CONTROLLABILITY OF HYPERBOLIC AND DEGENERATE PARABOLIC

EQUATIONS IN ONE DIMENSION

By

Jonathan Matthew Bohn

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2018

ABSTRACT

CONTROLLABILITY OF HYPERBOLIC AND DEGENERATE

PARABOLIC EQUATIONS IN ONE DIMENSION

By

Jonathan Matthew Bohn

In this thesis, we study the controllability problem for two systems of partial differential

equations. We will first consider the wave equation with variable coefficients and potential in
one dimension, utt − (a(x)ux)x + pu = 0, with control function v(t) acting on the boundary.

We consider a class of functions corresponding to a special weight function that contains the

variable coefficient a(x). From here, we derive a global Carleman estimate for this system,

and establish the controllability property. We then later extend the class of admissible

functions a(x) for which the controllability property holds true.

We then study the controllability problem for the degenerate heat equation in one di-
mension. For 0 ≤ α < 1, on (0, 1) × (0, T ), we consider wt − (xαwx)x = f . This equation is
degenerate because the diffusion coefficient xα is positive in the interior of the domain and

vanishes at the boundary. We consider this problem under the Robin boundary conditions.

Again, we derive a Carleman estimate for this system, taking into account the new boundary

terms that arise from the Robin conditions.

To those who helped me along the way.

iii

ACKNOWLEDGMENTS

I am deeply grateful to my advisor, Dr. Zhengfang Zhou.

I thank him for his insight,

encouragement, and time. I enjoyed our meetings and have learned much from him, not only

in mathematics, but from his exceptional personality. I am fortunate to have studied under

his guidance and support.

I would like to thank the Mathematics Chair Dr. Keith Promislow, Director of Graduate

Studies Dr. Jeffrey Schenker, and past supervisors for providing me with the opportunity to

study at Michigan State.

I would like to give thank to all the faculty and staff at MSU that helped me along the

way. I include special thanks to Dr. Baisheng Yan, Dr. Casim Abbas, Dr. Gabriel Nagy,

and Dr. Sheldon Newhouse, each of who went beyond their normal duties in a memorable

way for me. I would like to show special appreciation to Ms. Barbara Miller, Ms. Debra

Lecato, and Mr. James Chang.

Special thanks to Lianzhang Bao. I again enjoyed our meetings and learned much from

our correspondence. In particular, my experience in China was one of the most memorable

of my life due to your planning and diligence.

Special thanks to Seonghak Kim. You were the best office mate I could have. I give

thanks to Xin Yang and Khaldoun Al-Yasiri, who shared some challenges with me and

provided support when I needed it.

To James, Steve, and Logan, you helped me more than you know. Thanks to Nick and

Erick for scheduling help and Chris for proofreading.

I would like to thank my family, who has always supported me and provided me with

guidance.

iv

Finally, I thank all of the students I taught that helped to keep me sharp. Teaching was

one of the most enjoyable aspects for me. I hope that I in turn helped you in your chosen

careers.

v

TABLE OF CONTENTS

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Regularity of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Transposition Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Hyperbolic Equations

3.1

3.2 Carleman Estimates

. . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Carleman Estimate for large T . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Pointwise Carleman Estimate at −T . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Increased Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 Control
3.4

Chapter 4 Degenerate Parabolic Control

. . . . . . . . . . . . . . . . . . . .
4.1 Robin Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Carleman Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
10
14

19
19
23
25
34
40
42
46

51
52
57

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

vi

KEY TO SYMBOLS

1. Rn = n-dimensional real Euclidean space, where R = R1.
2. Ω ⊂ Rn = open set in Rn.

3. ∂Ω = boundary of Ω.

4. ν = ν(x) = unit outward normal.

5. For u : Ω → R, x ∈ Ω, then uxi = ∂u
6. For u : Ω → R, then (cid:52)u =(cid:80)n
∂xi

u(x+hei)−u(x)

h

= lim
h→0

, whenever this limit exists.

i=1 uxixi, the Laplacian of u.

7. AC(Ω) = {u : Ω ⊂ R → R|u is absolutely continuous} .
8. Ck(Ω) = {u : Ω → R|u is k − times continuously differentiable} .
c (Ω), etc., are those functions in C(Ω), Ck(Ω), etc., with compact support.
9. Cc(Ω), Ck
10. Lp(Ω) = {u : Ω → R|u is measurable and ||u||Lp(Ω) < ∞}, for 1 ≤ p < ∞, where

||u||Lp(Ω) ˙=(cid:0)(cid:82)

Ω |u|pdx(cid:1) 1

p .

esssupΩ |u|.

11. L∞(Ω) = {u : Ω → R|u is measurable and ||u||L∞(Ω) < ∞}, where ||u||L∞(Ω) ˙=

12. W k,p(Ω) = {u ∈ Lp(Ω)|Dαu ∈ Lp(Ω),∀|α| ≤ k}, the Sobolev spaces.

13. W k,p

0

(Ω) is the closure of C∞

c

in W k,p(Ω).

0 (Ω) = W k,2
14. Hk(Ω) = W k,2(Ω), Hk
15. H−1(Ω) is the dual space to H1
16. For X a real Banach space, with norm || · ||, then

0 (Ω).

(Ω).

0

(cid:16)(cid:82) T
(cid:17) 1
0 ||u(t)||pdt

Lp(0, T ; X) =
for 1 ≤ p < ∞, where ||u||Lp(0,T ;X) =

(cid:110)
u : [0, T ] → X|u is strongly measurable and ||u||Lp(0,T ;X) < ∞(cid:111)
(cid:110)
u : [0, T ] → X|u is continuous, and ||u||C([0,T ];X)<∞
||u||C([0,T ];X) = max
0≤t≤T

18. W 1,p(0, T ; X) =(cid:8)u ∈ Lp(0, T ; X)|u(cid:48) exists in the weak sense, and u(cid:48) ∈ Lp(0, T ; X)(cid:9) .
19. For m > 0, L∞≤m(Ω) = {q ∈ L∞(Ω),||q||L∞(Ω) ≤ m}.

17. C ([0, T ]; X) =

||u(t)||.

p .

,

(cid:111)

, where

vii

22. H2

a(0, 1) = {u ∈ L2(0, 1)|u ∈ AC([0, 1]) and
a(0, 1) = {u ∈ H1
a(0, 1)|aux ∈ H1(0, 1)}.

20. For m > 0, L∞≤m(Ω × (0, T )) = {p ∈ L∞(Ω × (0, T )),||p||L∞ ≤ m}.
21. H1
23. B((0, 1) × (0, T )) = C(cid:0)[0, T ]; L2(0, 1)(cid:1) ∩ L2(cid:0)0, T ; H1
H ((0, 1) × (0, T )) = L2 (0, T ; D(A)) ∩ H1(0, T ; L2(0, 1)) ∩ C(cid:0)[0, T ]; H1

24. For operator (A, D(A)),

√
aux ∈ L2(0, 1)}.

a(0, 1)(cid:1) .

a(0, 1)(cid:1).

viii

Chapter 1

Introduction

In this study, we present some applications of the problem of controllability in partial differ-

ential equations. The controllability problem in the context of this work may be described

in summary as follows: Consider a evolution system described in terms of partial differential

equations and a time T > 0. Allow ourselves to influence this system through means of a

control function. Now, given a set of initial states and final states, the goal is to find such

a control function, so that by means of the control, the system is driven from the initial

states to the final states in time T . The natural question arises then; Which systems are

controllable?

We will consider this question for two different systems. An answer can be developed

depending on what space the initial and final states reside. In Chapter 2, we develop the

framework necessary for the solution space to study the hyperbolic system of Chapter 3. Of
interest then, let Ω ∈ Rn be a domain, ω ⊂ Ω, T > 0, and consider the linear wave equation
with interior control function f in Ω × (0, T ),



utt − (cid:52)u = f (x, t)χω,

u = 0,

u(x, 0) = u0(x), ut(x, 0) = u1(x),

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω.

(1.1)

It is well known that for f ∈ L2(Ω), (u0, u1) ∈ H1

0 (Ω) × L2(Ω), then (1.1) is well posed,

1

with a unique finite energy solution u ∈ C(cid:0)[0, T ]; H1

0 (Ω)(cid:1)∩ C1(cid:0)[0, T ]; L2(Ω)(cid:1) . However, the

controllability question is most naturally posed in the larger class (u0, u1) ∈ L2(Ω)×H−1(Ω).

In Chapter 2, we present the Transposition Solutions, arising from this form of the initial

states. Then, in Chapter 3, the control function for the hyperbolic equation can be developed

in terms of these solutions.

In Chapter 3, we study the variable coefficient wave equation. For control v(t), and

potential function p(x, t), consider



utt − (a(x)ux)x + p(x, t)u = 0,

u(0, t) = 0,

u(1, t) = v(t),

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

ut(x, 0) = u1(x),

(x, t) ∈ (0, 1) × (0, T ),
t ∈ (0, T ),
x ∈ (0, 1).

(1.2)

By the Hilbert Uniqueness Method [38], the controllability of such a system is equivalent

to the observability of the adjoint problem. That is, for an appropriately defined subset of

the boundary, Γ(x0), one can positively show controllability by obtaining an inequality of

the form

||(u0, u1)||2

L2(Ω)×H−1(Ω)

≤ C

(cid:90) T

(cid:90)

0

Γ(x0)

(cid:12)(cid:12)(cid:12)(cid:12) ∂u

∂n

(cid:12)(cid:12)(cid:12)(cid:12)2

dΓdt.

(1.3)

A more refined inequality is the Carleman Estimate, which contains the necessary infor-

mation of observability. In Section 3.2 we develop the Carleman Estimate for a certain class
of functions a(x), assumed to be C2(Ω), and in the class B defined in (3.7). Specifically, for
these a(x), let T > 0, and assume that x0 < 0, a0 > 0. There exists β ∈ (0, 1) such that

T >

1
β

max
x∈[0,1]

(x − x0)(cid:112)a(x)

.

2

(1.4)

Then for any m > 0, we will show there exists λ > 0 independent of m, s0 = s0(m) > 0

and a positive constant M = M (m) such that for ψ and ϕ as defined as in (3.10), for

a(x) ∈ B(x0, a0) defined in (3.7), for all p(x, t) ∈ L∞≤m((0, 1) × (−T, T )) and for all s ≥ s0,

s

(cid:90) T

(cid:90) T
(cid:90) 1
(cid:90) 1
e2sϕ(|wt|2 + |wx|2)dxdt + s3
e2sϕ(−T )(|wt(−T )|2 + |wx(−T )|2)dx + s3
(cid:90) 1
(cid:90) T

(cid:90) T

(cid:90) 1

−T

0

−T

0

e2sϕ|Lw + pw|2dxdt + M s

0

+ s

≤ M

−T

0

e2sϕ|w|2dxdt

(cid:90) 1

0

e2sϕ(−T )|w(−T )|2dx

(1.5)

e2sϕ|wx(1, t)|2dt,

−T

for all w ∈ L2(−T, T ; H1
and wx(1, t) ∈ L2(−T, T ).

0 (0, 1)) also satisfying wtt − (a(x)wx)x + pw ∈ L2((0, 1) × (−T, T ))

In Section 3.3, we show the connection between the Carleman Estimate and controlla-

bility. We will show that with the Carleman Estimate as above, controllability holds true.

In Section 3.4, we expand the class of functions a(x) for which controllability holds. For

a(x) defined up to the base point x0, we extend the admissible class of functions to those
a(x) ∈ C1(Ω).

In Chapter 4, we pose the same questions for the degenerate parabolic equation. Consider

the parabolic equation ut − (a(x)ux)x = f (x, t)χω,

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

(x, t) ∈ (0, 1) × (0, T ),
x ∈ (0, 1).

(1.6)

Then if a(0) = 0, this system is said to be degenerate.

In the case a(x) = xα, the

Carleman Estimate is known to hold under the appropriate Dirichlet or Neumann boundary
conditions, for 0 ≤ α < 2. In Section 4.1, we recall and apply a recent work necessary for

3

well-posedness for Section 4.2. In Section 4.2, with this chosen a(x), we state and prove the

analogous Carleman Estimate with Robin boundary conditions.

4

Chapter 2

Preliminaries

2.1 Regularity of the Wave Equation

The goal of the first part of this section is to introduce the relevant existence and uniqueness

results for the wave equation with less regular data. It is well known that the hyperbolic

system is well posed with suitable regularity on initial conditions, for example, data in

0 (Ω) × L2(Ω). However, the system can be studied with less regular initial conditions.
H1

The corresponding transposition solutions can only be understood in the transposition sense

of Lions [39]. In turn, we will apply the transposition solutions to the controllability problem

for the wave equation of section 3.1, as in a very natural sense, this is the largest space for

which the problem is well-posed.

Consider the system

ϕtt − (cid:52)ϕ = f,

ϕ = 0,

ϕ(x, 0) = ϕ0(x),

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω,

ϕt(x, 0) = ϕ1(x),

where f ∈ L1(cid:0)0, T ; L2(Ω)(cid:1) , and the initial conditions ϕ0(x) ∈ H1

0 (Ω), and ϕ1(x) ∈ L2(Ω).

(2.1)

5

Then, with these regularity assumptions, the solutions ϕ of (2.1) satisfy

(cid:16)

(cid:17) ∩ C1(cid:16)

(cid:17)

ϕ ∈ C

[0, T ]; H1

0 (Ω)

[0, T ]; L2(Ω)

.

(2.2)

For the proof, see for example [3] or [16]. However, the finite energy property does not

that ∂ϕ

apriori show L2 regularity of ∂ϕ

∂ν by classical trace inequalities. The result is in fact true
∂ν ∈ L2 (∂Ω × (0, T )) . This is not straightforward, and requires careful analysis of the
∂ν is

resulting terms after using a multiplier technique. For this reason, the regularity for ∂ϕ

often referred to as Hidden Regularity. The correct multiplier for this particular problem is

due to Ho [26] (cf.

[36], [37], [38]). The regularity result will be needed for the definition

of the transposition solutions to the wave equation in Section 2.2.1. The Hidden Regularity

proof below is adapted from [37].

Theorem 1. (Hidden Regularity). Let Ω ∈ Rn be a domain with smooth boundary, and let

ϕ solve

where f ∈ L1(cid:0)0, T ; L2(Ω)(cid:1) , and initial conditions ϕ0(x) ∈ H1

ϕt(x, 0) = ϕ1(x),

0 (Ω), and ϕ1(x) ∈ L2(Ω).

Then ∂ϕ

∂ν ∈ L2 (∂Ω × (0, T )), and furthermore,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ

∂ν

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(∂Ω×(0,T ))

(cid:18)
||f||L1(0,T ;L2(Ω)) + ||ϕ0||

≤ C

(cid:19)

,

+ ||ϕ1||L2(Ω)

(2.4)

H1

0 (Ω)

for some constant C.

6



ϕtt − (cid:52)ϕ = f,

ϕ = 0,

ϕ(x, 0) = ϕ0(x),

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω,

(2.3)

Note: In the proof, always use the Einstein summation convention when applicable. That

is, for functions {fi}m

i=1,{gi}m

i=1, then

m(cid:88)

i=1

figi.

figi =

(2.5)

Proof. By the standard approximation argument, the problem reduces to proving the result

for regular functions f, ϕ0, and ϕ1.

This will use a function multiplier from [26], for which we always assume the Einstein
summation notation of (2.5). For k = 1, . . . , n, let hk(x) be functions such that hk ∈ C1( ¯Ω)

and hk = νk on ∂Ω. (Such functions exist because of the assumed regularity of the boundary).

Now, we incorporate the multiplier from [26]. Multiply both sides of (2.3) by hk

integrate to obtain

(cid:90)

Ω

ϕtthk

dx −

∂ϕ
∂xk

(cid:90)

Ω

(cid:52)ϕhk

∂ϕ
∂xk

dx =

(cid:90)

Ω

f hk

∂ϕ
∂xk

dx.

∂ϕ
∂xk

and

(2.6)

We will integrate by parts, considering terms separately. First, look at the second term

of (2.6). We have

(cid:90)

Ω

−(cid:52)ϕhk

∂ϕ
∂xk

dx =

=

(cid:90)

(cid:90)
(cid:90)

Ω

Ω

∂ϕ
∂xi
∂ϕ
∂xi

∂
∂xi

hk

(cid:19)

∂ϕ
∂xk

(cid:18)

hk

∂2ϕ

dx +

∂xi∂xk

dx −

(cid:90)

∂ϕ
∂xi

Ω

∂ϕ
∂ν
∂Ω
∂hk
∂xi

hk

∂ϕ
∂xk

(cid:90)

dS

∂ϕ
∂xk
dx −

(cid:18) ∂ϕ

(cid:19)2

(2.7)

dS,

(2.8)

∂Ω

∂ν

where we have used the definition of hk = νk on the boundary. Now, the first term of (2.8)

7

becomes

(cid:90)

Ω

∂ϕ
∂xi

hk

∂2ϕ

∂xi∂xk

dx =

1
2

(cid:90)

Ω

hk

∂
∂xk

(cid:16)|∇ϕ|2(cid:17)

dx.

