Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms

Nonlinear

Analysis.

Theory,

&Applications, Vol. 30. No. 1, pp. 11 l-122.1997 Pm. 2nd World Con ~CII o Nonlinar Analysts

Methods

Q1997Ltier Science Lrd Rinccd in Crest Britsin. All rights reserved 0362-546X/97517.00+0.00

PII: SO362-546X(97)00004-7

EXACT CONTROLLABILITY FOR SECOND-ORDER HYPERBOLIC EQUATIONS WITH VARIABLE COEFFICIENT-PRINCIPAL PART AND FIRST-ORDER TERMS I. LASIECKA, IApplied 1. INTRODUCTION.

DUAL

R. TRIGGIANI,

Mathematics,

University

PROBLEMS:

CONTINUOUS

AND P. F. YAOt

of Virginia,

Charlottesville,

OBSERVABILITY

VA 22901 INEQUALITIES.

Standing assumptions. (H.l): Let 0 c Rm be a bounded, open domain of class C2. Let l?c and Fr be open subsets of P with I’ = l?c U l?l. Let

LITERATURE

with boundary

I’ = 80

111 Jtw= i,jcl c z8% dij(S)g j ) x= [Xl,...,XJ ( 1 be a second-order uniform ellipticity

differential condition:

operator,

2

with

real

1 a2

%j(X)tEj

i,j=l

for some positive

constant i$l.ij(x)fi
aij

=

aji

of class C4, satisfying

El,

(1.2a)

v x E Rn,

that E = (G752,. .

.,6)’

(H.2): Let F~(uI) be a linear, first-order differential operator w with L,(Q)-coefficients, thus satisfying the following pointwise CT > 0 such that I&(w)12 < cT[w; + IhI + w2], \d

E Rn,

E #

(1.2b)

0.

in all variables {t, xi,. . . ,z,} on estimate: there exists a constant t,x

E Q,

(1.3)

where Q = (O,T] x R and w(t,x) E C’(Q). Let (O,T] x I’i = Ci, i = 0,l; (O,T] x l? = C. Dirichlet control. We consider the Dirichlet mixed second-order hyperbolic problem unknown w(t, x) and its dual homogeneous problem in $(t, z):

Acknowledgement: Grand DAAH04-96-1-0059, was partially supported

I. Lake&a’s

the

i=l

a > 0. Assume further > 0,

coefficients

(1.1)

and Ft. Triggiani’s

and by the National by the National Science

research

was partially

Science Foundation under Foundation of China.

111

supported Grant

by Army

DMS-9504822.

Research

in the

Office under

F. F. Yao’s

research

112

Second World

wtt + dw = FI(w) ~(0,

. ) = WO, wt(0,

in

.) = WI

Congress

(1.4a) in Q;

+ti + d$ = PC@,)

in 0;

$(T,

* ) = $0, $0,

(1.4b) . ) = $1

in 0;

(1.4c)

in C,

Illlc = 0

in Cl;

= u

Analysts

Q;

in CO; wlc,

of Nonlinear

(1.4d)

with control function u E Lz(O,T; &(I’r)) E Ls(C r ) in the Dirichlet B.C., where F(q) able first-order differential operator, depending on the original operator 4, and satisfying pointwise bound as (1.3).

is a suitthe same

Continuous observability inequality in the Dirichlet case. As our first goal, we seek to establish-under a suitable additional assumption-the following a-priori inequality for the homogeneous Dirichlet $-problem (1.4): there exists a constant TO > 0, depending upon the triple ($2, rs, I’r}, such that for all T > TO, there is a constant CT > 0 for which (1.5) [one obtains cT = c(T - TO)]. In (1.5), & = C&i oij g pi is the co-normal derivative, where u= [VI,..., vn] is the unit outward normal on I. Eqn. (1.5) ii the continuous observability inequality for the $-problem (1.4) [D-R.11. As is well-known, e.g., [La.l], [L-T.l], [Tr.l] inequality (1.5) for the +-problem (1.4) is, by duality or transposition, equivalent to the exact controllability property of the non-homogeneous w-problem (1.4) at time T, on the space &(R) x H-‘(n), within the class of &(O,T; Ls(I’i))-controls; in other words, to the property that the map LT: {u, 200 = 0, wl = 0) + LTt‘ = {w(T, { is surjective with

{w(T,

. ), wt(T,

Remark 1.2 smooth coeficients

. )} solution

from Lz(O,T;

L2(l?l))

of the w-problem

. ), wt(T,

. )}

(1-6)

onto L2(!2) x H-l(R),

(1.4) at t = T.

