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I: Main result II: A = I d and b =0 III: Links IV: Log convexity VI: Carleman commutator Observation estimate at one point in time for parabolic equations Kim Dang PHUNG UniversitØ dOrlØans. April 2017, Paris

Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

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Page 1: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Observation estimate at one point in timefor parabolic equations

Kim Dang PHUNG

Université d�Orléans.

April 2017, Paris

Page 2: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

The parabolic equations

smooth bounded domain in Rn, T > 0,A 2 C2

�� [0; T ]

�n� n symmetric positive-de�nite matrix,

b = (b0; b1) 2 (L1 (� (0; T )))n+1,

8<:@tu� div (Aru) + b1 � ru+ b0u = 0 in � (0; T ) ,

u = 0 on @� (0; T ) ,u (�; 0) = u0 2 L2 () ,

ku (�; t)kL2() � e�CAteCbt ku0kL2()

Page 3: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Observation at one point in time

smooth bounded domain in Rn, T > 0,A 2 C2

�� [0; T ]

�n� n symmetric positive-de�nite matrix,

Cb;T = 1 + eCbT , ! non-empty open subset of ,

8<:@tu� div (Aru) + b1 � ru+ b0u = 0 in � (0; T ) ,

u = 0 on @� (0; T ) ,u (�; 0) = u0 2 L2 () ,

ku (�; T )kL2() ��Ce

CT ku (�; T )kL2(!)

�� �Cb;T ku0kL2()

�1��

Page 4: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Plan of the talk

I. Applications to control problems and inverse problems

II. Links with known inequalities

III. Proof by Log Convexity and Frequency Function

IV. Proof by Carleman Commutator

Page 5: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

References

Observability : Fursikov-Imanuvilov, Lebeau-Robbiano,...Measurable sets : Apraiz-Escauriaza-Wang-Zhang, VessellaCarleman commutator : Escauriaza-Kenig-Ponce-Vega(JEMS 08)Backward estimate : Bardos-Tartar (ARMA 1973)Inverse problem : Seidman (SINUM 1996)

Collaboration with :Claude Bardos, Nhat Vo, Gengsheng Wang, Yashan Xu

Page 6: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Strategy for applications

To have in mind that for the model wave equation

Observability estimate () Controllability

() Stabilization =) Inverse Source

Page 7: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Application 1 : Pulse Control and Cost

8T1 < T2 < T3 8" > 0 8ye 2 L2 () 9f 2 L2 ()8>>>><>>>>:@ty ��y = 0 in � (T1; T3) nfT2g ,

y = 0 on @� (T1; T3) ,y (�; T1) = ye in ,

y (�; T2) = y (�; T2�) + �!f in ,yd = y (�; T3) in ,

kfkL2() � C" kyekL2() and kydkL2() � " kyekL2()

Minimize J ('0) =cost2

R! j' (x; T3 + T1 � T2)j

2 dx

+ "2

R j'0 (x)j

2 dx�R ye (x)' (x; T3) dx

Page 8: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Application 2 : Inverse Source problemLet f 2 L2 (!) and 0 < � < 1 be such that8<:

@tu��u = 0 in � (0; T ) ,u = 0 on @� (0; T ) ,u (�; 0) 2 H1

0 () ,

ku (�; T )� fkL2(!) � � ku (�; 0)kL2()

=) 9g 2 L2 ()

ku (�; 0)� gkL2() � C1qln1�

ku (�; 0)kH10 ()

Page 9: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Application 3 : Fast Impulse Stabilization

8 > 0 9F : L2 (!1)! L2 (!) 8y0 2 L2 ()8>>>>>><>>>>>>:@ty ��y = 0 in � (0;+1) nfT + TNg ,

y = 0 on @� (0;+1) ,y (�; 0) = y0 in ,

8n 2 Ny (�; (n+ 1)T ) = y (�; (n+ 1)T�) + �!F

��!1y

��;�n+ 1

2

�T��

ky (�; t)kL2() � Ce� t ky0kL2() and kFk � CeC

Page 10: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Observability

ku (�; t)kL2() � ku (�; 0)kL2() 8t � 0

ku (�; T )kL2() ��C1e

C2T� ku (�; T )kL2(!)