(2.9)

Upon integration by parts,

∂ϕ
∂xi

hk

∂2ϕ

∂xi∂xk

dx =

1
2

Ω

hkνk |∇ϕ|2 dS − 1
2

(cid:90)

Ω

|∇ϕ|2 dx.

∂hk
∂xk

(2.10)

Since ϕ = 0 on ∂Ω, we have that |∇ϕ|2 =

. In combination with (2.8), we discover

that the second term from (2.6) can be written as

(cid:90)

∂Ω

(cid:16) ∂ϕ

(cid:17)2

∂ν

(cid:90)

(cid:90)

−

(2.11)

(2.12)

dx

(cid:90)
(cid:52)ϕhk

Ω

= −1
2

Ω

∂ϕ
∂xk
∂hk
∂xk

|∇ϕ|2 dx +

(cid:90)

Ω

∂ϕ
∂xi

∂hk
∂xi

∂ϕ
∂xk

dx − 1
2

(cid:90)

(cid:18) ∂ϕ

(cid:19)2

∂Ω

∂ν

dS.

Now, return to (2.6). Integrating by parts with respect to time gives

(cid:19)

(cid:90) T

0

= −

(cid:18)
(cid:90)
(cid:90)
(cid:90) T
(cid:90) T

Ω

0

= −1
2

ϕtthk

∂ϕ
∂xk

dxdt

ϕthk

∂ϕt
∂xk

(cid:90)

Ω

dxdt +

(cid:90)

(cid:90)

Ω

ϕthk

∂hk
∂xk

(ϕt)2 +

ϕthk

∂ϕ
∂xk

Ω

0

Ω

(cid:12)(cid:12)(cid:12)(cid:12)T

0

dx

dx.

∂ϕ
∂xk

(cid:12)(cid:12)(cid:12)(cid:12)T

0

Integrating, and combining (2.6) and (2.12), we arrive at the expression

(cid:90)

(cid:90)
(cid:90) T
(cid:90) T

0

(cid:90)

Ω

− 1
2

= −1
2

∂hk
∂xk

(ϕt)2 +

ϕthk

Ω

|∇ϕ|2 dxdt +

∂hk
∂xk

0

Ω

(cid:12)(cid:12)(cid:12)(cid:12)T
(cid:90)

0

∂ϕ
∂xk

(cid:90) T

0

Ω

dx

∂ϕ
∂xi

∂hk
∂xi

∂ϕ
∂xk

dxdt − 1
2

(cid:90) T

(cid:90)

(cid:18) ∂ϕ

(cid:19)2

0

∂Ω

∂ν

(2.13)

dSdt

8

(cid:90) T

(cid:90)

0

Ω

+

f hk

∂ϕ
∂xk

dxdt.

Now, since h ∈ C1(Ω), there exists constants c and ˜c such that

|h(x)| ≤ c, and

n(cid:88)

(cid:12)(cid:12)∂ihj(x)(cid:12)(cid:12) ≤ ˜c.

Solving (2.13) for (cid:82) T

0

expression for || ∂ϕ

∂ν

∂Ω

(cid:82)

(cid:16) ∂ϕ
∂ν ||L2(∂Ω×(0,T ).
(cid:90)

E(t) =

1
2

Ω

(cid:17)2

i,j=1

dSdt and using the above bounds, we arrive at the

Now, we can recover the desired inequality (2.4). Define the energy

t (x, t) + |∇ϕ(x, t)|2dx,
ϕ2

0 ≤ t ≤ T.

(2.14)

Then, by computing the derivative, we easily see

(cid:90)

dE
dt

=

Ω

ϕt(ϕtt − (cid:52)ϕ)dx = 0.

That is, the energy is independent of t. In particular, for t = 0, we use (2.13) and (2.14) to

obtain the inequality in terms of the initial data, as desired,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∂ϕ

∂ν

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(∂Ω×(0,T ))

≤ C

(cid:18)
||f||L1(0,T ;L2(Ω)) + ||ϕ0||

H1

0 (Ω)

(cid:19)

.

+ ||ϕ1||L2(Ω)

(2.15)

9

2.2 Transposition Solutions

Now, we introduce the notion of transposition solutions. Because of the decreased regularity

of the initial conditions, the solution to the wave equation does not exist in the classical

sense. Rather, the solution must be defined through an intermediate system. The general

method for the transposition solutions can be found in volumes 1 and 2 of [39]. The short

introduction for our problem is adapted from [45]. The main idea is that the solution of the

less understood equation can be interpreted by a different well understood one.

Consider the system

utt − (cid:52)u = f,

u = 0,

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

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω,

(2.16)

ut(x, 0) = u1(x),

where (u0, u1) ∈ L2(Ω) × H−1(Ω), and f ∈ L2(cid:0)0, T ; H−1(Ω)(cid:1). For ease of notation, let us

set

Y = L2(Ω) × H−1(Ω) and D(A) = H1

0 (Ω) × L2(Ω).

Then, for (y1, y2) ∈ D(A), we define A by its action

Ay = A(y1, y2) = (y2,−(cid:52)y1).

Here, −∆ : H1

0 (Ω) → H−1(Ω), and so we may also associate the scalar product on H−1(Ω)

10

in the usual fashion, defined by

(∆y1, ξ)H−1(Ω) = −

(cid:90)

Ω

∇y1 · ∇(−(cid:52))−1ξdx.

(2.17)

In (2.17) above, (−(cid:52))−1ξ is that solution w of the equation −(cid:52)w = ξ in Ω (for w ∈

H1

0 (Ω)). As is standard then, the method of proof for well-posedness will be to construct a

semigroup of contractions.

Theorem 2. The operator (A, D(A)) is the generator of a semigroup of contractions on Y .

The following short lemma will prove to be useful in the proof of Theorem 2.

Lemma 1. The map f (cid:55)→ ||f||H−1(Ω) =(cid:10)f, (−(cid:52))−1f(cid:11) 1

2

H−1(Ω)×H1

0 (Ω)

is a norm in H−1(Ω)

equivalent to the usual norm.

Proof. Since (−(cid:52))−1 : H−1(Ω) → H1
||(−(cid:52))−1f||

0 (Ω) is an isomorphism, then also the map f (cid:55)→
is a norm in H−1 equivalent to the usual norm. Now, let f ∈ H−1(Ω),

and test the equation −(cid:52)(cid:0)(−(cid:52))−1f(cid:1) = f with (−(cid:52))−1f . We get upon integration by

0 (Ω)

H1

parts that

|∇((−(cid:52))−1f )|2dx =

(cid:68)

f, (−(cid:52))−1f

(cid:69)

(cid:90)

Ω

H−1(Ω)×H1

0 (Ω)

.

(2.18)

By Poincar´e’s inequality, we have that the left hand side of the inequality is equivalent to
in H−1 is also an equivalent

the H1

Ω

11

norm. Therefore,

0 norm. By our first statement, f (cid:55)→ ||(−(cid:52))−1f||
(cid:68)

(cid:90)

|∇((−(cid:52))−1f )|2dx =

c||f||2

≤

H−1(Ω)

H1

0 (Ω)

f, (−(cid:52))−1f

(cid:69)

H−1(Ω)×H1

0 (Ω)

≤ ||f||H−1(Ω)||(−(cid:52))−1f||
≤ C||f||2

H−1.

H1

0 (Ω)

(2.19)

Taking square roots, the lemma is proved.

With this lemma, we continue with the proof of the theorem.

Proof of Theorem 2. The proof will involve showing the hypotheses for the Hille-Yosida The-

orem, which is standard. The statement of the Hille-Yosida Theorem can be found in [16].

First, A must be shown to be densely defined and closed. It is clear that D(A) is dense in Y .
Let us show that A is closed on D(A). Let yn = (y1n, y2n) ∈ D(A) be a sequence converging
to some (y1, y2) ∈ D(A), where Ayn converges to some (f, g) ∈ L2(Ω) × H−1(Ω). By the
definition of A, we have that y2 = f . Now, because (−(cid:52))−1 is continuous as an operator
from H−1(Ω) to H1
0 (Ω), we have that (−(cid:52))−1(−(cid:52)y1n) = y1n → (−(cid:52))−1g. From here, it
0 (Ω) with g = −(cid:52)y1.
follows that y1 ∈ H1

For the second part of the Hille-Yosida Theorem, it must be shown that every positive

λ belongs to the resolvent set and that the resolvent operator defined by Rλu ˙=(λI − A)−1u
satisfies ||Rλ|| ≤ 1
λ for λ > 0. Recall λ belongs to the resolvent set if the operator λI − A is

one-to-one and onto.

Now, let λ > 0. For f ∈ L2(Ω), g ∈ H−1(Ω), consider the system

 λy1 − y2 = f,

λy2 − (cid:52)y1 = g,

in Ω,

in Ω.

(2.20)

Notice then that Rλ(f, g) = (λI − A)−1(f, g) = (y1, y2). By multiplying the top equation

12

in (2.20) by λ and adding the two equations, we obtain

λ2y1 − (cid:52)y1 = λf + g, in Ω.

(2.21)

By the standard elliptic theory (see [25] for example), there exists a unique weak solution

to (2.21) in H1

0 (Ω). Now, we need to obtain a suitable estimate. By applying (−(cid:52))−1 to

the first equation of (2.20), we get

λ(−(cid:52))−1y1 − (−(cid:52))−1y2 = (−(cid:52))−1f.

(2.22)

Next, by testing (2.20) with (−(cid:52))−1y2 and substituting by (2.22), we obtain

(cid:68)
(cid:69)
y2, (−(cid:52))−1y2
(cid:68)
(cid:69)
g, (−(cid:52))−1y2

=

λ

(cid:69)

(cid:68)−(cid:52)y1, (−(cid:52))−1y1
(cid:68)−(cid:52)y1, (−(cid:52))−1f
(cid:69)

H−1(Ω)×H1

0 (Ω)

+ λ

H−1(Ω)×H1

0 (Ω)

+

H−1(Ω)×H1

0 (Ω)

(2.23)

H−1(Ω)×H1

.

0 (Ω)

Using Lemma 1, the first term on the right hand side of (2.23) can be estimated,

H−1(Ω)×H1

0 (Ω)

≤ ||g||H−1(Ω)||(−(cid:52))−1y2||
H1
≤ C||g||H−1(Ω)||y2||H−1(Ω).

0 (Ω)

(2.24)

(2.25)

(cid:16)

(cid:17)
(−(cid:52)y1)((−(cid:52))−1y1)

(cid:90)
(cid:90)

Ω

dx

(2.26)

H−1(Ω)×H1

0 (Ω)

=

=

y2
1dx.

Ω

13

(cid:68)

g, (−(cid:52))−1y2

(cid:69)

(cid:68)−(cid:52)y1, (−(cid:52))−1y1

(cid:69)

Also, we can see by an application of Green’s formula that

The same is true replacing y1 with y2. Also,

(cid:68)−(cid:52)y1, (−(cid:52))−1f

(cid:69)

H−1(Ω)×H1

0 (Ω)

(cid:90)
(cid:90)

Ω

Ω

=

=

(cid:16)

(−(cid:52)y1)((−(cid:52))−1f

(cid:17)

y1f dx.

dx

(2.27)

From (2.23), (2.24), (2.26), and (2.27), we finally obtain

λ||y1||2

L2(Ω)

+ λ||y2||2

H−1(Ω)

≤ C

(cid:16)||f||L2(Ω)||y1||L2(Ω) + ||g||H−1(Ω)||y2||H−1(Ω)

(cid:17)

. (2.28)

This bound is valid for all f ∈ L2(Ω), g ∈ H−1(Ω), and so we can obtain the required

estimate by taking the norm on L2(Ω)× H−1(Ω) as ||(y1, y2)||L2(Ω)×H−1(Ω) = ||y1||2
||y2||2

. By the Hille-Yosida Theorem, A is the generator of a semigroup of contractions

L2(Ω)

+

H−1(Ω)

on Y .

2.2.1 Application

Here we apply the method to our specific problem of interest. We will use the transposition

solutions defined in this section to the problem outlined in Chapter 3.

Consider the problem

ztt − (cid:52)z = f,

z = u,

z(x, 0) = z0(x),

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω,

where f ∈ L1(cid:0)0, T ; H−1(Ω)(cid:1), the control function u(x, t) ∈ L2 (∂Ω × (0, T )), and the initial

zt(x, 0) = z1(x),

(2.29)

14

conditions z0(x) ∈ L2(Ω), z1(x) ∈ H−1(Ω).

By Theorem 2, we know that problem (2.29) has a solution when u is identically 0. But,

to study the control problem, one must make sense of what it means for u to control the

boundary. This is made precise in the method of transposition. We now introduce the

notion of solutions for (2.29). This will be the definition used when proving existence and

uniqueness.

Definition 2.2.1. (Transposition Solution) A function z = z(x, t) ∈ C(cid:0)[0, T ]; L2(Ω)(cid:1) ∩
C1(cid:0)[0, T ]; H−1(Ω)(cid:1) is a transposition solution if
(cid:28) ∂z

(cid:90) T

(cid:29)

(cid:90)

(cid:90)

(cid:90) T
(cid:28) ∂z
(cid:90)
(cid:90) T

∂t

0

f ydxdt =

0

Ω

−

+

zϕdxdt +

Ω

(T ), yT

∂t

(cid:29)

(0), y(0)

H−1(Ω)×H1

0 (Ω)

∂y
∂ν

udsdt,

(cid:90)

H−1(Ω)×H1
−

z(T )γT dx +

0 (Ω)

(cid:90)

Ω

Ω

z(0)

∂y
∂t

(0)dx

(2.30)

for all (ϕ, yT , γT ) ∈ L1(cid:0)0, T ; L2(Ω)(cid:1) × H1

∂Ω

0

0 (Ω) × L2(Ω), where y solves the problem



ytt − (cid:52)y = ϕ,

y = 0,

y(x, T ) = yT ,

yt(x, T ) = γT ,

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω.

(2.31)

Now, we introduce the main existence/uniqueness theorem, in the sense of transposition,

due to [38].

Theorem 3. (Existence and Uniqueness of Transposition Solutions). For every set of func-
tions (f, u, z0, z1) ∈ L1(0, T ; H−1(Ω)) × L2(∂Ω × (0, T )) × L2(Ω) × H−1(Ω), the equation

15

(2.29) has a unique transposition solution z. The map to z is continuous as a map into
C([0, T ]; L2(Ω)) ∩ C1([0, T ]; H−1(Ω)).

Proof. By Theorem 2, because of the semigroup theory, given initial data (f, z0, z1) ∈
L1(0, T ; H−1(Ω)) × L2(Ω) × H−1(Ω), there is a unique solution z = z(f, 0, z0, z1) to system
(2.29). The map to z is linear and continuous into C([0, T ]; L2(Ω))∩C1(0, T ; H−1(Ω)). Thus,

without loss of generality, by linearity we can consider the specific case where (f, z0, z1) =
(0, 0, 0). For, if z solves (2.29) for u ≡ 0, and ˜z solves (2.29) for (f, z0, z1) = (0, 0, 0), then

w = z + ˜z solves (2.29).

Now, let Λ be defined by Λ(ϕ) = ∂y

∂ν , where y solves



ytt − (cid:52)y = ϕ,

y = 0,

y(x, T ) = 0,

yt(x, T ) = 0,

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω.

(2.32)

By the Hidden Regularity estimate from Theorem 1, we have that ∂y

∂ν ∈ L2 (∂Ω × (0, T )).
Thus Λ defines a map Λ : L2(Ω × (0, T )) → L2 (∂Ω × (0, T )). Now, set z = Λ∗u, where
∂ν ds = (cid:104)Λ∗u, ϕ(cid:105)L2(Ω). Then, z ∈
L2(Ω × (0, T )). By multiplying (2.29) by y and integrating by parts in space in time, we

Λ∗ is defined in the sense of (cid:104)u, Λϕ(cid:105)L2(∂Ω) = (cid:82)

∂Ω u ∂y

arrive at

(cid:28) ∂z

∂t

(cid:29)

(0), y(0)

H−1(Ω)×H1

0 (Ω)

+

(cid:90)

Ω

z(0)

∂y
∂t

(0)dx +

(cid:90) T

(cid:90)

0

∂Ω

∂y
∂ν

udsdt = 0.