The converse of inequality (1.5) is always true, for any T > 0 and for suficiently aq, possibly depending on t as well, and sufficiently smooth I?, see [L-L-T]. 0

Neumann control. Here we let I’s # 8, l?s nl?r = 8, and consider the Neumann mixed secondorder hyperbolic problem in the unknown w(t, z) and is dual homogeneous version in $(t, z):

wtt + dw = FI(w); w(0, *) = wo, wt(0,

ht + & .) = w1;

$(T, I

- ) = $0, tW’,

4+3:0= 0

in Q;

= F(+,) . ) = $1

in Q in CO

(1.7a) (1.7b) (1.7c)

with control function u E Lz(O,T; Lz(I’l)) E L2(C 1) in the Neumann B.C. where F is a suitable first-order differential operator depending on 4, and satisfying a pointwise estimate such as (1.3) for FI and /3 is a suitable function, also depending on 4.

Second World Congress of Nonlinear Analysts

113

Continuous observability inequality in the Neumann case. As our second goal, we seek to establish-under a suitable additional assumption-the following o-priori inequality for the homogeneous Neumann $-problem (1.7): there exists a constant 2’0 > 0, depending upon the triple (0, rs, Ii}, such that for all T > TO, there is a constant or > 0 for which

where H&(n) = {f E H’(Q) : fir,, = 0}, w h enever the left-hand side is finite. This is the continuous observability inequality for the $-problem (1.7). Again, by duality or transposition, inequality (1.8) is equivalent (see e.g., [La.l], [L-T.l], [L-T.41) to the exact controllability property of the non-homogeneous w-problem (1.7) at time T, on the space H&(n) x &(R), within the class of Lz(O, T; &(I’r))-controls; in other words, to the property that the map LT: (u,wo i with

{w(T,

. ), wt(T,

= 0, to1 = 0) + Lpi

is surjective . )} solution

from Lz(O,T;

E {w(T, Lz(I’,))

. ),wt(T,

. )}

H&(R)

x

onto

of the w-problem (1.7) at t = T. (1.8) is false for dimfl

Remark 1.3 The converse of inequality [L-T.+

(1.9)

Lz(Ci),

2 2, and is true for dimR

= 1

0

Literature. observability

In short, various techniques have been used to establish the validity inequalities such as (1.5) and (1.8), notably:

(i) differential

multipliers

(ii) pseudo-differential

[H.l],

[K], [L.l],

multipliers,

[L-T.l],

micro-local

[T.l],

of the continuous

etc.;

analysis, and propagation

of singularities

[B-L-R];

(iii)

general pseudo-differential multipliers derived from pseudo-convex Carleman estimates for general evolution equations, including-but order hyperbolic equations [Ta];

(iv)

(related to (iii)) specific differential multipliers constructed from pseudo-convex functions and yielding Carleman estimates for second-order hyperbolic equations [L-T.41, [I], [F-I], along with sharp trace estimates via micro-local analysis [L-T.21;

(v) differential There are servability geometric sharpness,

geometry

methods

combined

with differential

multipliers

functions and related to not limited to-second-

Ty].

various issues that intervene in describing the degree of generality of the continuous obinequalities such as (1.5) and (1.8), once established by a given method. These are both as well as analytical factors, and their combination defines the range of applicability and against which a given method may be measured. These issues include:

(1) The ability

to treat variable

(2) The ability

to include

coefficients

first-order

oij of the principal

part A in (1.1).

energy level terms such as F in (1.3).

(3) Sharp geometrical conditions on the triple {fl,I’o,I’i}; in particular, the ability to determine a sort of ‘minimal’ active (controlled) portion I’1 of the boundary, and a corresponding ‘good estimate’ of the exact controllability time TO. In short, a sharp estimate of l?r x [0, TO]. In the case where ri = l?, so that the control u is active on the entire boundary I’ and A = -A, one should recover the optimal time TO = 2R, R being the radius of the smallest ball in Rn containing fl2; see [K], and Huyghen’s principle for n = odd = 3,5,. . .. This is related to the issue of obtaining sharp trace estimates.