�� �ku (�; 0)kL2()

�1��

=)

ku (�; T )kL2() �C1C2eC�

C2T�

Z T

0ku (�; t)kL2(!) dt

Page 11: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of observability and cost

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��=) ku (�; T )k � 1

"1���

C1eC2T�

u (�; T )j! + " ku (�; 0)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k � ku (�; t)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

Next, integrate over (Tm+1; Tm)

Page 12: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of observability and cost

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��=) ku (�; T )k � 1

"1���

C1eC2T�

u (�; T )j! + " ku (�; 0)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k � ku (�; t)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

Next, integrate over (Tm+1; Tm)

Page 13: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of observability and cost

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��=) ku (�; T )k � 1

"1���

C1eC2T�

u (�; T )j! + " ku (�; 0)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k � ku (�; t)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

Next, integrate over (Tm+1; Tm)

Page 14: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of observability and cost

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��=) ku (�; T )k � 1

"1���

C1eC2T�

u (�; T )j! + " ku (�; 0)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k � ku (�; t)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

Next, integrate over (Tm+1; Tm)

Page 15: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of observability and cost

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��=) ku (�; T )k � 1

"1���

C1eC2T�

u (�; T )j! + " ku (�; 0)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k � ku (�; t)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

Next, integrate over (Tm+1; Tm)

Page 16: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

ku (�; Tm)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

=)Z Tm

Tm+1

ku (�; Tm)k dt

� 1

"1���

C1

Z Tm

Tm+1

e

C2

(t�Tm+2)� u (�; t)j! dt+ "Z Tm

Tm+1

ku (�; Tm+2)k dt

=) ku (�; Tm)k

� 1

"1���

C1Tm � Tm+1

e

C2

(Tm+1�Tm+2)�

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k �1

"1���

C1z�C2

e2C2

(z�1)�T�z�(m+2)

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k

Page 17: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

ku (�; Tm)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

=)Z Tm

Tm+1

ku (�; Tm)k dt

� 1

"1���

C1

Z Tm

Tm+1

e

C2

(t�Tm+2)� u (�; t)j! dt+ "Z Tm

Tm+1

ku (�; Tm+2)k dt

=) ku (�; Tm)k

� 1

"1���

C1Tm � Tm+1

e

C2

(Tm+1�Tm+2)�

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k �1

"1���

C1z�C2

e2C2

(z�1)�T�z�(m+2)

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k

Page 18: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

ku (�; Tm)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

=)Z Tm

Tm+1

ku (�; Tm)k dt

� 1

"1���

C1

Z Tm

Tm+1

e

C2

(t�Tm+2)� u (�; t)j! dt+ "Z Tm

Tm+1

ku (�; Tm+2)k dt

=) ku (�; Tm)k

� 1

"1���

C1Tm � Tm+1

e

C2

(Tm+1�Tm+2)�

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k �1

"1���

C1z�C2

e2C2

(z�1)�T�z�(m+2)

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k

Page 19: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

ku (�; Tm)k �1

"1���

C1e

C2

(t�Tm+2)� u (�; t)j! + " ku (�; Tm+2)k

=)Z Tm

Tm+1

ku (�; Tm)k dt

� 1

"1���

C1

Z Tm

Tm+1

e

C2

(t�Tm+2)� u (�; t)j! dt+ "Z Tm

Tm+1

ku (�; Tm+2)k dt

=) ku (�; Tm)k

� 1

"1���

C1Tm � Tm+1

e

C2

(Tm+1�Tm+2)�

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k0 < Tm+2 < Tm+1 � t < Tm < � � � < T with Tm =

T

zmand z> 1

=) ku (�; Tm)k �1

"1���

C1z�C2

e2C2

(z�1)�T�z�(m+2)

Z Tm

Tm+1

uj! dt+ " ku (�; Tm+2)k

Page 20: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

"1��� e

� 2C2(z�1)�T�

z�(m+2) ku (�; Tm)k

�" 1� e�2C2

(z�1)�T�z�(m+2) ku (Tm+2)k �

C1z�C2

Z Tm

Tm+1

uj! dt=) "

1��� e

� 2C2(z�1)�T�

z�(2m+2) ku (T2m)k

�" 1� e�2C2

(z�1)�T�z�(2m+2) ku (T2m+2)k �

C1z�C2

Z T2m

T2m+1

uj! dt" = e

� 2C2(z�1)�T�

z�(2m+2)and Am = e

� 2C2(z�1)�T�

z2�m=)

Xmz }| {(Am)

z2� 1� ku (�; T2m)k �

Xm+1z }| {(Am+1)

1�+1 u ��; T2(m+1)�

� C1z�C2

Z T2m

T2m+1

u (�; t)j! dtNext, take z2� 1� =

1� + 1 and sum the telescoping series from

m = 0 to +1 to complete the proof. with C� =2(1+�)

�(1+�)

12� �1

�� .