(2.33)

−

That is, z solves (2.29) in the transposition sense (uniquely) with zero initial data, in the

case yT = 0, γT = 0.

16

Now, we must show that the solution z actually does belong to z ∈ C(cid:0)[0, T ]; L2(Ω)(cid:1) ∩
C1(cid:0)[0, T ]; H−1(Ω)(cid:1). Let (un)∞
0 (Ω)(cid:1)∩C1(cid:0)[0, T ]; L2(Ω)(cid:1)

0 (Ω)×L2(Ω). Then there is a solution y ∈ C(cid:0)[0, T ]; H1

n=1 be a regular approximating sequence. Let τ ∈ [0, T ], and

(yτ , γτ ) ∈ H1

to



ytt − (cid:52)y = 0,

y = 0,

y(x, τ ) = yτ ,

yt(x, τ ) = γτ ,

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,
x ∈ Ω.

(2.34)

Since the initial data has enough regularity, the Hidden Regularity estimate of Theorem

1 applies, and we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂y(yτ , γτ )

∂ν

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)L2(∂Ω×(0,T ))

(cid:18)

≤ C

||yτ||

H1

0 (Ω)

(cid:19)

.

+ ||γτ||L2(Ω)

(2.35)

Now, recall definition 2.2.1 of the transposition solution. Take γτ = 0 and yτ = 0

(cid:28) ∂zn

∂t

and

respectively. From (2.34), the two equations are satisfied by zn ˙=z(0, un, 0, 0):

(cid:29)

(cid:90)

(cid:90) τ

(cid:90)

(τ ), yτ

H−1(Ω)×H1

0 (Ω)

=

Ω

(τ )yτ dx = −

∂zn
∂t

∂y(yτ , 0)

0

∂Ω

∂ν

undsdt

(2.36)

(cid:90)

Ω

zn(τ )γτ dx =

(cid:90) τ

(cid:90)

∂y(0, γτ )

0

∂Ω

∂ν

undsdt.

(2.37)

From here, we can easily obtain the Cauchy estimates. By using H¨older’s inequality and

(2.35),

||zn − zm||C([0,T ];L2(Ω)) = sup
τ∈[0,T ]

||γτ||

sup
L2(Ω)

=1

17

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) τ

0

(cid:90)

∂y(0, γτ )

∂Ω

∂ν

(cid:12)(cid:12)(cid:12)(cid:12)

(un − um)dsdt

≤ C||un − um||L2(∂Ω×(0,T )),

(2.38)

and

||(zn)t − (zm)t||C([0,T ];H−1(Ω)) = sup
τ∈[0,T ]

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) τ

0

=1

(cid:90)

∂y(yτ , 0)

∂Ω

∂ν

(cid:12)(cid:12)(cid:12)(cid:12)

(un − um)dsdt

||yτ||

sup
H1

0 (Ω)

≤ C||un − um||L2(∂Ω×(0,T )).

(2.39)

Taking limits, the theorem is proved.

18

Chapter 3

Hyperbolic Equations

3.1

Introduction

In this section, we consider global Carleman type estimates for the one-dimensional wave

equation with variable coefficients and a potential term. Specifically, the system of study is

the following:

utt − (a(x)ux)x + p(x, t)u = 0,

u(0, t) = 0,

u(1, t) = v(t),

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

ut(x, 0) = u1(x),

(x, t) ∈ (0, 1) × (0, T ),
t ∈ (0, T ),
x ∈ (0, 1).

(3.1)

The end time is fixed with T > 0. Also, we assume that a(x) ∈ C2([0, 1]) with a(x) ≥
a0 > 0 in [0, 1], the potential p(x) ∈ L∞((0, 1) × (0, T )), and the initial conditions u0 ∈
L2(0, 1) and u1 ∈ H−1(0, 1). Here, v = v(t) ∈ L2(0, T ) is called the control function and

u = u(x, t) is the associated state.

From Theorem 3, for (u0, u1) ∈ L2(0, 1) × H−1(0, 1), there is a unique transposition

solution u(x, t) to (3.1), such that

u ∈ C0([0, T ]; L2(0, 1)) ∩ C1([0, T ]; H−1(0, 1)).

(3.2)

The exact controllability problem for (3.1) at time T > 0 is then the following: Given

19

initial conditions (u0, u1) and final conditions (z0, z1), find a control function v ∈ L2(0, 1)

such that the corresponding solution u satisfies

u(x, T ) = z0, ut(x, T ) = z1, for x ∈ (0, 1).

(3.3)

In other words, through the control function v(t), is it possible to drive a set of given

initial conditions to a prescribed final state? Because of linearity and time reversibility

of the operator of study, exact controllability of (3.1) at time t = T is equivalent to null

controllability at T . That is, under the same initial conditions, the corresponding solution u

satisfies

u(x, T ) = 0, ut(x, T ) = 0, for x ∈ (0, 1).

(3.4)

The controllability problem for the wave equation has been studied for both one dimension

and higher dimensions. It is known that (3.1) is null controllable for a large enough time T
depending on a, although the majority of previous works focus mainly on the case a ≡ 1.

By the Hilbert Uniqueness Method [38], the null controllability of (3.1) is equivalent to the

observability of the adjoint problem under certain geometric conditions from [4] on subsets

Γ(x0) of the boundary Γ. These geometric conditions are taken into account because of

possible Gaussian beam solutions from [41], [43], [44], that can propagate along curves.
Thus, for x0 ∈ Rn, if one considers subsets of the form

Γ(x0) = {x ∈ Γ : (x − x0) · n(x) > 0},

20

then the boundary observability inequality for the controllability problem is

||(u0, u1)||2

L2(Ω)×H−1(Ω)

≤ C

dΓdt.

(3.5)

(cid:90) T

(cid:90)

0

Γ(x0)

(cid:12)(cid:12)(cid:12)(cid:12) ∂u

∂n

(cid:12)(cid:12)(cid:12)(cid:12)2

Previous authors have developed a variety of tools to derive this inequality. Multiplier

methods, [32], [38], microlocal analysis, [4], finite difference, [56], and Carleman estimates [5],

[6], [13], [22], [24], [27], [29], [49], among other techniques have been developed. In addition,

for particular cases, sharp results about the observability constant C, and the time T are

known; see for example [12], [15], [50], [52], and [53]. The problem has been studied in the

context of unique continuation in [2], [14], [28], [35], [30], [31], [47], and [48]. Controllability

results for a subset of functions a(x) for higher dimensions was considered in [40]. A survey

paper for these results can be found in [54].

We note here that we consider control on the boundary. One could also consider a control

function as acting in the interior of the domain (as will be done in Chapter 4). In some sense,

the boundary control is connected to internal control in the limit, as in [17], [18].

Carleman estimates (developed in section 3.2), the focus of this chapter, offer more

information and utility than just the existence of the observability inequality. They give, for
example, the advantage of being able to easily deal with L∞ potential terms, while requiring

less regularity than the microlocal analysis technique. This flexibility means that studying

the effects that the variable coefficient a(x) introduces is of interest.

Once the Carleman estimate is proved, a variational technique adapted from [27] naturally

produces the control function required. Thus, the question becomes: For which functions

a(x) does the Carleman estimate hold?

In the recent work [13], the authors study the same control problem (3.1) for the one

21

dimensional case. They prove Carleman estimates and controllability for (3.1), but with a

restrictive requirement on the function a(x). Specifically, they require a(x) to belong to the

family

(cid:40)
a ∈ C3([0, 1]) : a(x) ≥ a0 > 0,

A(x0, a0) =

− min

[0,1]

(a(x) + (x − x0)ax(x)) < min

[0,1]

(cid:18)

a(x) +

1
2

(cid:19)(cid:41)

,

(x − x0)ax(x)

(3.6)

where x0 < 0 and a0 > 0 is a positive constant.
It can be easily seen that all constant
functions are in this family. Also, since x0 < 0, A contains all strictly positive, monotonically
increasing functions. Furthermore, if the derivative is small, |ax| << 1, the functions will

belong to this family. However, if the function decreases too rapidly in (0,1), this condition

may fail to be true.

In Section 3.2, we consider a different condition on a(x), providing more functions for

which controllability of (3.1) will hold, as well as an computationally easier condition to

check. Let us take x0 < 0 and a0 > 0 a positive constant. We introduce the family

(cid:26)
a ∈ C2([0, 1]) : a(x) ≥ a0 > 0, 2 − ax(x)(x − x0)

a(x)

B(x0, a0) =

(cid:27)

> 0

.

(3.7)

It will be seen that this is a natural choice of family, given the weight function that will

be chosen in Section 3.1.1. Besides the decreased regularity needed on a(x), notice that
B(x0, a0) also contains all constant functions, as well as any strictly positive, monotonically

decreasing function. From now on, we shall always assume that the function a(x) belongs to
B(x0, a0). Also, for the remainder of the section, by a time translation we will use the time
interval (−T, T ) to make the notation and proofs easier. With this assumption, the main

22

result of this section is the existence of a control function v(t) so that the controllability for

(3.1) holds true.

Theorem 4. Given initial conditions u0(x) ∈ L2(0, 1), u1(x) ∈ H−1(0, 1), for T large
enough, there exists a control function v(t) ∈ L2 ({1} × (−T, T )) such that the solution

u(x, t) of

utt − (a(x)ux)x + p(x, t)u = 0,

u(1, t) = v(t),

u(0, t) = 0,
u(x,−T ) = u0(x),

ut(x,−T ) = u1(x),

(x, t) ∈ (0, 1) × (−T, T ),
t ∈ (−T, T ),
x ∈ (0, 1),

(3.8)

satisfies u(x, T ) = ut(x, T ) = 0.

The rest of this chapter is organized as follows. In Section 3.1.1, we outline the particular

weight functions used for the Carleman estimate. In Section 3.2, we state and prove various

Carleman estimates. First, the estimate is proved in the special case that the functions

vanish at the time endpoints, and then this restriction is eventually removed. Here we

give a precise time for the T condition necessary in Theorem 4. In Section 3.3, after some

preliminary definitions, we use the results of Section 3.2 to prove Theorem 4. This theorem

will then be extended in Section 3.4, where we prove an increased regularity result.

3.1.1 Weight Functions

Carleman Estimates give weighted bounds on functions and their derivatives. As in [13],
often the weight functions are defined in the following fashion: Let x0 /∈ Ω ∈ Rn and

23

β ∈ (0, 1). Define for (x, t) ∈ Ω × (−T, T ),

˜ψ(x, t) = |x − x0|2 − βt2 + C0, and for λ > 0, ˜ϕ(x, t) = eλ ˜ψ(x,t),

(3.9)

where C0 > 0 is chosen such that ˜ψ ≥ 1 in Ω × (−T, T ).

However, since the system of study (3.1) contains the function a(x), it is reasonable that

this should be incorporated into the weight function. Thus, in this system, we consider a

special test function for the one-dimensional case. Fix x0 < 0, and then define the weight

function

ψ(x, t) =

(cid:90) x

0

s − x0
a(s)

ds − βt2 + C0, and for λ > 0, ϕ(x, t) = eλψ(x,t),

(3.10)

where C0 > 0 is chosen such that ψ ≥ 1 in (0, 1) × (−T, T ). Note that by the choice of
a(x) ∈ B(x0, a0) from (3.7), then (3.10) is well defined. Here ϕ(x, t) depends on β and λ,

but for simplicity of notation these dependencies are omitted. Also, as in [5] we define for

m > 0, the spaces

L∞≤m(0, 1) = {q ∈ L∞(0, 1),||q||L∞(0,1) ≤ m},

L∞≤m((0, 1) × (−T, T )) = {p ∈ L∞((0, 1) × (−T, T )),||p||L∞ ≤ m}.

We will also use the following notation when appropriate:

Lu := utt − (a(x)ux)x.

(3.11)

24

3.2 Carleman Estimates

The method of deriving the Carleman Estimate comes from incorporating the weight function

into a series of integration by parts. This type of argument was used in the method from

[22]. Also, see [5], [27], [53]. The novelty comes from the new weight function, where ψ now

has a(x) in its definition. First, a new state w is considered from the original state and the

weight function. From here, the estimate can be derived for w. The main result is the global

Carleman estimate.

Theorem 5. (Carleman Estimate). Assume that x0 < 0, a0 > 0. There exists β ∈ (0, 1)

such that

T >

1
β

max
x∈[0,1]

(x − x0)(cid:112)a(x)

.

(3.12)

Then for any m > 0, there exists λ > 0 independent of m, s0 = s0(m) > 0 and a positive
constant M = M (m) such that for ψ and ϕ as defined as in (3.10), for a(x) ∈ B(x0, a0), for
all p(x, t) ∈ L∞≤m((0, 1) × (−T, T )) and for all s ≥ s0,
(cid:90) T
(cid:90) 1

e2sϕ|w|2dxdt

(3.13)

(cid:90) T
(cid:90) 1
(cid:90) 1
e2sϕ(|wt|2 + |wx|2)dxdt + s3
e2sϕ(−T )(|wt(−T )|2 + |wx(−T )|2)dx + s3
(cid:90) 1
(cid:90) T

(cid:90) T

−T

−T

0

0

0

(cid:90) 1

0

s

+ s

≤ M

e2sϕ|Lw + pw|2dxdt + M s

e2sϕ|wx(1, t)|2dt,

−T

−T

0

e2sϕ(−T )|w(−T )|2dx

for all w ∈ L2(−T, T ; H1
and wx(1, t) ∈ L2(−T, T ).

0 (0, 1)) also satisfying wtt − (a(x)wx)x + pw ∈ L2((0, 1) × (−T, T ))

Before showing Theorem 5, we first prove a Carleman estimate in the special case where
the function and its derivative vanishes at the endpoints, so that v(·,±T ) = vt(·,±T ) = 0.

25

Note here that T can be arbitrary.

Theorem 6. Let ϕ(x, t) be defined as in (3.10). Also, let a(x) ∈ B(x0, a0) as defined in
(3.7). Then, there exists positive constants s0, and M , depending only on x0, a0,||a||, and T
such that, for all s ≥ s0,
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) 1
(cid:90) T
e2sϕ(|vt|2 + |vx|2)dxdt + s3
0 (0, 1)(cid:1) also satisfying Lv ∈ L2 ((0, 1) × (−T, T )) ,
for any v ∈ L2(cid:0)−T, T ; H1

−T
e2sϕ|vx(1, t)|2dt

e2sϕ|Lv|2dxdt + M s

e2sϕ|v|2dxdt

s

−T
≤ M

(cid:90) 1

0

0

−T

0

−T

vx(1, t) ∈ L2(−T, T ), and v(·,±T ) = vt(·,±T ) = 0.

Remark: All computations can be done for smooth functions v. By classical results using

density arguments, the result will hold true for all v(x, t) satisfying the conditions of Theorem

6.

Proof. As noted previously, the proof follows similar arguments to that in [13]. We indicate

in the proof especially where the new admissibility condition of (3.7) and weight function of

(3.10) are used.

Consider the function

w = esϕv,

and define P w as the conjugation

P w = esϕL(e−sϕw) = esϕ(cid:0)(e−sϕw)tt − (a(e−sϕw)x)x

(cid:1) .

(3.14)

(3.15)

26

By the admissibility condition (3.7), we can find constants α, β such that

0 <

4β

2β + 1

< α <

2 − axψx
2β + 1

.

Then, we can decompose the function as P w = P1w + P2w + Rw, where

P1w = wtt − (awx)x + s2λ2ϕ2w(|ψt|2 − a|ψx|2),
P2w = (α − 1)sλϕw(ψtt − (aψx)x) − sλ2ϕw(|ψt|2 − a|ψx|2)

− 2sλϕ(ψtwt − aψxwx),

Rw = −αsλϕw(ψtt − (aψx)x).

(3.16)

(3.17)

Let us note here that by the choice of ψ from (3.10), the term (ψtt− (aψx)x) = −2β− 1 is

constant. This will simplify some of the expressions below. However, we keep this expression

in its generic form, so that one may use different possible weight functions not considered in

this work. We will first estimate the integral

(cid:90) T

(cid:90) 1

−T

0

I =

(P1w)(P2w)dxdt =

3(cid:88)

i,j=1

Iij.