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Second World Congress of Nonlinear Analysts

(4) Smoothness of the analytical of the boundary l?.

data, particularly

of the coefficients

oij of the principal

part; and

In addition, ease in verifying the various (sufficient) conditions in application to specific classes of examples is also an issued of great importance. In all this context, at present, no method appears to stand out inequivocably exempt from potential drawbacks or limitions. As to (i). The original differential multipliers for $-problems (1.4) and (1.7)-i.e., h . Vll, and 11,div h, where h(z) is a coercive vector field-have been successful in proving the continuous observability inequalities (1.5) and (1.8) in the case where A = -A (or in the case of constant coefficients oij of the principal part) [H.l], [L.l], [L-T.11, [T.l], h owever, these original differential multipliers tolerate additional terms only below the energy level; i.e., a zero-order operator F is fine, but a truly first-order operator F causes the method to fail. As to (ii). The micro-local/propagation of singularities approach used in [BLR] is very general; however, it is not an easy matter to verify in applications and examples the (sharp) sufficient condition that all rays of geometric optics hit the effective controlled part Cr = (O,T] x l?r of the lateral boundary C of the cylinder Q at a non-diffractive point. Moreover, the method uses Coo data and I’, at least at present. Extension to other models seems a serious issue. As to (iii). The techniques with pseudodifferential Carleman multipliers proposed in [Ta] are unifying across general evolution equations. However, they require the existence of a pseudo-convex function, a property which, essentially can be verified mostly if not exclusively in the case of constant cofficients oij of the principal part d. Moreover, at least in [Ta], the control is taken to be active on the entire boundary l?. As to (iv). In the specific concrete analysis of differential Carleman multipliers tuned to secondorder hyperbolic equations, as in (L-T.41, [I], [F-I], the drawback of the existence of a pseudo-convex function remains, of course, while now a more detailed analysis-this time at the differential rather than pseudo-differential level-allow the control to act on a suitable part of the boundary. These differential Carleman multipliers-see below in (4.2), (4.3) -can be viewed as a non-trivial generalization of the original differential multipliers h . V$, 1c,div h in (i), over which they possess an added flexibility via the parameter 7 below, which allows to handle also those first-order terms F as in (1.3), that original multipliers could not deal with. As to (v). Even more recently, Riemann geometric tools and analysis have been brought to bear in proving the continuous observability inequalities such as (1.5) and (1.8) [Yl. This method has the virtue to allow variable coefficients aij(z) of the principal part A, subject to certain verifiable assumptions. However, in its present form p], this approach also cannot handle genuine first-order, energy level terms F. The reason will be explained below in Section 4. Contribution of the present paper. In this paper we wish to announce and sketch, within the allowed space, a successful combination of three key ingredients which allow to establish the validity of the continuous observability inequalities (1.5) and (1.8) in the case of (a) variable coefficients oij(z) of the principal part d, subject to verifiable conditions, and (b) genuine first-order, energy level terms F. These three ingredients are: (1) the Riemann

geometric

approach

of [Y];

(2) the Carleman differential multipliers used in [L-T.41, which now replace the original multipliers of [Y], though in the Riemann metric; (3) the pseudo-differential approach in [L-T.21 which led to an &-estimate tive (gradient) of the solution w in terms of La-boundary estimates lower-order terms; see Theorem 4.1 below.

differential

of the tangential of wt and e,

derivamodulo

Second World Congress of Nonlinear Analysts Ingredient (2) permits to add Ingredient (3), moreover, permits in M. The present approach provides inequalities (1.5) and (1.8), as is 2. RIEMANNIAN

Recalling

the optimal time T for the validity of the continuous the case with pseudo-convex functions. GENERATED

and G(z)

A(Z) = (aij(Z)); Both are n x n matrices.

a born&de first-order operator Fr, as in (1.3), to the results of [Y]. to eliminate geometrical conditions present in the Neumann case

METRIC

a~ = oji, let A(z)

BY THE

b e, respectively,

G(Z) = [A(Z)]-1

A(s)

115

is positive

PRINCIPAL

the coefficient

A

matrix

and its inverse

i, j = 1,. . . , n; z E Rn

= gij(x)),

definite

PART

observability

(1.1)

by (1.2b).

