice that π_r^x₀⁰_,κ^,γ0,l extends asκgrows.

Varying parameters of admissible paraboloids one can verify that they exhaust the subdomain Ω^σ,T_s .