(3.18)

There are nine integrals to compute. Throughout, we will repeatedly use the assumption
that w(·,±T ) = wt(·,±T ) = 0. Then, integration by parts in time, space, or both gives the

following explicit representations. Because of the complexity of the expressions, only the

final representations will be provided, without intermediate computation.

I11 = (α − 1)sλ

(cid:90) T

(cid:90) 1

−T

0

wttϕw(ψtt − (aψx)x)dxdt

27

0

0

0

−T

0

2

2

−T

−T

−T

sλ2

sλ3

= sλ2

ϕ|w|2|ψt|2(ψtt − (aψx)x)dxdt.

ϕ|w|2ψtt(ψtt − (aψx)x)dxdt

(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
ϕ|wt|2(ψtt − (aψx)x)dxdt
= (1 − α)sλ
− (1 − α)
(cid:90) 1
(cid:90) T
− (1 − α)
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) T
I12 = −sλ2
wttϕw(|ψt|2 − a|ψx|2)dxdt
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) 1
ϕ|wt|2(|ψt|2 − a|ψx|2)dxdt − sλ2
− 5sλ3
(cid:90) T
(cid:90) 1
−T
2
− sλ4
ϕ|w|2|ψt|2(|ψt|2 − a|ψx|2)dxdt.
2
(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
I13 = −2sλ
(cid:90) 1
(cid:90) T
ϕ|wt|2ψttdxdt + sλ2
−T
(cid:90) 1
(cid:90) T
ϕ|wt|2(aψx)xdxdt + sλ2

(cid:90) T
(cid:90) 1
wttϕ(ψtwt − aψxwx)dxdt
(cid:90) T

ϕ|w|2|ψt|2ψttdxdt +

sλ3
2

= sλ

+ sλ

0

0

−T

0

−T

0

−T

0

−T

0

−T

0

−T

−T

0

− 2sλ2

−T

0

ϕaψxψtwxwtdxdt.

(cid:90) 1
ϕ|wt|2|ψt|2dxdt

0

ϕ|wt|2a|ψx|2dxdt

(cid:90) 1

ϕ|w|2|ψtt|2dxdt

0

−T
ϕ|w|2a|ψx|2ψttdxdt

Making use of the fact that ψtt − (aψx)x is constant,

0

−T

(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) 1
(awx)xϕw(ψtt − (aψx)x)dxdt
(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) 1

ϕa|wx|2(ψtt − (aψx)x)dxdt

ϕ|w|2(aψx)x(ψtt − (aψx)x)dxdt

ϕa|w|2|ψx|2(ψtt − (aψx)x)dxdt

−T

−T

0

0

−T

0

I21 = (1 − α)sλ

= −(1 − α)sλ

(1 − α)

2

(1 − α)

2

+

+

sλ2

sλ3

28

(|ax|2 + aaxx)|ψx|2 + 4aaxψxψxx + 2a(aψx)xψxx

(cid:17)

dxdt

ϕ|w|2(ax(aψx)xx + a(aψx)xxx)dxdt.

I22 = sλ2

−T

sλ

2

−T

0

−T

0

−T

0

= −sλ2

(cid:90) 1
(cid:90) 1

0

ϕ|w|2aψx(aψx)xxdxdt

ϕa|wx|2(|ψt|2 − a|ψx|2)dxdt

(cid:90) T
(cid:90) T
− (1 − α)sλ2
− (1 − α)
(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
(awx)xϕw(|ψt|2 − a|ψx|2)dxdt
(cid:90) 1
(cid:90) T
ϕ|w|2(cid:16)
(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) 1
ϕa|wx|2(ψtt + aψxx)dxdt
(cid:90) T

− sλ2
2
sλ3
2
sλ4
2
− sλ3

(awx)xϕ(ψtwt − aψxwx)dxdt

+ sλ2

+

+

−T

0

−T

0

−T

0

−T

0

0

0

0

−T

−T

−T

= sλ

ϕa|wx|2(|ψt|2 + a|ψx|2)dxdt − 2sλ2

ϕ|w|2(aψx)x(|ψt|2 − a|ψx|2)dxdt

ϕ|w|2a|ψx|2(|ψt|2 − a|ψx|2)dxdt

ϕ|w|2aψx(ax|ψx|2 + 2aψxψxx)dxdt.

I23 = 2sλ

(cid:90) T

(cid:90) 1

−T

0

ϕaψxψtwxwtdxdt

− 2sλ

[a(1)2|wx(1, t)|2ϕ(1, t)ψx(1, t) − a(0)2|wx(0, t)|2ϕ(0, t)ψx(0, t)]dt.

(cid:90) 1

0

ϕ3|w|2(|ψt|2 − a|ψx|2)(ψtt − (aψx)x)dxdt.

−T

(cid:90) T
I31 = (α − 1)s3λ3
(cid:90) 1
(cid:90) T

−T

I32 = −s3λ4

−T

0

ϕ3|w|2(|ψt|2 − a|ψx|2)2dxdt.

29

−T

ϕ3w(|ψt|2 − a|ψx|2)(ψtwt − aψxwx)dxdt

(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
I33 = −2s3λ3
(cid:90) 1
(cid:90) T
ϕ3|w|2(|ψt|2 − a|ψx|2)(ψtt − (aψx)x)dxdt
(cid:90) T
(cid:90) 1

(cid:18)
|ψt|2ψtt +

ϕ3|w|2

+ 2s3λ3

= s3λ3

−T

−T

1
2

aaxψx|ψx|2 + a2|ψx|2ψxx

0

0

0

ϕ3|w|2(|ψt|2 − a|ψx|2)2dxdt.

+ 3s3λ4

−T

0

(cid:19)

dxdt

Collecting all the terms and simplifying, we find that

I =

= sλ

+ sλ

0

0

0

−T

−T

−T

(P1w)(P2w)dxdt

(cid:90) T
(cid:90) 1
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
ϕ|wt|2(2ψtt − α(ψtt − (aψx)x)dxdt
(cid:90) T
(cid:90) 1
ϕa|wx|2(α(ψtt − (aψx)x) + 2(aψx)x − axψx)dxdt
(cid:90) 1
(cid:90) T
ϕ(|wt|2|ψt|2 − 2aψxψtwxwt + a2|wx|2|ψx|2)dxdt
(cid:90) 1
(cid:90) T
ϕ3|w|2(|ψt|2 − a|ψx|2)2dxdt
(cid:90) T
(cid:90) 1
ϕ3|w|2(|2ψt|2ψtt + aaxψx|ψx|2 + 2a2|ψx|2ψxx)dxdt
(cid:90) T
(cid:16)

ϕ3|w|2(|ψt|2 − a|ψx|2)(ψtt − (aψx)x)dxdt

−T

−T

−T

−T

0

0

0

0

+ 2sλ2

+ 2s3λ4

+ s3λ3

a(1)2|wx(1, t)|2ϕ(1, t)ψx(1, t) − a(0)2|wx(0, t)|2ϕ(0, t)ψx(0, t)

dt

(cid:17)

+ αs3λ3

− 2sλ

+ X,

−T

where X is the sum of the remaining terms, satisfying

|X| ≤ M sλ4

ϕ3|w|2dxdt.

(cid:90) T

(cid:90) 1

−T

0

30

Now, we will take advantage of the new class of admissible functions of B(x0, a0) in

estimating I. First, let us notice that the third term above is a perfect square, and so we

have

2sλ2

(cid:90) T

(cid:90) 1

−T

0

(cid:16)|wt|2|ψt|2 − 2aψxψtwxwt + a2|wx|2|ψx|2(cid:17)

ϕ

dxdt ≥ 0.

Consider next the terms of order sλ. Let us briefly mention here that estimation of these
sλ terms is how the authors from [13] arrived at the class A defined in (3.6). For us, since

α and β were chosen such that

4β
2β+1 < α by (3.16), we obtain by computation that the

integrand of the first term in I satisfies

2ψtt − α(ψtt − (aψx)x) > 0.

Similarly, since α < 2−axψx

2β+1 by (3.16), we obtain that the integrand of the second term

satisfies

α(ψtt − (aψx)x) + 2(aψx)x − axψx > 0.

Thus, the order sλ terms can be estimated as follows:

(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
ϕ|wt|2(2ψtt − α(ψtt − (aψx)x))dxdt
(cid:90) 1
(cid:90) T
ϕa|wx|2(α(ψtt − (aψx)x) + 2(aψx)x − axψx)dxdt

(cid:90) T

(cid:90) 1

−T

0

0

ϕ|wt|2dxdt + M sλ

ϕ|wx|2dxdt.

sλ

+sλ

−T
≥M sλ

−T

0

−T

0

It remains to estimate the higher order terms. We can rewrite these terms in the following

31

fashion:

2s3λ4

+s3λ3

+αs3λ3

0

−T

(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
ϕ3|w|2(|ψt|2 − a|ψx|2)2dxdt
(cid:90) T
(cid:90) 1
ϕ3|w|2(|2ψt|2ψtt + aaxψx|ψx|2 + 2a2|ψx|2ψxx)dxdt
(cid:90) T
(cid:90) 1

ϕ3|w|2(|ψt|2 − a|ψx|2)(ψtt − (aψx)x)dxdt

−T

0

−T

0

ϕ3|w|2Fλ (x, Y (x, t)) dxdt,

=s3λ3

−T

0

where we have let Y = |ψt|2 − a|ψx|2 and

Fλ = 2λY 2 + (2ψtt + α(ψtt − (aψx)x))Y + a|ψx|2(2ψtt + axψx + 2aψxx)

= 2λY 2 − (4β + α(2β + 1))Y − a(x)|ψx|2(4β + axψx − 2).

This is a quadratic polynomial in Y . As such, since λ > 0, we can estimate by the critical
− a(x)|ψx|2(4β + axψx − 2). Once again, by (3.16),
point to get Fλ(x, Y ) ≥ − (4β+α(2β+1))2
we have 4β + axψx − 2 < 0. Thus, for λ large enough the function Fλ is nonnegative, and

8λ

we can conclude

s3λ3

(cid:90) T

(cid:90) 1

−T

0

ϕ3|w|2Fλ(x)dxdt ≥ M s3λ3

(cid:90) T

(cid:90) 1

−T

0

ϕ3|w|2dxdt.

Combining all of the estimates so far gives

(cid:90) T

(cid:90) 1
(cid:90) T

0

I =

−T
≥ M sλ

(cid:90) 1

−T

0

(P1w)(P2w)dxdt

ϕ(|wt|2 + |wx|2)dxdt + M s3λ3

32

(cid:90) T

(cid:90) 1

−T

0

ϕ3|w|2dxdt

(3.19)

(cid:90) T

(cid:90) 1

−T

0

− M sλ4

ϕ3|w|2dxdt − M sλ

(cid:90) T

−T

|wx(1, t)|2dt.

Let us use (3.19) to finally obtain the desired estimate. Recall the decomposition (3.17) of

P w to write

(cid:90) T

(cid:90) 1

−T

0

(cid:16)|P1w|2 + |P2w|2 + 2(P1w)(P2w)
(cid:17)

dxdt =

(cid:90) T

(cid:90) 1

−T

0

|P w − Rw|2dxdt.

By the Cauchy estimate, we have that

(cid:90) T

(cid:90) 1

−T

0

|P w − Rw|2dxdt ≤ M

(cid:90) T

(cid:90) 1

−T

0

|P w|2dxdt + M s2λ2

(cid:90) T

(cid:90) 1

−T

0

ϕ2|w|2dxdt.

Thus, we obtain

(cid:90) T

(cid:90) 1

−T

0

(P1w)(P2w)dxdt ≤ M

(cid:90) T

(cid:90) 1

−T

0

|P w|2dxdt + M s2λ2

(cid:90) T

(cid:90) 1

−T

0

ϕ2|w|2dxdt.

(3.20)

Combining (3.19) and (3.20) gives

(cid:90) T
(cid:90) T

−T

−T

(cid:90) 1
(cid:90) 1
(cid:90) T

0

0

sλ

≤M

+M sλ4

−T

0

(cid:90) T

(cid:90) 1

−T

0

ϕ3|w|2dxdt

(cid:90) T

ϕ(|wt|2 + |wx|2)dxdt + s3λ3
(cid:90) T
(cid:90) 1

−T
ϕ3|w|2dxdt + M s2λ2

|P w|2dxdt + M sλ

|wx(1, t)|2dt

(cid:90) 1

ϕ2|w|2dxdt.

−T

0

When s is large, the last two terms on the right hand side can be absorbed to the left.

Finally, we can use w = vesϕ, P w = esϕLv to rewrite the equation in terms of v, so that

33

Theorem 6 holds. We obtain

(cid:90) T

s

−T
≤ M

(cid:90) 1
(cid:90) T

0

(cid:90) T
(cid:90) T
(cid:90) 1
e2sϕ(|vt|2 + |vx|2)dxdt + s3

e2sϕ|Lv|2dxdt + M s

(cid:90) 1

−T

0

e2sϕ|v|2dxdt

0

−T
e2sϕ|vx(1, t)|2dt

−T

3.2.1 Carleman Estimate for large T

Theorem 6 was valid for functions that vanished at the endpoints. If the time T is large

enough, this condition can be removed. The argument will combine the previous Carleman

inequality along with an energy estimate. The major ideas of the proof can be found in [5],

although some modification is needed.

First, we recall the weighted Poincar´e inequality from [5], as applied to this case:

Lemma 2. (Poincar´e): Let φ ∈ C2([0, 1]) and assume that

|φx| ≥ δ > 0.

inf
[0,1]

Then, there exists some constants s0 > 0 and M > 0 such that for all s ≥ s0, and for all
z ∈ H1

0 ([0, 1]),

s2

e2sφ|z|2dx ≤ M

0

0

e2sφ|zx|2dx.

(3.21)

(cid:90) 1

(cid:90) 1

For the following, we will need to use a slightly modified form of this inequality, with

34

(x−x0)√

a(x)

that T > 1
β max
x∈[0,1]
that for all s ≥ s0,
(cid:90) T
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) T
e2sϕ(|wt|2 + |wx|2)dxdt + s3
for all w ∈ L2(cid:0)(−T, T ); H1

e2sϕ|Lw|2dxdt + M s

≤M

−T

−T

−T

0

s

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|w|2dxdt

(3.23)

e2sϕ|wx(1, t)|2dt,

a(x) ∈ B(x0, a0), which is immediately seen.

(cid:90) 1

0

s2

e2sφ|z|2dx ≤ M

(cid:90) 1

0

e2sφa(x)|zx|2dx.

(3.22)

Note that by direct computation, our weight function ϕ(x, t) defined in (3.10) indeed

satisfies the hypothesis for Lemma 2. Then, we can state the Carleman estimate for large

enough T .

Theorem 7. Let ϕ(x, t) be defined as in (3.10). Also, let β satisfy (3.16) and T be such

. Then there exists some s0 > 0 and a positive constant M , such

0

0 (0, 1)(cid:1) satisfying Lw ∈ L2 ((0, 1) × (−T, T )) and wx(1, t) ∈

L2(−T, T ).

Proof. By the assumption on T , there exists some η ∈ (0, T ), and 
 > 0 such that

(1 − 
)(T − η)β ≥ max
x∈[0,1]

x − x0(cid:112)a(x)

.

(3.24)

To facilitate work near the time endpoints, we introduce a cutoff function χ ∈ C∞
that 0 ≤ χ ≤ 1 and

c (R) such

1,

0,

χ(t) =

if |t| ≤ T − η,

if |t| ≥ T.

35

Set v = χw. This function satisfies the hypotheses v(·,±T ) = vt(·,±T ) = 0 of Theorem 6,

and so the Carleman estimate from there applies:

(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) 1
e2sϕ(|vt|2 + |vx|2)dxdt + s3

(cid:90) T

−T

0

e2sϕ|Lv|2 + M s

s

≤M

−T

0

e2sϕ|vx(1, t)|2dt.

−T

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|v|2dxdt

Now, since Lv = χLw + χttw + 2χtwt, it follows that

(cid:90) T−η
(cid:90) 1
(cid:90) T
(cid:90) 1
(cid:90) −T +η

−T +η

−T

0

0

s

≤M

+M

−T

0

e2sϕ(|wt|2 + |wx|2)dxdt + s3
(cid:90) 1

e2sϕ|Lw|2dxdt + M s

(cid:90) T

e2sϕ(|wt|2 + |w|2)dxdt + M

−T

e2sϕ|wx(1, t)|2dt

(cid:90) T

(cid:90) 1

T−η

0

(cid:90) T−η

(cid:90) 1

−T +η

0

e2sϕ|w|2dxdt

(3.25)

e2sϕ(|wt|2 + |w|2)dxdt.