Riemannian metric. Let Rn have the usual topology and 2 = (~1, 22, . . . , zn)r be the natural coordinate system. For each x E R”, define the inner product and the norm on the tangent space R; = R” by S(X,y)

1x1, = (X,X)i’2,

9.

= (X,Vg

=

VX = 2

2 ij=l

~yid

j=l

Y = 2

8%’

JXJO=(X.X)~‘~,

the Euclidean

manifold

VX=kai&,

i=l

(1.3)

with the Riemannian

Y=~fi~$y~R~, i=l

metric on R”. For z E R”, and with

E RE.

pi &

i=l

It is easily checked from (1.2b) that (Rn,g) is a Riemannian We shall denote 9 = C&i gijdxidxj. Euclidean metric. For each x E Rn, denote by X’Y=kcUiP,y

(1.2)

gij(X)dj7

I

i=l

2

metric

(1.4)

reference to (2.1), set

(1.5) Let X = CF=r (Y~(Z)& be a vector field on Rn. Denote metric by dive(X). Then ’

the divergence

of X in the Euclidean

(1.6) Further

relationships. If f E C’($?), denote by

Vof

and V,f,

(1.7)

the gradients off in the Euclidean metric and in the Riemannian metric g, respectively. The following lemma provides further relationships. Lemma 2.1 Let 2 = (x1, x2,. . . , x,)~ be the natural coordinate system in Rfl. Let f, h E C1(fi). Finally, let H, X be vector fields. Then with reference to the above notation, we have (1) (H(x),

A(z)X(z)),

= H(z)

. X(z),

2 E Rn;

(1.8)

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Second World Congress of Nodnear Analysts

(l-9) (3) (V,f, V&j,

=

V,f (h) = Vof

. A(+‘&,

z E Rn;

(1.10)

(4) W,f, V,(Wf )Ng = DH(V,f? V,f) + f d~~o(lV,f I$b) - flvgf I@, ~~vo(~i)(~), 2 E R”, where DH

is the covariant

diflerential

A=-g&

Covariant

discussed below;

(gaijb)$

differential.

(1.11)

Denote

=-divo(Vpj,

the Levi-Civita

connection

(1.12)

WEC~(W.

in the Riemannian

metric

g by D.

Let H=kh&;

(1.13) k

k=l

be vector fields on (Rn, g). The covariant for each x E Rn, defined by DH(X, where DxH is the covariant notation of (2.13), DxHe

=

differential

DH of H determines

Y) = (DxH,Y),,

derivative

of H with

V X,Y respect to X.

a bilinear

E R;,

form

on

R,” xRz, (1.14)

This is computed

as follows, in the

n

cDx k=l

=

kx(hk)

&

k=l

-I.2

M$alazi

(&)

7

(1.15)

k,i=l

where by definition, X(hk)

= X . Vohk = 2

& 2,

Daiazi

(&)

= g

r6

jj$?

(1.16)

i=l

I’g being the connection

coefficients (1.17)

Inserting DxH

(2.17) into (2.16), and then (2.16) into (2.15) yields = 5 k=l

X(hk)

&

+

(1.18)

Second World

Finally,

inserting

Congress

of Nonlinear

(2.18) into (2.14), we obtain

via (2.2),

DH(X,

= g e,j=l

X)

=

(DxH,X)g

117

Analysts

(1.20)

Thus, in Rg x Rg, DH(

. , . ) is equivalent

to the n x n matrix (1.21)

Hessian in the Riemannian respect to the metric g is

metric

g. Let f E C2(Rn).

By definition,

the Hessian off

with

(1.22)

by

(2.2),

(2.9),

(2.18),

where

!=

ft=eabg,

1,2 ,...,

n.

(1.23)

P

p=l

Thus, by (2.22), we have that D2f is positive the n i,j

on R!j

x

Rt if and only if

mij = CF=l ( = 1 , . . . , n is positive. x

n matrix

3. MAIN

$

SCj

+ CE,&l

hkScjrfk)

j

(1.24)

RESULTS

Let the domain Sl and the elliptic operator A be given satisfying the standing assumption (H.l). The additional hypothesis which we shall need to establish the continuous observability inequalities (1.5) and (1.8) is the following Main assumption (H.3): We assume that there exists a function v : Q + R of class C2 which is strictly convex on a, with respect to the Riemannian metric g defined in Section 2: this means that the Hessian of v in the Riemannian metric g is positive on fi, as defined by (2.24):

D2v(X, X)(z) > 0, Since fi is compact,

it follows from (3.la)

D2v(X,X)

Vz~i=i,

XER;.

that there exists a positive

1 plX&

V z E a, X E R;.