Denote by Es = Es(t) the energy correlated with L,

e2sϕ(|wt|2 + a(x)|wx|2)dx, for all t ∈ (−T, T ).

(cid:90) 1

0

Es(t) =

1
2

(cid:90) 1

0

(cid:90) 1
(cid:90) 1

0

−s

dEs
dt

Then, taking derivatives,

dEs
dt

= s

e2sϕϕt(|wt|2 + a|wx|2)dx +

(cid:90) 1

0

e2sϕ(wtwtt + awxtwx)dx.

Through integration by parts,

e2sϕϕt(|wt|2 + a|wx|2)dx + 2s

(cid:90) 1

0

e2sϕaϕxwtwxdx

(3.26)

e2sϕwtLwdx.

=

0

36

We must now estimate these terms. Let us first restrict the value of t to work in one specific

time interval.
Estimate for t ∈ (T − η, T ):

We claim that

− s

dEs
dt

(cid:90) 1

0

e2sϕ(ϕt +

√
aϕx)(|wt|2 + a|wx|2)dx ≤

(cid:90) 1

0

e2sϕwtLwdx.

(3.27)

To see this, first note that since a(x) and ϕx are strictly positive, we have

2sϕxwtwxa ≥ −sϕx

√

a(|wt|2 + a|wx|2).

So, from (3.26) above, this gives

(cid:90) 1
(cid:90) 1

0

0

− s

dEs
dt

≤

e2sϕϕt(|wt|2 + a|wx|2)dx − s

e2sϕwtLwdx,

(cid:90) 1

0

√
a(|wt|2 + a|wx|2)dx

e2sϕϕx

from which

(cid:90) 1

0

− s

dEs
dt

√

e2sϕ(ϕt +

aϕx)(|wt|2 + a|wx|2) ≤

(cid:90) 1

0

e2sϕwtLwdx.

(3.28)

Due to (3.24), the definition of ϕ from (3.10), and the restriction of t ∈ (T − η, T ), we notice

that

−(ϕt +

√

aϕx) ≥

(cid:18)
2β(T − η) − √

(cid:19)

x − x0
a(x)

a

eλψ ≥ c > 0.

37

Thus, we arrive at

dEs
dt

+ sc

(cid:90) 1

0

e2sϕ(|wt|2 + a|wx|2)dx ≤

(cid:90) 1

0

e2sϕwtLwdx.

(3.29)

The right hand side can also be bounded,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) 1

0

(cid:12)(cid:12)(cid:12)(cid:12) ≤ sc

2

(cid:90) 1

0

(cid:90) 1

0

|Lw|2dx.

e2sϕ|wt|2dx +

1
2sc

e2sϕwtLwdx

By absorbing the first term above to the left hand side of (3.29), and using the definition of

the energy, we have

dEs
dt

+

1
2

scEs ≤ 1
2sc

e2sϕ|Lw|2dx.

Then, by the Gronwall Lemma, for all t ∈ (T − η, T ),

(cid:90) 1

0

Es(t) ≤ Es(T − η)esc(T−η−t) +
≤ Es(T − η)esc(T−η−t) +

1
2sc

1
2sc

(cid:90) t
(cid:90) T

T−η

T−η

0

esc(τ−t)
(cid:90) 1

(cid:90) 1

0

e2sϕ(τ )|Lw(τ )|2dxdτ

e2sϕ(τ )|Lw(τ )|2dxdτ.

Integration in the t variable gives

(cid:90) T

T−η

(cid:90) T

(cid:90) T

(cid:90) 1

−T

0

η
2sc

e2sϕ|Lw|2dxdt

Es(t)dt ≤ Es(T − η)

T−η
Es(T − η) +

≤ M
s

esc(T−η−t)dt +
(cid:90) 1
(cid:90) T

M
s

−T

0

e2sϕ|Lw|2dxdt.

(3.30)

(3.31)

It remains then to get a pointwise bound on Es(T − η), in terms of Es(τ ), where τ ∈
(−T + η, T − η). To do this, we recall (3.26), and integrate both sides in t from τ to T − η,

38

producing

(cid:90) T−η

(cid:90) 1

τ

0

Es(T − η) − Es(τ ) = s

(cid:90) T−η

(cid:90) 1

− 2s

τ

0

(cid:90) T−η
e2sϕϕt(|wt|2 + a|wx|2)

(cid:90) 1

e2sϕaϕxwtwxdxdt +

e2sϕwtLwdxdt.

τ

0

Here we use Cauchy-Schwarz to estimate. Since the resulting integrands are positive, we

may integrate over the larger time domain to arrive at

Es(T − η) − Es(τ ) ≤ M s

Es(t)dt +

M
s

(cid:90) T−η

−T +η

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|Lw|2dxdt.

Once more we integrate, this time in τ between −T + η and T − η. Since s is large enough,

the second term on the left can be absorbed to obtain

(cid:90) T−η

−T +η

(cid:90) T

(cid:90) 1

−T

0

Es(T − η) ≤ M s

Es(t)dt +

M
s

e2sϕ|Lw|2dxdt.

(3.32)

Using the relation (3.32) on the right hand side of (3.31), the fact that s is large, and

positivity of the integrands gives

(cid:90) T
(cid:90) 1
(cid:90) T−η

T−η

0

≤M

−T +η

0

e2sϕ(|wt|2 + a|wx|2)dxdt
(cid:90) 1

e2sϕ(|wt|2 + |wx|2)dxdt +

(cid:90) T

(cid:90) 1

−T

0

M
s

e2sϕ|Lw|2dxdt.

(3.33)

By the Weighted Poincar´e estimate from Lemma 2, we finally get

(cid:90) T

(cid:90) 1

s

T−η

0

e2sϕ(|wt|2 + a|wx|2 + s2|w|2)dxdt

39

(cid:90) T−η

−T +η

≤M s

Es(t)dt + M

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|Lw|2dxdt.

(3.34)

Estimate for t ∈ (−T,−T + η)
By the change of variable t → −t, we obtain similar estimates on (−T,−T + η). By the

same procedure, we obtain

(cid:90) −T +η
(cid:90) 1
(cid:90) 1
(cid:90) T−η
e2sϕ(|wt|2 + a|wx|2)dxdt

−T

0

e2sϕ(|wt|2 + |wx|2)dxdt +

≤M

−T +η

0

(cid:90) T

(cid:90) 1

−T

0

M
s

e2sϕ|Lw|2dxdt,

(3.35)

and

(cid:90) −T +η
(cid:90) 1
(cid:90) T−η

0

s

−T
≤M s

−T +η

e2sϕ(|wt|2 + a|wx|2 + s2|w|2)dxdt

(cid:90) T

(cid:90) 1

−T

0

Es(t)dt + M

e2sϕ|Lw|2dxdt.

(3.36)

To conclude, combining estimates (3.25), (3.33), and (3.35), for s large enough, we can say

(cid:90) 1
(cid:90) T−η
(cid:90) 1
(cid:90) T

−T +η

0

−T

0

s

≤M

e2sϕ(|wt|2 + |wx|2 + s2|w|2)dxdt

(cid:90) T

−T

e2sϕ|Lw|2dxdt + M s

e2sϕ|wx(1, t)|2dt.

(3.37)

3.2.2 Pointwise Carleman Estimate at −T

Following similar steps as in Theorem 7, we obtain an additional estimate at time −T .

40

Corollary 1. Under the conditions of Theorem 7, we obtain

(cid:90) 1
e2sϕ(−T )(|wt(−T )|2 + |wx(−T )|2)dx + s3
(cid:90) 1
(cid:90) T

(cid:90) T

s

0

(cid:90) 1

0

e2sϕ|Lw|2dxdt + M s

e2sϕ|wx(1, t)|2dt,

≤M

−T

0

−T

e2sϕ(−T )|w(−T )|2dxdt

(3.38)

for all w ∈ L2((−T, T ); H1
L2(−T, T ).

0 (0, 1)) satisfying Lw ∈ L2((0, 1) × (−T, T )) and wx(1, t) ∈

Proof. By the change of variable t → −t, we obtain similar estimates to (3.30) and (3.32).
Evaluating these at t = −T , we get an estimate for Es(−T ). Then, by combining the

Poincar´e inequality and Theorem 7, we obtain the desired estimate (3.38).

With the Carleman estimates for the operator Lw, we can obtain the Carleman estimate

for the operator L + p for bounded p. With the results proved so far, we can finally complete

the proof of the first theorem.

Proof. (Theorem 5). Notice that p(x, t) ∈ L∞≤m((0, 1) × (−T, T )) so that

|Lw|2 ≤ 2|Lw + pw|2 + 2(cid:107)p(cid:107)2

L∞((0,1)×(−T,T ))|w|2 ≤ 2|Lw + pw|2 + 2m2|w|2.

Taking s0 large enough, the term

2M m2

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|w|2dxdt

can be absorbed by the left hand side of (3.23). Then combining (3.23) and (3.38), we obtain

41

the result of (3.13),

s

(cid:90) T

(cid:90) T
(cid:90) 1
(cid:90) 1
e2sϕ(|wt|2 + |wx|2)dxdt + s3
e2sϕ(−T )(|wt(−T )|2 + |wx(−T )|2)dx + s3
(cid:90) 1
(cid:90) T

(cid:90) T

(cid:90) 1

−T

0

−T

0

e2sϕ|Lw + pw|2dxdt + M s

0

+ s

≤ M

−T

0

e2sϕ|wx(1, t)|2dt.

−T

e2sϕ|w|2dxdt

(cid:90) 1

0

e2sϕ(−T )|w(−T )|2dx

3.3 Control

For the remainder of the section, assume that the time condition (3.12) holds true. We now

would like to use the Carleman estimates to prove controllability for the original system

(3.1). The technique used is based on [5] and [22]. We restate here the main theorem to

prove.

Theorem 8. Let u0(x) ∈ L2(0, 1), u1(x) ∈ H−1(0, 1) be initial conditions. Then there exists
a control v(t) ∈ L2({1} × (−T, T )) such that the solution u(x, t) of

utt − (a(x)ux)x + p(x, t)u = 0,

u(1, t) = v(t),

u(0, t) = 0,
u(x,−T ) = u0(x),

ut(x,−T ) = u1(x),

(x, t) ∈ (0, 1) × (−T, T ),
t ∈ (−T, T ),
x ∈ (0, 1),

(3.39)



satisfies u(x, T ) = ut(x, T ) = 0.

The control will arise as part of the minimizer of some functional Fs,p to be specified

in the next theorem. Let us first classify where this functional should be defined. For a

42

potential p ∈ L∞((0, 1) × (−T, T )), define the space

(cid:110)

T (p) =

u ∈ L2(−T, T ; H1

0 (0, 1)), where Lu + pu ∈ L2((0, 1) × (−T, T ))

and ux(1, t) ∈ L2((−T, T ))

.

(3.40)

(cid:111)

One can note that this is the space of functions that are well-defined for the Carleman
estimate (3.13). Also, since pu ∈ L2((0, 1) × (−T, T )) whenever p ∈ L∞((0, 1) × (−T, T ))
and u ∈ L2(−T, T ; H1
0 (0, 1)), we see that this space does not depend on p. Let us then drop
the dependency and label in the future T (p) ≡ T . We can then define the norm on T as

follows :

||z||2

s,p =

1
s

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|ztt − (a(x)zx)x + pz|2dxdt +

(cid:90) T

−T

e2sϕ|zx(1, t)|2dt.

(3.41)

Because of the Carleman estimate (3.13), this is a norm for s ≥ s0. As this quantity is
the (scaled) right hand side of the estimate (3.13), we see that if ||z||s,p = 0, then z = 0

identically. Furthermore, e2sϕ is bounded from above and below on this domain by positive
constants depending on s, so that || · ||s,p is a norm for all s > 0.

Next, we define the natural norms to be used for the initial values. On H1

0 (0, 1)×L2(0, 1),

introduce || · ||−T,s defined by

||(z0, z1)||2−T,s =

(cid:90) 1

0

e2sϕ(−T )(|(z0)x|2 + |z1|2)dx.

(3.42)

One can see that by the estimate (3.13), and by bounding functions depending on s, for

43

all s > 0, there is a constant C depending on p and s such that

||(z(−T ), zt(−T ))||−T,s ≤ C(s, p)||z||s,p

(3.43)

for those functions with the regularity needed for the Carleman estimate. Also, on the dual
space L2(0, 1) × H−1(0, 1), take as a family of norms

||(u0, u1)||2−T,s,∗ =

e−2sϕ(−T )

(−(cid:52)d)−1u1

dx.

(3.44)

(cid:90) 1

0

(cid:32)
|u0|2 +

(cid:12)(cid:12)(cid:12)(cid:12) d

dx

(cid:12)(cid:12)(cid:12)(cid:12)2(cid:33)

Here, the notation (−(cid:52)d)−1u1 is the one dimensional version of the definition as in

(2.17). Now, we can define the correct functional and produce the desired control function

as minimizer of that functional.

Theorem 9. Assume that the time condition from Theorem 7 holds. Then the functional

e2sϕ|ztt − (a(x)zx)x + pz|2dxdt +

e2sϕ|zx(1, t)|2dt

Fs,p(z) =

+

1
2s

(cid:90) 1

0

0

−T
u0zt(−T ) − (cid:104)u1, z(−T )(cid:105)

H−1×H1

0

dx

(3.45)

is continuous, convex, and coercive on (T ,||·||s,p), and thus has a unique minimizer Z[s, p] =
Z ∈ T .

Proof. Fix s > 0 and p ∈ L∞((0, 1)×(−T, T )). Convexity is straightforward to check. Then,
by the estimate (3.43), for (u0, u1) ∈ L2(0, 1)× H−1(0, 1), the functional Fs,p is defined and
continuous on T . The same estimate also implies the coercivity condition, since we have the

44

(cid:90) T

(cid:90) 1

(cid:90) T

−T

1
2

inequality

Fs,p(z) ≥ 1
2
≥ 1
2

||z||2
||z||2

s,p − ||(z(−T ), zt(−T )||−T,s||(u0, u1)||−T,s,∗
s,p − C(s, p)||z||s,p||(u0, u1)||−T,s,∗.

Thus, there is a unique minimizer Z[s, p].

We now use this minimizer to prove the existence of the control.

Proof. (Theorem 8). Fix s > 0 and p. Let Z = Z[s, p] be that minimizer from Theorem 9.

We claim that by taking

U [s, p] =

1
s

e2sϕ(Ztt − (a(x)Zx)x + pZ) and V [s, p] =

1

a(1)

e2sϕZx(1, t),

then U [s, p] solves (3.39) with the corresponding control V [s, p], and also satisfies U (x, T ) =

Ut(x, T ) = 0. We will check the control is as claimed. The Euler-Lagrange equation can be

computed directly.

(cid:90) 1

0

(cid:90) T
(cid:90) T

−T

−T

1
s

+

(cid:90) 1

0

e2sϕ(ztt − (a(x)zx)x + pz)(Ztt − (a(x)Zx)x + pZ)dxdt

(3.46)

e2sϕ(zx(1, t))(Zx(1, t))dt +

u0zt(−T ) − (cid:104)u1, z(−T )(cid:105)

H−1×H1

0

dx = 0.

Let z(x, t) ∈ C∞

by computation that with U = 1

C ((0, 1)×(−T, T )). Then, after integration by parts, we can check directly
s e2sϕ(Ztt−(a(x)Zx)x +pZ), we have Utt−(a(x)Ux)x +pU =

0.

Next, let us check the initial and boundary conditions of U . Choose now z(x, t) such that

z, zx vanishes on the boundary. Then, noting that U is a solution to (3.39), the first term

45

in (3.46) gives, after integration by parts,

U (ztt − (a(x)zx)x + pz)dxdt =

(cid:90) 1

0

(ztU − zUt)|T−T dx.

(3.47)

But then, using (3.47) in (3.46), what remains is just

(cid:90) T

(cid:90) 1

−T

0

(cid:90) 1

0

(cid:90) 1

0

(ztU − zUt)|T−T +

u0zt(−T ) − (cid:104)u1, z(−T )(cid:105)

H−1×H1

0

dx = 0.

(3.48)

Since z was arbitrary, we obtain the boundary/initial conditions U (T ) = Ut(T ) = 0, and
also U (−T ) = u0, Ut(−T ) = u1. Finally, using these conditions and (3.48) in (3.46), we see

that

−

+

(cid:90) T
(cid:90) T
(cid:90) T

−T

−T

−T

a(1)(z(1, t)Ux(1, t) − zx(1, t)U (1, t))dt

(3.49)

a(0) (z(0, t)Ux(0, t) − zx(0, t)U (0, t)) dt

e2sϕ(zx(1, t))(Zx(1, t))dt = 0.