(l.la)

•I constant

p > 0 such that (l.lb)

118

Second World

Congress

of Nonlinear

Analysts

Under assumption (H.3), we take the vector field (1.2) defined as the gradient of w(z) with respect to the Riemannian metric g, see (2.9). Section 5 below will illustrate some situations where (the standing assumption (H.l) as well as) the main assumption (H.3) are guaranteed to hold true. Continuous observability inequalities. We are now in the position to state our main results concerning the validity of the continuous observability inequalities (1.6) and (1.11). First, define p as in (3.lb).

(1.3)

Theorem 3.1 (Dirichlet case) Let SI, d, and F satisfy the standing assumptions(H.l), (H.2). Let assumption (H.3) hold true and define h(x) by (3.2). Let T > TO, see (3.3). Assume that h(x) . V(X) 5 0 for x E TO, where we recall that V(X) = [VI(X), . . . , vn(x)] is the unit outward normal vector to r, and where h(x). u(x) = CE1 hi(x)ui(x) is the dot product in Rn. Then, inequality (1.5) 0 for the Dirichlet +-problem (1.4) holds true. Theorem 3.2 (Neumann case)Let 0, A, and F satisfy the standing assumption (H.l), (H.2). Let assumption(H.3) hold true and define h(x) by (3.2). Let I?0 and rr be given, I? = I’s u I’l, F. n i?r = 0, and h(x) . V(X) 5 0 for x E To. Let T > To, see (3.3). Then, inequality (1.8) for the 0 Neumann $-problem (1.10) holds true. Remark 3.1 Both Theorems3.1 and 3.2 require a uniquenesscontinuation result for the hyperbolic problem (l.da-b-c), respectively (1.7a-b-d), with over-determinedB.C.: z[xl

Th eorem

~Ofor

3.1; $1~ E 0 for Theorem

3.2.

(1.4)

This, in turn, can be reduced[B-L-R] to a correspondingsecond-orderover-determined elliptic problem to which we apply IHor.1, Theorem 17.26, p. 141. 0 Carleman estimates. The results of Theorems3.1 and 3.2 can be shown as a consequenceof

suitable Carleman estimatesfor Eqn. (1.4a) [with no boundary conditions imposed], which we now describe. Let v : 0 + R be the strictly convex function, with respect to the Riemannianmetric g, provided by the main assumption (H.3). Define the function 4 : fl x R + R by q5(x,t) SW(~)

-clt-

:I’,

0
p, pas in (3.3),

where T > To, see(3.3) and c are chosenso that CT > To. Theorem 3.3 (Carleman estimates)Assume (H.l), (H.2), and (H.3). a solution of the second-orderhyperbolic equation wtt+dw=Fr(w)+f [with no boundary conditions], within

I

w E HI?‘(Q)

the following = L2(0,T;

‘I ‘Wtr-& ELz(O,T;

Lzu-1).

in&

Let f E Ls(Q).

(1.5)

Let w be

(l-6)

class: H’(0))

n H’(O,T;

Lz(0))

(1.7a) (1.7b)

Second World

Congress

of Nonlinear

119

Analysts

Let 4(x, t) be the function defined by (3.5). Then for r > 0 suficiently large, the following one-parameterfamily of estimates (i) With $(x,0)

< -6 < 0, 4(x,T)

< -6 < 0 uniformly

(Wlc + $ Le”+‘f’dQ + TCT 2 (ii)

p-c-cr

IVsw12+ wz] dQ - CIT~-‘~[E(T)

7

where the boundary

terms (BT)lc

I-

4. CONCEPTUAL

SKETCH

W3)

t1

>J eT

const,llwI12C([O;~;Lz(n))

to

E(t)dt

- ClTe-“[E(T)

(1.9)

+ E(O)],

over C = [O,T] x I’ are given by

+A

with ~(x, t) a suitablefunction

+ E(O)],

x E a:

W)Ic + F s, @f2dQ + TCT p-c-CT

in x E R:

const,llwl12C([O;~;Lz(n))

With 4(x, t) > 1 for t E [to, tl] c (O,T),

2

holds true.

depending

on 4.