Thus, U|{1}×(−T,T ) = 1

a(1) e2sϕZx(1, t), and U|{0}×(−T,T ) = 0, as claimed.

3.4

Increased Regularity

In proving the Carleman estimate, there are two main considerations. First, one must

choose the weight function ψ(x, t). Then, one must prove that through a specified time

condition determined by ψ, this weight function will actually satisfy the inequalities. This

is determined through the integration by parts. In turn, when using “direct” computation

the weight function ψ(x, t) determines the class of functions that will be allowed.

46

We saw that the choice of ψ from (3.10) determined the family B from (3.7). In particular,

this choice of weight function also determines the regularity needed on a(x). In this section,

our main goal is to show that the Carleman estimate still holds true with only C1 regularity.

This in fact is true with just the knowledge that the estimate holds with the “standard”

weight ˜ψ(x, t) from (3.9), assuming that a(x) is defined up to the base point x0. Note this

is used in the new time condition.

Theorem 10. Assume that x0 < 0, a0 > 0, and let β ∈ (0, 1) be such that

T >

1
β

max
x∈[0,1]

1(cid:112)a(s)

ds.

(3.50)

(cid:90) x

x0

Then for any m > 0, there exists λ > 0 independent of m, s0 = s0(m) > 0 and a positive
constant M = M (m) such that for ϕ defined as in (3.10), for a(x) ∈ C1[0, 1] with a(x) ≥
a0 > 0, for all p ∈ L∞≤m((0, 1) × (−T, T )) and for all s ≥ s0,
(cid:90) T
(cid:90) T
(cid:90) 1
e2sϕ(|ut|2 + a(x)|ux|2)dxdt + s3

(cid:90) 1
(cid:90) T

e2sϕ|u|2dxdt

(cid:90) T

(cid:90) 1

e2sϕ|Lu + pu|2dxdt + M s

s

−T
≤ M

0

−T

0

0

−T
e2sϕ|ux(1, t)|2dt,

−T

(3.51)

for all u ∈ L2(−T, T ; H1
and ux(1, t) ∈ L2(−T, T ).

0 (0, 1)) also satisfying utt − (a(x)ux)x + pu ∈ L2((0, 1) × (−T, T ))

As before, note that all computations in the proof can be done for smooth functions u.

By classical results using density arguments, the result will hold true for u satisfying the

conditions of Theorem 10.

Proof of Theorem 10: Let us introduce a change of variable. For a given function b(x),

47

let

(cid:90) x

0

y(x) =

b(s)ds =: B(x).

(3.52)

Without loss of generality, we can scale y by a constant, so that y(0) = 0, y(1) = 1, and
y(cid:48)(x) = b(x). In the particular case of

1(cid:112)a(x)

b =

,

v(y, t) = u(x, t),

(3.53)

vy = ux(B−1(y))(cid:48) = ux

1

B(cid:48)(B−1(y))

= ux

1

b(B−1(y))

= ux

1

b(x)

,

(3.54)

then we have

so that

a(x)ux = a(x)b(x)vy =(cid:112)a(x)vy.

Taking one more derivative gives

(a(x)ux)x = vyyy(cid:48)(x)(cid:112)a(x) +
2(cid:112)a(x)

= vyy +

2(cid:112)a(x)

1

a(cid:48)(x)vy

1

a(cid:48)(x)vy.

(3.55)

(3.56)

Consider now the Carleman estimate from Theorem 5 with “standard” weight as in (3.9),

defined by

ψ = |y − y0|2 − βt2 + C, ϕ = eλψ,

(3.57)

where y0 < 0 and C is chosen so ψ ≥ 1. Because a(cid:48)√

a

∈ L∞, and p ∈ L∞, then denoting

48

(cid:3)v = vtt − vxx, we have that

(cid:90) 1

e2sϕ|v|2dydt

(cid:90) T

s

−T
≤ M

≤ M

+M

0

e2sϕ|(cid:3)v|2dydt + M s

(cid:90) 1
(cid:90) T
(cid:90) T
(cid:90) T
(cid:90) 1
e2sϕ(|vt|2 + |vy|2)dydt + s3
(cid:90) T
(cid:90) 1
(cid:12)(cid:12)(cid:12)(cid:12) a(cid:48)
(cid:12)(cid:12)(cid:12)(cid:12)L∞
(cid:90) T

(cid:90) T
e2sϕ|vtt − vyy + pv +

−T
√

(cid:90) 1

−T

−T

a

0

0

0

e2sϕ|vy(1, t)|2dt.

+M s

−T

0

−T
|vy(1, t)|2dt
−T
a(cid:48)
√
1
2
a

vy|2dydt

(cid:90) T

(cid:90) 1

−T

0

e2sϕ|vy|2dydt + M|p|L∞

e2sϕ|v|2dydt

For s large enough, the terms can be absorbed, and we obtain

(cid:90) T

(cid:90) 1

(cid:90) T

s

(cid:90) 1

−T

0

≤ M

0

−T
e2sϕ|(cid:3)v + pv +

e2sϕ(|vt|2 + |vy|2)dydt + s3

a(cid:48)
√
2
a

vy|2dydt + M s

(cid:90) 1

0

e2sϕ|v|2dydt

e2sϕ|vy(1, t)|2dt.

(cid:90) T
(cid:90) T

−T

−T

Now, we go back to the u variable. The space portion of the weight function from (3.57)

(cid:90) x0

0

1(cid:112)a(s)

ds

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

(3.58)

changes to

|y − y0|2 = |B(x) − y0|2 =

0

(cid:90) x
(cid:90) x

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
e2sQ(cid:16)|ut|2 + a(x)|ux|2(cid:17) dx√
(cid:90) 1

x0

=

a

e2sQ√
a

−T

0

[(L + p)u]2 dxdt + M s

1(cid:112)a(s)
1(cid:112)a(s)

√

ds

.

ds −

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:90) T

(cid:90) 1

(cid:90) 1
(cid:90) T

0

(cid:90) T

s

−T
≤ M

Let Q = |B(x) − y0|2 − βt2 + C. Then vy = ux

a, and we have

dt + s3

(cid:90) T

−T

e2sQ|u|2 dx√
a

−T
e2sQa(x)|ux(1, t)|2dt.

dt

0

49

After scaling again, we attain the estimate (3.13).

50

Chapter 4

Degenerate Parabolic Control

In the previous chapters, we have studied the control problem for the hyperbolic equation.

Here, the variable coefficient a(x) was assumed to be strictly positive. We now consider one

such analogue for the parabolic problem.

Let Ω = (0, 1). Let α ∈ [0, 2), and consider the function a(x) = xα. Then, a(x) vanishes
at x = 0. Let h(x, t) ∈ L2((0, 1) × (0, T )) be a control function, and let u0 be an initial

state in Ω. Then, the basic form for the interior degenerate parabolic system with control

function h is



ut − (aux)x = hχω,

(x, t) ∈ Ω × (0, T ),

u(1, t) = 0,

u(0, t) = 0,

(aux)(0, t) = 0,

t ∈ (0, T ),

for 0 ≤ α < 1, t ∈ (0, T ),

for 1 ≤ α < 2, t ∈ (0, T ),

(4.1)

u(x, 0) = u0,

x ∈ Ω.

In this chapter, we study the controllability properties for the degenerate heat equation.

We recall the criterion of null controllability for (4.1): Given T > 0, and initial state u0,

the goal is to find a control function h(x, t) such that the solution u(x, t) to (4.1) satisfies

u(T ) = 0.

The properties of the degenerate heat equation of the form (4.1) are less well known than

51

nondegenerate parabolic equations. An early study was done in [8], where a Carleman-type

estimate was established. It is shown in [1], [8], [10], that null controllability holds in the
case 0 ≤ α < 2, and in [11] that null controllability does not hold in the case α ≥ 2. The

study has been expanded to contain multiplicative control [7] and approximate control for

the nonlinear degenerate Cauchy-Neumann case [20]. Also, under certain hypothesis on the

degeneracy, control in higher dimensions was considered in the recent work [9].

4.1 Robin Boundary Conditions

The goal of this section is to show that the Carleman estimates are also true for the degenerate

parabolic equation with Robin boundary conditions.

Consider the degenerate parabolic system with Robin boundary conditions,

ut − (a(x)ux)x = p(x, t)u + f (x, t, u),

β0u(0, t) + β1a(0)ux(0, t) = 0,

γ0u(1, t) + γ1a(1)ux(1, t) = 0,

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



(x, t) ∈ (0, 1) × (0, T ),

t ∈ (0, T ),

t ∈ (0, T ),

x ∈ (0, 1).

(4.2)

As in [7] and [21], we define the spaces appropriate for the degenerate parabolic equation.

For a(x) ∈ C0[0, 1], define

a(0, 1) ˙={u ∈ L2(0, 1)|u ∈ AC([0, 1]) and
H1

√
aux ∈ L2(0, 1)},

(4.3)

a(0, 1) ˙={u ∈ H1
H2

a(0, 1)|aux ∈ H1(0, 1)}.

52

Then, for 0 ≤ α < 1, it can be shown that these are Hilbert spaces under the respective

natural norms of

and

||u||2

1,a ˙=||u||2

L2(0,1)

+ |u|2

1,a,

||u||2

2,a ˙=||u||2

1,a + ||(aux)x||2

L2(0,1)

,

where |u|2

1,a ˙=||√

aux||2

L2(0,1)

is a seminorm.

Furthermore, as in [7], consider the operator (A, D(A)) defined by

u ∈ H2

D(A) =

a(0, 1)

β0u(0) + β1a(0)ux(0) = 0,

γ0u(1) + γ1a(1)ux(1) = 0,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(4.4)

(4.5)

(4.6)





Au = (aux)x + pu,

∀u ∈ D(A),

for p ∈ L∞ ((0, 1)). It is shown in [7] that A is a closed self-adjoint dissipative operator
with dense domain in L2(0, 1). Then A is the infinitesimal generator of a C0−semigroup of
contractions in L2(0, 1), which will give rise to the uniqueness/existence theorem.

As the stated goal of this section is to show the Carleman estimate holds true for the

original system with the Robin boundary conditions, we shall not need to take advantage

of the full nonlinear term in which [21] provides well-posedness in more generality. See [21],

Definition 1.1 for the full problem formulation. The important conditions necessary and

applicable to this section are the degeneracy condition on a(x) and the sign condition for

the boundary condition coefficients.

1. a(x) ∈ C0([0, 1]) satisfies a(0) = 0.

2. a(x) ∈ C1(0, 1) is such that

1

a(x) ∈ L1(0, 1).

53

3. (Sign condition) : The constants β0, β1, γ0, γ1 ∈ R satisfy β2

0 + β2

1 > 0, γ2

0 + γ2

1 > 0,

and the sign conditions β0β1 ≤ 0, and γ0γ1 ≥ 0.

The first two conditions combined are called the weak degenerate condition, correspond-

ing to the diffusion coefficient a(x). Notice that the function a(x) = xα does indeed satisfy
the first two conditions, when α ∈ [0, 1). Since the function degenerates at 0 in (4.2), this

equality is in the limit sense. For completeness, and to make sense of the solution space
later, we include the conditions on f = f (x, t, u) from [21] here. Define QT = (0, 1)× (0, T ).

Then,

4. f : QT × R → R is such that

• (x, t) (cid:55)→ f (x, t, u) is measurable, for all u ∈ R,

• u (cid:55)→ f (x, t, u) is continuous, for almost every (x, t) ∈ (0, 1) × (0, T ),

• t (cid:55)→ f (x, t, u) is locally absolutely continuous for a.e x ∈ (0, 1), and for every

u ∈ R.

• There exists constants κ ≥ 0,O ∈ [1, 4), and ν ≥ 0 such that

|f (x, t, u) − f (x, t, v)| ≤ ν

(cid:16)

1 + |u|O−1 + |v|O−1(cid:17)|u − v|,

(f (x, t, u) − f (x, t, v)) (u − v) ≤ ν(u − v)2,

|f (x, t, u)| ≤ κ|u|O,

ft(x, t, u)u ≥ −νu2,

for a.e. (x, t) ∈ QT ,∀u, v,∈ R.

54

(4.7)

(4.8)

(4.9)

(4.10)

Now we must define the Banach spaces in time for the well-posedness result. As in [7]

and [21], given T > 0, define

B(QT ) ˙=C

(cid:16)

[0, T ]; L2(0, 1)

(cid:17) ∩ L2(cid:16)

(cid:17)

.

0, T ; H1

a(0, 1)

(4.11)

This has the norm

||u||2B(QT ) = sup
t∈[0,T ]

||u(·, t)||2 + 2

(cid:90) T

(cid:90) 1

0

0

a(x)u2

xdxdt.

(4.12)

Also, define

H(QT ) ˙=L2 (0, T ; D(A)) ∩ H1(cid:16)

(cid:17) ∩ C

0, T ; L2(0, 1)

(cid:16)

(cid:17)

.

[0, T ]; H1

a(0, 1)

(4.13)

This stronger space has norm

||u||2H(QT ) ˙= sup
t∈[0,T ]

(cid:16)||u||2 + ||√

aux||2(cid:17)

(cid:90) T

0

+

(cid:16)||ut||2 + ||(aux)x||2(cid:17)

dt.

(4.14)

Finally, we provide for completeness the two notions from [21] of solutions of this problem,

providing the necessary well-posedness result. These are the strict solutions and strong

solutions, corresponding to the respective solution spaces.

55

Definition 4.1.1. For u0 ∈ H1

a(0, 1), u is a strict solution of (4.2) if u ∈ H(QT ) and

ut − (a(x)ux)x = p(x, t)u + f (x, t, u(x, t)),

β0u(0, t) + β1a(0)ux(0, t) = 0,

γ0u(1, t) + γ1a(1)ux(1, t) = 0,

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

a.e. (x, t) ∈ (0, 1) × (0, T ),

a.e. t ∈ (0, T ),

a.e. t ∈ (0, T ),

x ∈ (0, 1).

(4.15)





Definition 4.1.2. Let u0 ∈ L2(0, 1). Then, u is a strong solution of (4.2), if u ∈ B(QT ),
u(·, 0) = u0 and there exists a sequence {uk}k∈N in H(QT ) such that as k → ∞, uk → u in
B(QT ), and for every k ∈ N, uk is the strict solution of

ukt − (a(x)ukx)x = p(x, t)uk + f (x, t, uk(x, t)),

β0uk(0, t) + β1a(0)ukx(0, t) = 0,

γ0uk(1, t) + γ1a(1)ukx(1, t) = 0,

uk(x, 0) = uk0

(x),

a.e. (x, t) ∈ (0, 1) × (0, T ),

a.e. t ∈ (0, T ),

a.e. t ∈ (0, T ),

x ∈ (0, 1).

(4.16)

With all conditions as defined above, the well-posedness results are the major results of

[21]. The two cases are as follows.

Theorem 11. For all u0 ∈ H1

a(0, 1), there exists a unique strict solution u ∈ H(QT ) in the

sense of Definition 4.1.1 to (4.2).

Theorem 12. For each u0 ∈ L2(0, 1), there exists a unique strong solution u ∈ B(QT ) in

the sense of Definition 4.1.2 to (4.2).

56

4.2 Carleman Estimate

With the well-posedness result, we can now prove the Carleman estimate associated to the

degenerate parabolic equation with Robin boundary conditions, extending the work of [8].