OF THE PROOF

2Jx

er$[wi - IV,w12]h . v dC,

(1.10)

cl

OF THEOREM

3.3 (CARLEMAN

ESTIMATES)

The idea of the proof of Theorem 3.3 is as follows: Step 1. By usingthe Riemanngeometricanalysisin M, Eqn. (3.7) on (0, T] x n = Q is converted into the equation on M wtt-Aw=h(w)+fl (1.1) on a Riemann manifold M, where A in (4.1) is the Laplacian, pr is a first-order differential operator. Step 2. To Eqn. (4.1) we apply the counterpart, in the Riemann matric g, of the Carleman estimatesapproach of [L-T.41 in the caseof Euclidean metric [i.e., with A in (1.4a) replaced by -A]: i.e., we usethe main multipliers x) - 4(x, t)wt(t, x)1,

(l-2)

t) - -$ (eT+4t))] ; w e7+,

(1.3)

eTd(“*t)[Vg4(x, t) . V&t, as well as the additional multipliers w [div (e’+V&(x,

where 4(x, t) is the function defined in (3.5). Step 3. We absorb tangential gradient e of w by using the following result from [L-T.21. Theorem 4.1 Let w be a solution of Eqn. (3.6), in the class (3.7).

Second World

120 (i)

Given

6 > 0 and

Congress

of Nonlinear

Analysts

small and given T > 0, there exists a constant

co > 0 arbitratily

C&,,~

> 0

such that

(ii)

If, moreover, w satisfies the boundary byrl. 0 4. ILLUSTRATIONS

Example

5.1

WHERE

condition

w(r,

ASSUMPTIONS

(H.1)

Let Sz C FL2 be a bounded domain.

XY3 -- a ay ( i+22+y6

E 0, then (4,4) holds true with !C’replaced

AND

(H.3)

ON A HOLD

TRUE

Assume that A is defined by

au -- a i+x2 au ZG1 ay ( 1+~2+~6 &i > .

(1-l)

Set

Then G(x, y) = (gij) = A-‘(qy) Consider the Riemannian manifold natural coordinate system (x, y) by g= Consider

(R2,g),

=

‘-:;;

-xy3 1 +y6

where the Riemannian

.

(1.3)

metric

g is defined in the

(1+~2)dzdz-zy3d~dy-zy3dydz+(1+y6)dydy.

the surface in R3 given by M=

{

(x,y,t)(z=;x2-~y4

>

)

= (x,y), for any (z,y,z) E M, with the induced Riemannian metric gM. Then the map @(z,y,z) determines an isometry from M to (R2,g). We obtain the Gaussian curvature of (R2,g) at (x, y), J&Y)

Since by (5.6) the Gaussian

=

the Gaussian

=

-3y2 (1 + x2 + y6J2 5 0,

curvature

curvature

is non-positive,

of M at (x, y, z) V (x,Y) E R2. then the function

x0 fixed

E R2,

defined by (1.7)

Second World Congress of Nonhex

Analysts

121

i.e., as the square of the distance d,(x, xe), in the Riemann metric of (2.3), from x to a given ftxed Thus, assumptions (H.l) and (H.3) hold true in this xo E R2, is in fact strictly convex on (R”,g). case. Example 5.2 Let fl C R* be a bounded domain and ai > 0 constants, i = 1,2,. . . , n. Consider the operator on Rn,

Set

-01022122

1 + Cyz”=2a:xf A(x)

=

1 1 + CE1 a:xi

1+&L2afx;

-a2anxlxn

.. .

.. .

-a,alx+l Then

G(x) = A-l(x)

a2anx1xn

=

. ..

-~1%$1%

...

-a2a,x2xn

.. .

..*

. . - 1 + Crzl

--ana2xnx2

1+ a:xf

.**

ala22122

. .*

Q%XlGl

1 + a$x$

...

020,x2x,

.. .

.. .

afxf

I*

(1.10)

.. .

1+a2x2n n Imetric g is determined Consider the Riemannian manifold (R”,g), where the Riemannian natural coordinate system x = (x1, x2,. . . , xCn) by i analxnxl

(14

...

ana2xnx2

in the

(1.11) Then 2 i,j=l

(6,

+ aiajxixj)Mj

>-

ltli

7

V x,<

=

(J1,E2

7. . .,&J

E Rn.

(1.12)

Riemannian It is easly checked from the above inequality that (R”,g) as . a complete non-compact manifold. Let al, a2,. . . , a, be positve number and let M be the hypersurface of Rn+’ defined by Xn,

Xn+l)I

%+l

=

f

$ 2=1

&Xf

3

with the induced Riemannian metric in R n+l. Then, by [Y, Lemma 3.11, M is of everywhere sectional curvature. It is easily verified from (5.11) that the map Cp : M --t (Rn,g), defined @(PI = $9

V~=h,x2,...,~n+1)

(1.13) positive

EM,

is an isometry between M and (Rn,g). Thus, (Rn,g) is of everywhere positive sectional curvature. Since (Rn,g) is a non-compact, complete Riemannian manifold of everywhere positive sectional curvature, then there exists a Coo strictly convex function v(x) on (R”,g) by [G-W]. Assumptions (H.l) and (H.3) are verified.

122

Second World Congress of Nonlinear Analysts

References [B-L-R]

C. Bardos, G. Lebeau, and J. Rauch, Sharp efficient conditions for the observation, control, and stabilization of wave from the boundary, SIAM J. Co&r. and Optim. 30(5) (September 1992), 1024-1065.

P-RI

S. Dolecki and D. Russell, A general theory of observation and control, SIAM J. Control 15(1977), 185-220.

[F-1.1]

A. V. Fursikov and 0. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes N. 34, Research Institute of Mathematics, Seoul National University Seoul, 151-742, Korea.

[G-WI

R. E. Greene and H. Wu, Cm convex functions and manifolds of positive curvature, 137(1976), 209-245.

WI

F. L. Ho, Observabilite 302( 1986)) 443446.

[Hor.l]

L. Hormander,

PI

0. Yu. Imanuvilov, Exact controllability of hyperbolic (1990), 10-13; and n. 4 (1990), 31-39 (in Russian).

WI

V. Komornik, Exact Controllability and Stabilization, Research in Applied Mathematics Editors: P. G. Ciarlet and J. Lions), 1994, Masson/John Wiley co-publication.

[La.l]

J. Lagnese, The Hilbert Uniqueness Method: A Retrospective, Lecture Notes in Control Information (edited by M. Thoma and A. Wyner), 148 (1990, 158-181.

[L-L-T]

I. Lasiecka, J. L. Lions, and R. Triggiani, Nonhomogeneous boundary order hyperbolic operators, J. Math. Pures Appl. 65 (1981), 149-192.

[L-T.11

I. Lasiecka and R. Triggiani, Exact controllability control, Appl. Math. Optim. 19 (1989), 243-290.

[L-T.21

I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim. 25 (1992), 189-244.

[L-T.31

I. Lasiecka and R. Triggiani, Sharp regularity for mixed second-order hyperbolic equations of Neumann type, Part I: The &boundary case, Annal. di Matemat. Pura e Appl. (IV) Vol. CLVII (1990), 285-367.

[L-T.41

I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, non-conservative second order hyperbolic equations, Marcel Dekker, Lecture Notes in Pure and Applied Mathematics, to appear; Proceedings of P.D.E. Conference held at Scuola Normale Superiore, Pisa, Italy, December 1994.

PJ.11

J. L. Lions, Contr8labilitB exacte, stabilisation Paris. Vol. 1, Contr6labilitB exacte, 1988.

PI

D. Tataru, Boundary 295.

PI

R. Triggiani, Exact boundary controllability on &(R) x H-l(Q) of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems, Appl. Math. Optim. 18(1988), 241-277.

frontiere de l’equation

Acta Math.

des ondes, C. R. Acad. Sci. Paris, Ser. I Math

The analysis of linear partial differential operators, Vol. III, Springer-Verlag,

controllability

P. F. Yao, Exact controllability preprint, 1996.

1985.

equations, Part I & II, Automat&a,

n. 3 (Series and

value problems for second-

of the wave equation with Neumann boundary

et perturbations

des systbmes distribuks,

Masson,

of Conservative PDEs, Appl. Math. Optim. 31 (1995), 257-

of wave equations with variable coefficients in a bounded region,