Theorem 13. Let 0 ≤ α < 1, and let T > 0. Let a(x) = xα. Consider the adjoint degenerate

parabolic problem with Robin boundary conditions,



wt + (a(x)wx)x = f (x, t),

β0w(x, t) + β1a(x)wx(x, t) = 0,

w(1, t) = 0,

w(x, T ) = wT (x),

(x, t) ∈ (0, 1) × (0, T ),

x = 0, t ∈ (0, T ),

t ∈ (0, T ),

x ∈ (0, 1),

(4.17)

0 + β2

1 > 0, and β0β1 ≤ 0. Then, there exists
where β0 and β1 satisfy the sign conditions β2
a function σ(x, t) : [0, 1] × (0, T ) → R of the form σ(x, t) = p(x)θ(t), where p(x) > 0 for all
x ∈ [0, 1], and θ(t) → ∞ as t → 0+, T−, and two positive constants C and R0 = R0(C, T ),
such that for all wT ∈ L2(0, 1), f ∈ L2((0, 1) × (0, T )), the solution w to (4.17) satisfies, for
(cid:90)(cid:90)

all R > R0, the Carleman Estimate

(cid:16)

x + R3θ3x2−αw2(cid:17)

Rθxαw2

QT

e−2Rσdxdt
(cid:90)(cid:90)

≤ C

(cid:90) T

(cid:110)
Rθe−2Rσw2

(cid:111)

x

(4.18)

.

x=1

e−2Rσf 2dxdt + C

QT

0

57

Proof. Consider the Robin boundary conditions

β0w(x, t) + β1a(x)wx(x, t) = 0,

w(1, t) = 0.

x = 0,

(4.19)

For simplicity, by scaling let us assume that β1 = 1. Then, call β0 = β. According to the

sign condition 3, we have that β < 0. Similar to the procedure for the hyperbolic equation,
as in (3.14), we make the change of variables z = e−Rσ(x,t)w(x, t). Then the space derivative

satisfies

zx = e−Rσ(x,t)w(x, t)(−Rσx) + wxe−Rσ(x,t)

(4.20)

= z(−Rσx) + wxe−Rσ(x,t).

The boundary conditions for z(x, t) satisfy

βz(x, t) + a(x)zx(x, t)

(4.21)

= −a(x)zRσx + a(x)wxe−Rσ + βe−Rσw
= −a(x)e−RσwRσx + a(x)wxe−Rσ + βe−Rσw
= −a(x)e−RσwRσx
= −a(x)zRσx(x, t).

Then we also have the relation for the z derivative,

zx = e−Rσ(x,t)w(x, t)(−Rσx) + wxe−Rσ(x,t)

(4.22)

58

= z(−Rσx) + wxe−Rσ(x,t).

The same method as from Chapter 3, (3.15) can be utilized. That is, we decompose the

equation into 2 parts, and compute the product. Define the relations

P1z = Rσtz + R2aσ2

xz + (azx)x.

P2z = zt + R(aσx)xz + 2Raσxzx.

Then, we have

P1z + P2z = f e−Rσ.

(4.23)

(4.24)

(4.25)

From [8] (Lemma 3.1), by computing (cid:104)P1z, P2z(cid:105), the following identity holds from the de-

composition,

(cid:104)P1z, P2z(cid:105) =

(cid:90) T

(cid:104)

azxzt + R2aσtσxz2 + R3a2σ3

xz2

(4.26)

(cid:105)1
(cid:90) (cid:90)

0

dt

x + Ra(aσx)xzzx
θttpz2 − 2R2

R

0
+Rσxa2z2
− 1
2
− R3

(cid:90) (cid:90)
(cid:90) (cid:90)
(cid:90) (cid:90)

− R

QT

QT

QT

θθtxαp2

xz2

QT

θ3x2α−1(2xpxx + αpx)p2

xz2

θx2α−1(2xpxx + αpx)z2

x − R

(cid:90) (cid:90)

QT

θxα(xαpx)xxzzx.

This splits the product into two parts, the boundary terms, and the space-time terms.

The interior terms will be evaluated second. The boundary terms now obey the condition

59

of (4.21). The boundary terms are

(cid:90) T
(cid:90) T

0

(cid:104)

+ R

0

(cid:105)1

0

Boundary =

[azxzt]1

0 dt + R2

a2σ3

dt

(4.27)

(cid:90) T
(cid:104)
aσtσxz2(cid:105)1
(cid:90) T

0

0

(cid:90) T

0

dt + R3

(cid:104)

xz2(cid:105)1

0

σxa2z2
x

dt + R

[a(aσx)xzzx]1

0 dt.

0

To avoid confusion, let us evaluate each of these terms separately. Let us here also define

the function p(x) by

p(x) =

2 − x2−α
(2 − α)2 .

Note that p(x) > 0. Then the derivative satisfies

−x1−α
(2 − α)

.

px =

Using (4.21), the first term from (4.27) gives

(cid:90) T

0

0

(cid:90) T
(cid:90) T
[(−βz − aRθpxz)zt]1
0 dt
(cid:90) T
{(βz + aRθpxz)zt}x=0 dt
(cid:110)
(cid:111)
(cid:26) R
(cid:90) T

β(z2)t + aRθpx(z2)t

0
1
2

(cid:27)

0

2 − α

θtxz2

dt,

x=0

0

x=0

[azxzt]1

0 dt =

=

=

=

(4.28)

(4.29)

(4.30)

dt

where we have used the decay in the limit from the definition of z. The second term of (4.27)

gives

(cid:90) T

0

aσtσxz2(cid:105)1
(cid:104)

0

R2

dt = −R2

(cid:90) T

0

aσtσxz2(cid:111)
(cid:110)

x=0

dt

(4.31)

60

(cid:26)

θθt

(cid:90) T

0

= R2

(cid:27)

(cid:26)

θθt

(cid:90) T

0

(cid:27)

dt.

x=0

x2−α
(2 − α)3 xz2

dt − R2

x=0

2

(2 − α)3 xz2

The third term from (4.27) gives

(cid:90) T

(cid:104)

0

xz2(cid:105)1

0

R3

a2σ3

dt = −R3

(cid:90) T
xz2(cid:111)
(cid:110)
(cid:26) x2−α
(cid:90) T

a2σ3

0

x=0

dt

(cid:27)

= R3

0

(2 − α)3 θ3xz2

dt.

x=0

The fourth term from (4.27) gives

R

σxa2z2
x

(cid:90) T
(cid:104)
(cid:90) T
(cid:110)
(cid:90) T

0

0

(cid:105)1
(cid:26) 1

σxa2z2
x

0

dt

(cid:111)

= R

= −R

(cid:110)

(cid:90) T

0

dt − R

(cid:111)

x=0

dt

(cid:111)

σxa2z2
x

(cid:90) T

(cid:110)

x=1

dt − R

(cid:27)

(2 − α)

θz2
x

0

x=1

0

θa2pxz2
x

dt.

x=0

(4.32)

(4.33)

Now, once again using the Robin boundary conditions from (4.21), we continue the equality

from (4.33),

R

σxa2z2
x

(cid:90) T

0

= −R

= −R

(cid:104)
(cid:90) T
(cid:90) T

0

+

2R

(2 − α)

(cid:90) T

= −R

0

0

− 2R2
(2 − α)2

0

x

dt

x=1

dt +

(2 − α)

(cid:105)1
(cid:26) θz2
(cid:27)
(cid:26) θz2
(cid:27)
(cid:90) T
(cid:110)
x1−αβRθ(apx)z2(cid:111)
(cid:26) θz2
(cid:27)
(cid:90) T
x2−αβθ2z2(cid:111)
(cid:110)

(2 − α)

(2 − α)

dt +

dt +

x=1

x=1

x

x

0

R

(2 − α)

R

(2 − α)

x=0

R

(2 − α)

0

(cid:90) T
(cid:90) T

0

0

dt +

(cid:90) T

0

(cid:110)
θx1−α (−βz − aRθpxz)2(cid:111)
β2θx1−αz2(cid:111)
(cid:110)
(cid:90) T
(cid:110)
x1−αa2R2θ3p2
β2θx1−αz2(cid:111)
(cid:110)
(cid:90) T
x3−αθ3z2(cid:111)
(cid:110)

(2 − α)

x=0

x=0

dt.

dt

dt

R

0

x=0

x=0

xz2(cid:111)

(4.34)

dt

x=0

dt

(4.35)

dt +

x=0

R3

(2 − α)3

0

61

The final term from (4.27) gives

(cid:90) T

0

= −R

R

[a(aσx)xzzx]1

(cid:90) T

0

0 dt

(cid:19)
(cid:18) −1
(cid:26)
(cid:90) T
(cid:110)
βθz2(cid:111)

2 − α

θ

= − R

(2 − α)

0

(cid:27)
(cid:90) T

0

z (−βz − aRθpxz)

+

R2

(2 − α)2

x=0

x=0

dt

(cid:110)
θ2xz2(cid:111)

x=0

(4.36)

dt.

Combining all of the results from (4.30) - (4.36), we arrive at the expression for the boundary

terms:

Boundary =

+ R2

+ R3

− R

(cid:27)

0

0

θθt

θtxz2

(cid:90) T

2 − α
2

(cid:26) R
(cid:26)
(cid:27)
(cid:90) T
(cid:26) x2−α
(cid:27)
(cid:90) T
(2 − α)3 xz2
(cid:26) θz2
(cid:27)
(cid:90) T
(2 − α)3 θ3xz2
(cid:90) T
(cid:110)
x2−αβθ2z2(cid:111)
(cid:90) T
(cid:110)
βθz2(cid:111)

(2 − α)

dt +

x=1

+

x

0

0

0

0

x=0

− 2R2
(2 − α)2

− R

(2 − α)

x=0

dt

x=0

R

(2 − α)

dt

x=0

dt − R2

(cid:26)

θθt

(cid:90) T

0

(cid:27)

x=0

x2−α
(2 − α)3 xz2

(4.37)

(cid:90) T

0

(cid:110)
β2θx1−αz2(cid:111)
(cid:90) T
(cid:110)
x3−αθ3z2(cid:111)
(cid:90) T
(cid:110)
θ2xz2(cid:111)

(2 − α)3

R3

x=0

dt.

dt

0

x=0

dt +

x=0
R2

(2 − α)2

0

dt.

x=0

By [8] (Lemma 3.1), we have that for all v ∈ H1

Because of the β sign condition, β < 0, every term involving β from (4.37) is nonnegative.
a(0, 1), then xv2(x) → 0 as x → 0+. Thus,
(cid:90) T

the remaining terms vanish, and we have

(cid:110)

(cid:111)

Boundary ≥ − R

(2 − α)

0

θz2
x

dt.

x=1

(4.38)

From the form of (4.26) and (4.37), we choose the weight function θ(t). We will prove

62

the theorem using a more general function θ(t) than that in [8]. For k > 0, consider the

function

θ(t) =

(cid:19)k

(cid:18)

1

t(t − T )

.

(4.39)

Note that the choice for θ(t) satisfies the limit conditions prescribed in Theorem 13 for
t → 0−, t → T +. Then the time derivatives are

θt =

k(2t − T )
(t(T − t))k+1

,

and

(cid:18)(4k + 2)t2 − 2t(2kT + T ) + (k + 1)T 2

(t(T − t))k+2

(cid:19)

.

θtt = k

From (4.26), the space-time terms satisfy

(4.40)

(4.41)

(4.42)

(cid:90)(cid:90)

R

(cid:90)(cid:90)
(cid:90)(cid:90)
(cid:90)(cid:90)

QT

QT

Space-time = −1
2
− R3

θttpz2dxdt − 2R2

θθtxαp2

xz2dxdt

QT
QT
θ3x2α−1(2xpxx + αpx)p2
xz2dxdt

− R

θx2α−1(2xpxx + αpx)z2

xdxdt − R

(cid:90)(cid:90)

QT

θ(xαpx)xxzzxdxdt.

By the choice of p(x), this simplifies to

Space-time = − R

(2 − α)2

(cid:90)(cid:90)
(cid:90)(cid:90)
(cid:90)(cid:90)

QT

− 2R2
(2 − α)2

+

R3

(2 − α)2

R

2(2 − α)2

θttz2dxdt +

QT
θθtx2−αz2dxdt

θ3x2−αz2dxdt + R

(cid:90)(cid:90)

θxαz2

xdxdt.

QT

(cid:90)(cid:90)

QT

θttx2−αz2dxdt

(4.43)

QT

63

Let us estimate these terms in (4.43). The leading term in z2 contains θ3. The leading

term in z2
x contains θ. Thus, we want to estimate the first three terms in (4.43) with respect
to the last two. Now, by the form of θ and θtt from (4.40), an estimate of the form |θtt| ≤ cθ3
is possible if and only if k ≥ 1. In this case, |θtt| ≤ cT 4k−2θ3. Thus, the second term from

θ3x2−αz2dxdt.

(4.44)

we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(4.43) gives

(cid:90)(cid:90)

QT

θttx2−αz2dxdt

R

2(2 − α)2

(cid:90)(cid:90)

(2 − α)2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Rc(k)T 4k−2
(cid:19)2k+1
(cid:18)

QT

For the third term from (4.43), we must estimate |θθt|. By definition from (4.39) and (4.40),

θθt = k(2t − T )

1

t(T − t)

,

(4.45)

so that once again |θθt| ≤ cθ3 if and only if k ≥ 1. If this is the case, then we can estimate

the third term as

(cid:90)(cid:90)

QT

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ R2c(k)T 2k−1

(2 − α)2

θθtx2−αz2dxdt

(cid:90)(cid:90)

QT

θ3x2−αz2dxdt.

(4.46)

2R2

(2 − α)2

Finally, we consider the first term from (4.43). By Young’s Inequality, we can estimate the

(cid:16)

θxp1z2(cid:17)r1(cid:16)

θ3x2−αz2(cid:17)r2

,

θtt ≤ cθm = c

second derivative,

where 0 < r2 ≤ r1 and

r1 + r2

= 1

r1 + 3r2

= m
(2 − α)r2 + p1r1 = 0.

(4.47)

(4.48)

The first two equations restrict m to 1 < m < 3. The inequality θtt ≤ cθm can be only be

64

satisfied if k(1 − m) + 2 ≤ 0. If this is the case, then

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:90)(cid:90)
(cid:90)(cid:90)

(cid:90)(cid:90)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

R

(cid:16)

QT

θttz2dxdt

(2 − α)2

(2 − α)2
≤ Rc(k)T−2+2k(m−1)
Rc(k)T−2+2k(m−1)
(cid:18)1

(cid:19) r1
(2 − α)2
QT
r2 Rc(k)T−2+2k(m−1)

≤ 


QT

θxp1z2(cid:17)r1(cid:16)
(cid:90)(cid:90)

θxp1z2dxdt

+




(2 − α)2

QT

θ3x2−αz2dxdt

θ3x2−αz2(cid:17)r2

dxdt

(4.49)

Since α < 2, and r2 ≤ r1, then

xp1 = x

(α−2)

r2
r1 ≤ xα−2.

If α (cid:54)= 1, then Hardy’s inequality ([8] Lemma 2.1) gives

(cid:90)(cid:90)

(cid:90)(cid:90)

θxp1z2dxdt ≤

QT

θxα−2z2dxdt ≤

C

(α − 1)2

QT

(cid:90)(cid:90)

QT

θxαz2

xdxdt.

(4.50)

Combining (4.44), (4.46), (4.49), and (4.50) gives

θ3x2−αz2dxdt + R

(cid:90)(cid:90)
(cid:90)(cid:90)
θ3x2−αz2dxdt − R2c(k)T 2k−1
(cid:90)(cid:90)
(2 − α)2

QT

(cid:90)(cid:90)

QT

θxαz2

xdxdt

(4.51)

θ3x2−αz2dxdt

QT

(2 − α)2

Space-time ≥ R3
(cid:90)(cid:90)
− Rc(k)T 4k−2
(2 − α)2
QT
Rc(k)T−2+2k(m−1)
(cid:18)1
(2 − α)2(1 − α)2

(cid:19) r1
QT
r2 Rc(k)T−2+2k(m−1)

− 


(2 − α)2

−




θxαz2

xdxdt

(cid:90)(cid:90)

θ3x2−αz2dxdt.

QT

65

To make the Space-time terms nonnegative, we must choose 
 = 
(T ) appropriately, and

then R = R(
, T, α) from there. The term containing 
 must be dominated then by the
second term, so 
 needs to be on the same order as T 2−2k(m−1)(2 − α)2(1 − α)2. In terms
r2 . From here, we can choose R.

of T , this means that

r2 is of order

(cid:17) r1

(cid:16)

(cid:16) 1




By comparing the first and last terms of (4.51),

T−2+2k(m−1)(cid:17) r1
(cid:18)

(cid:19) r1

r2

,

1

(1 − α)2(2 − α)2

R3 (cid:38) RT−2+2k(m−1)(cid:16)
(cid:17)

(cid:16)

r2

T−2+2k(m−1)(cid:17) r1
(k(m−1)−1)(cid:18)

R (cid:38) T

1+

r1
r2

(cid:19) r1

r2

.

1

|(1 − α)(2 − α)|

so that

Also, by comparing the first, third and fourth terms of (4.51),

R (cid:38) T 2k−1.

From these, choose R sufficiently large, so that

(cid:40)

(cid:16)

(cid:17)

R ≥ max

T

1+

r1
r2

(k(m−1)−1)

, T 2k−1

(cid:41)

.

(4.52)

Then, for the chosen 
 small enough and R large enough as in (4.52), from (4.51), the first

two terms will dominate, giving

Space-time ≥ R3

(2 − α)2

(cid:90)(cid:90)

θ3x2−αz2dxdt + R

QT

(cid:90)(cid:90)

QT

θxαz2

xdxdt ≥ 0.

(4.53)

66

Thus, with R large enough as in (4.52), we have from (4.25), (4.38), and (4.53),

(cid:90)(cid:90)

||f e−Rσ||2 ≥ (cid:104)P1z, P2z(cid:105) = Space-time + Boundary
(cid:90) T
(cid:110)

≥ R3
(cid:90)(cid:90)

θ3x2−αz2dxdt

(2 − α)2

QT
xdxdt − R

θxαz2

+ R

(2 − α)

0

θz2
x

QT

(4.54)

(4.55)

(cid:111)

x=1

dt.

Finally, we recall that w = eRσz. We use this to obtain the bounds in the original variable.

So,

R3θ3x2−αw2 + Rθxαw2

x = R3θ3x2−αe2Rσz2 + Rθxα(RσxeRσz + eRσzx)2
≤ c

R3θ3x2−αe2Rσz2 + Rθxαe2Rσz2

(cid:16)

(cid:17)

.

x

To match (4.55), multiply by e−2Rσ to get

R3θ3x2−αw2e−2Rσ + Rθxαw2

xe−2Rσ ≤ c

(cid:16)
R3θ3x2−αz2 + Rθxαz2

x

(cid:17)

.

(4.56)

(4.57)

(4.58)

(cid:111)

.

x=1

From (4.55) and (4.57), the Carleman Estimate follows.

(cid:90)(cid:90)

(cid:16)

QT

x + R3θ3x2−αw2(cid:17)

Rθxαw2

e−2Rσdxdt
(cid:90)(cid:90)

≤ C

QT

e−2Rσf 2dxdt + C

(cid:90) T

0

(cid:110)
Rθe−2Rσw2

x

67

APPENDIX

68

Appendix

Minimum Energy: Approximate

Control

In this appendix, we now consider one brief application regarding the energy for the non-
degenerate parabolic system. Let Ω be a domain, let ω ⊂ Ω. Consider the interior control

problem



ut − (cid:52)u = gχω,

u = 0,

u(x, 0) = 0,

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω.

(A.1)

Here, g is a control function satisfying the approximate controllability condition. That is,

instead of driving the solution to an exact final state at a given end time as we have observed

before, the control function should only have the restriction that it drives the solution close

to a specified final state.

Let u1 ∈ L2(Ω) be a final state for (A.1). For u0 ∈ L2(Ω), g ∈ L2(ω × (0, T )), there is a

unique solution u ∈ C(cid:0)[0, T ]; L2(Ω)(cid:1) ∩ L2(cid:0)0, T ; H1

0 (Ω)(cid:1) . Given 
 > 0, we say then that the

approximate controllability condition is satisfied if

||u(x, T ) − u1||L2 ≤ 
.

(A.2)

69

Then, we know (see [19] for example) that if u0 ∈ L2(Ω), u1 ∈ L2(Ω), that (A.1) is

approximately controllable in the sense of (A.2). Furthermore, ˜g is a control function for

(A.1), where ˜g is given by the solution to the adjoint equation



˜gt + (cid:52)˜g = 0,

˜g = 0,

˜g(x, T ) = g0

(x, t) ∈ Ω × (0, T ),
(x, t) ∈ ∂Ω × (0, T ),
x ∈ Ω,

(A.3)

where g0 is the (unique) minimizer of the functional

J
(u1, g) =

1
2

|g|2dxdt + 


0

ω

Ω

(cid:18)(cid:90)

(cid:19) 1

2 −

(cid:90)

Ω

g2dx

u1gdx,

(A.4)

(cid:90) T

(cid:90)

in the sense that J
(u1, g0) = min

g∈L2(Ω)

the minimum energy.

J
(u1, g). We will observe that in fact, this ˜g also has

Lemma 3. Let g0 be the minimizer of (A.4). Let ˜g(x, t) be the solution to (A.3) with initial

data g0. Then ˜g also minimizes the energy. That is, for any control function g for (A.1),

we have

(cid:90) T

(cid:90)

(cid:90) T

(cid:90)

|g|2dxdt ≥

|˜g|2dxdt.

(A.5)

0

ω

0

ω

Proof. Let g(x, t) be any control function for (A.1). Then, decompose g(x, t) as the sum of

functions

g(x, t) = ˜g(x, t) + h(x, t),

(A.6)

where ˜g(x, t) solves (A.3) with initial data g0, where g0 is the minimizer of (A.4). Then,

upon multiplication of ˜g and integration, we get

70

(cid:90) T

(cid:90)

0

ω

(˜g + h) ˜gdxdt =

(cid:90) T

(cid:90)

(cid:90) T

(cid:90)

|˜g|2dxdt +

0

ω

0

ω

˜ghdxdt.

(A.7)

On the other hand, using the definition of the decomposition and (A.1), we have the

relation

ω

0

(cid:90)
(cid:90) T
(cid:90) T
(cid:90) T
(cid:90)

=

=

0

0

(cid:90)
(cid:90)

(˜g + h) ˜gdxdt

(A.8)

(ut − (cid:52)u) ˜gdxdt

u (˜gt + ˜g) dxdt +

(cid:90)

Ω

u(T )˜g(T )dx

Ω

Ω

=

u(T )˜g(T )dx,

Ω

since ˜g solves the adjoint equation. Now, by using the definition for the approximate con-

trollability condition, (A.7), and (A.8), we obtain

ω

(cid:90) T
(cid:90)
(cid:90) T
(cid:90)
(cid:90) T
(cid:90) T

0
1
2

0

0

1
2

=

=

≥ 1
2

(cid:90)

ω
1
2

(cid:90)

ω

0

ω

(cid:90)
(cid:90)
(cid:90)

Ω

Ω

Ω

=

=

≥

|g|2dxdt

|˜g + h|2dxdt

˜g2 + ˜gh +

˜g2dxdt +

h2dxdt

1
2

(cid:90)
(cid:90) T
(cid:90) T
(cid:90)

0

ω

˜ghdxdt

˜g2dxdt, by (A.8)

(cid:90)

0

ω

u(T )˜g(T )dx − 1
2
(cid:90)
(u(T ) − u1)˜g(T )dx − 1
2

u1˜g(T )dx +
u1˜g(T )dx − 
||˜g(t)||L2(Ω) − 1

(cid:90) T

Ω

(cid:90) T

˜g2dxdt

0

ω

2

0

ω

(A.9)

(cid:90)

˜g2dxdt

= −J
(u1)

71

=

1
2

Thus, as claimed,

(cid:90) T

(cid:90)

|˜g(x, t)|2dxdt.

0

ω

(cid:90) T

(cid:90)

0

ω

|g(x, t)|2dxdt ≥

(cid:90) T

(cid:90)

0

ω

|˜g(x, t)|2dxdt.

(A.10)

72

BIBLIOGRAPHY

73

BIBLIOGRAPHY

[1] F. Alabau-Boussouira, P. Cannarsa, and G. Fragnelli Carleman estimates
for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6
(2006) no. 2 pp. 161-204.

[2] A. Amirov and M. Yamamoto, Unique continuation and an inverse problem for
hyperbolic equations across a general hypersurface, J. Phys.: Conf. Ser., 12 (2005) pp.
1-12.

[3] A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation

and Control of Infinite Dimensional Systems, Birkh¨auser Boston 2007.

[4] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation,
control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992)
no. 5 pp. 1024-1065.

[5] L. Baudouin, M. Buhan and S. Ervedoza, Global Carleman estimates for waves

and applications, Comm. Partial Differential Equations, 38 (2013) no. 5 pp. 823-859.

[6] L. Baudouin, E. Cr´epeau and J. Valein, Global Carleman estimate on a network
for the wave equation and application to an inverse problem, Math. Control Relat. Fields,
1 (2011) no. 3 pp. 307-330.

[7] P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degen-
erate parabolic problems with Robin boundary conditions, Commun. Appl. Ind. Math., 2
(2011) no. 2 DOI: 10.1685/journal.caim.376.

[8] P. Cannarsa, P. Martinez, and J. Vancostenoble, Carleman Estimates for a
class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008) no. 1 pp. 1-19.

[9] P. Cannarsa, P. Martinez, and J. Vancostenoble, Global Carleman Estimates
for Degenerate Parabolic Operators with Applications, American Mathematical Soc., 2015.

[10] P. Cannarsa, P. Martinez, and J. Vancostenoble, Null controllability of de-

generate heat equations, Adv. Differential Equations 10 (2005) no. 2 pp. 153-190.

74

[11] P. Cannarsa, P. Martinez, and J. Vancostenoble, Persistent regional control-
lability for a class of degenerate parabolic equations, Comm. Pure Appl. Math. 3 (2004)
no. 4 pp. 607-635.

[12] C. Castro and E. Zuazua, Concentration and lack of observability of waves in highly

heterogeneous media, Arch. Ration. Mech. Anal. 164 (2002) pp. 39-72.

[13] N. Cˆındea, E. Fern´andez-Cara and A. M¨unch, Numerical controllability of the
wave equation through primal methods and Carleman estimates, ESAIM Control Optim.
Calc. Var., 19 (2013) pp. 1076-1108.

[14] D. D. S. Ferreira, Sharp Lp Carleman estimates and unique continuation, Duke

Math. J., 129 (2005) no. 3 pp. 503-550.

[15] T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability
inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincar´e
Anal. Non Lin´eaire, 25 (2008) no. 1 pp. 1-41.

[16] L. C. Evans, Partial Differential Equations, American Mathematical Society, Provi-

dence, 2010.

[17] C. Fabre, Exact boundary controllability of the wave equation as the limit of internal

controllability, SIAM J. Control Optim., 30 (1992) no. 5 pp. 1066-1086.

[18] C. Fabre and J.-P. Puel, Pointwise controllability as limit of internal controllability
for the wave equation in one space dimension, Port. Math., 51 (1994) no. 3 pp. 335-350.

[19] E. Fern´andez-Cara and E. Zuazua, The cost of approximate controllability for

heat equations: the linear case, Adv. Differential Equations, 5 (2000) pp. 465-514.

[20] G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems

with bilinear control, J. Differential Equations, 257 (2014) no. 9 pp. 3382-3422.

[21] G. Floridia, Well-Posedness for a class of nonlinear degenerate parabolic equations,
in M. de Le´on, W. Feng, Z. Feng, J. Lopez-Gomez, X. Lu, J.M. Martell, J. Parcet, D.
Peralta-Salas, and W. Ruan, Eds., Dynamical Systems and Differential Equations, AIMS
Proceedings 2015, pp. 455-463, 2015.

[22] A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations,
Volume 34 of Lecture Notes Series. Seoul National University Research Institute of Math-
ematics Global Analysis Research Center, Seoul, 1996.

75

[23] X. Fu, A weighted identity for partial differential operators of second order and its

applications, C. R. Acad. Sci. Paris, Ser. I 342 (2006) pp. 579-584.

[24] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear

hyperbolic equations, SIAM J. Control Optim., 46 (2007) pp. 1578-1614.

[25] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second

Order, Springer-Verlag Berlin Heidelberg, 2001.

[26] L. F. Ho, Observabilit´e fronti`ere de l’´equation des ondes, C.R. Math. Acad. Sci. Paris

S´er. I Math. 304 (1987) pp. 367-370.

[27] O. Y. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymptot. Anal.,

32 (2002) no. 3-4 pp. 185-220.

[28] O. Y. Imanuvilov and M. Yamamoto, Global uniqueness and stability in deter-
mining coefficients of wave equations, Comm. Partial Diff. Eqns., 26 (2001) no. 7-8 pp.
1409-1425.

[29] O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary
Lam´e system and the application to an inverse problem, ESAIM Control Optim. Calc.
Var., 11 (2005) no. 1 pp. 1-56.

[30] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and
unique continuation in J. Garc´ıa-Cuerva, Ed., Harmonic Analysis and Partial Differential
Equations: Proceedings of the International Conference held in El Escorial, Spain, June
9–13, 1987, Springer Berlin Heidelberg, 1989.

[31] H. Koch and D. Tataru, Carleman estimates and unique continuation for second-
order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001)
no. 3, pp. 339-360.

[32] V. Komornik, Exact controllability and Stabilization, The Multiplier Method. RAM:

Research in Applied Mathematics. Masson, Paris, 1994.

[33] V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control

Optim., 29 (1991) no. 1 pp. 197-208.

[34] J. Lagnese, Note on boundary stabilization of the wave equation, SIAM J. Control

Optim., 26 (1988) no. 5 pp. 1250-1256.

76

[35] I. Lasiecka, R. Triggiani, and P.-F. Yao, Inverse observability estimates for
second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl. 235
(1999) pp. 13-57.

[36] J.-L. Lions, Contrˆolabilit´e exacte, perturbations et stabilisation de syst`emes distribu´es,

Recherches en Math´ematiques Appliqu´ees, Tomes 1 et 2, Masson, Paris, 1988.

[37] J.-L. Lions, Control of Distributed Singular Systems, Bordas, Paris, 1985.

[38] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed sys-

tems, SIAM Rev., 30 (1988) no. 1 pp. 1-68.

[39] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Ap-

plications 3 vols. Springer Berlin Heidelberg, 1972-1973.

[40] Y. Liu, Some sufficient conditions for the controllability of wave equations with variable

coefficients, Acta Appl. Math, 128 (2013) pp. 181-191.

[41] F. Maci`a and E. Zuazua, On the lack of observability for wave equations: a gaussian

beam approach, Asymptot. Anal., 32 (2002) no. 1 pp. 1-26.

[42] P. Martin, L. Rosier, and P. Rouchon, Null controllability of the structurally
damped wave equation with moving point control, SIAM J. Control Optim., 51 (2013) no.
1 pp. 660-684

[43] J. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl.

Math, 22 (1969) no. 6, 807-823.

[44] J. Ralston, Gaussian beams and the propagation of singularities in Walter Littman,
Ed., Studies in Partial Differential Equations, MAA Studies in Mathematics, 23 (1989)
pp. 206-248.

[45] J.-P. Raymond, Optimal Control of Partial Differential Equations, Online, retrieved

from https://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf .

[46] V.G. Romanov, Carleman estimates for second-order hyperbolic equations, Sib. Math.

J., 46 (2006) no. 1 pp. 131-151.

[47] C. Sogge, Oscillatory integrals and unique continuation for second order elliptic dif-

ferential equations, J. Amer. Math. Soc., 2 (1989) no. 3 pp. 491-515.

77

[48] C. Sogge, Uniqueness in Cauchy problems for hyperbolic differential operators, Trans.

Amer. Math. Soc., 333 (1992) no. 2 pp. 821-833.

[49] D. Tataru, Carleman estimates and unique continuation for solutions to boundary

value problems, J. Math. Pures Appl. (9), 78 (1996) no. 1 pp. 65-98.

[50] P.-F. Yao, On the inequalities for exact controllability of wave equations with variable

coefficients, SIAM J. Control. Optim., 37 (1999) pp. 1568-1599.

[51] J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control:

Foundations & Applications, Birkh¨auser Boston, Inc., Boston, MA, 1992.

[52] X. Zhang, Explicit observability inequalities for the wave equation with potential and

its application, Proc. R. Soc. A, 456 (2000) pp. 1101-1115.

[53] X. Zhang, Explicit observability inequalities for the wave equation with lower order
terms by means of Carleman inequalities, SIAM J. Control Optim., 39 (2000) no. 3 pp.
812-834.

[54] E. Zuazua, Controllability and Observability of Partial Differential Equations: Some
Results and Open Problems, in C.M. Dafermos and E. Feireisl, Eds., Handbook of Dif-
ferential Equations: Evolutionary Equations Vol. 3, Elsevier/North-Holland, Amsterdam,
(2007) pp. 527-621.

[55] E. Zuazua, Some results and open problems on the controllability of linear and semi-
linear heat equations, in F. Colombini and C. Zuily, Eds., Carleman Estimates and Appli-
cations to Uniqueness and Control Theory, Birkha¨user, Basel, (2001) pp. 191-211.

[56] E. Zuazua, Propagation, observation, and control of waves approximated by finite dif-

ference methods SIAM Rev., 47 (2005) no. 2 pp. 197-243.

[57] E. Zuazua, Propagation of waves: numerical approximation and control, Bol. Soc. Esp.

Math Apl., 25 (2003) pp. 55-126.

78