Page 21: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

"1��� e

� 2C2(z�1)�T�

z�(m+2) ku (�; Tm)k

�" 1� e�2C2

(z�1)�T�z�(m+2) ku (Tm+2)k �

C1z�C2

Z Tm

Tm+1

uj! dt=) "

1��� e

� 2C2(z�1)�T�

z�(2m+2) ku (T2m)k

�" 1� e�2C2

(z�1)�T�z�(2m+2) ku (T2m+2)k �

C1z�C2

Z T2m

T2m+1

uj! dt" = e

� 2C2(z�1)�T�

z�(2m+2)and Am = e

� 2C2(z�1)�T�

z2�m=)

Xmz }| {(Am)

z2� 1� ku (�; T2m)k �

Xm+1z }| {(Am+1)

1�+1 u ��; T2(m+1)�

� C1z�C2

Z T2m

T2m+1

u (�; t)j! dtNext, take z2� 1� =

1� + 1 and sum the telescoping series from

m = 0 to +1 to complete the proof. with C� =2(1+�)

�(1+�)

12� �1

�� .

Page 22: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

"1��� e

� 2C2(z�1)�T�

z�(m+2) ku (�; Tm)k

�" 1� e�2C2

(z�1)�T�z�(m+2) ku (Tm+2)k �

C1z�C2

Z Tm

Tm+1

uj! dt=) "

1��� e

� 2C2(z�1)�T�

z�(2m+2) ku (T2m)k

�" 1� e�2C2

(z�1)�T�z�(2m+2) ku (T2m+2)k �

C1z�C2

Z T2m

T2m+1

uj! dt" = e

� 2C2(z�1)�T�

z�(2m+2)and Am = e

� 2C2(z�1)�T�

z2�m=)

Xmz }| {(Am)

z2� 1� ku (�; T2m)k �

Xm+1z }| {(Am+1)

1�+1 u ��; T2(m+1)�

� C1z�C2

Z T2m

T2m+1

u (�; t)j! dtNext, take z2� 1� =

1� + 1 and sum the telescoping series from

m = 0 to +1 to complete the proof. with C� =2(1+�)

�(1+�)

12� �1

�� .

Page 23: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

"1��� e

� 2C2(z�1)�T�

z�(m+2) ku (�; Tm)k

�" 1� e�2C2

(z�1)�T�z�(m+2) ku (Tm+2)k �

C1z�C2

Z Tm

Tm+1

uj! dt=) "

1��� e

� 2C2(z�1)�T�

z�(2m+2) ku (T2m)k

�" 1� e�2C2

(z�1)�T�z�(2m+2) ku (T2m+2)k �

C1z�C2

Z T2m

T2m+1

uj! dt" = e

� 2C2(z�1)�T�

z�(2m+2)and Am = e

� 2C2(z�1)�T�

z2�m=)

Xmz }| {(Am)

z2� 1� ku (�; T2m)k �

Xm+1z }| {(Am+1)

1�+1 u ��; T2(m+1)�

� C1z�C2

Z T2m

T2m+1

u (�; t)j! dtNext, take z2� 1� =

1� + 1 and sum the telescoping series from

m = 0 to +1 to complete the proof. with C� =2(1+�)

�(1+�)

12� �1

�� .

Page 24: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

"1��� e

� 2C2(z�1)�T�

z�(m+2) ku (�; Tm)k

�" 1� e�2C2

(z�1)�T�z�(m+2) ku (Tm+2)k �

C1z�C2

Z Tm

Tm+1

uj! dt=) "

1��� e

� 2C2(z�1)�T�

z�(2m+2) ku (T2m)k

�" 1� e�2C2

(z�1)�T�z�(2m+2) ku (T2m+2)k �

C1z�C2

Z T2m

T2m+1

uj! dt" = e

� 2C2(z�1)�T�

z�(2m+2)and Am = e

� 2C2(z�1)�T�

z2�m=)

Xmz }| {(Am)

z2� 1� ku (�; T2m)k �

Xm+1z }| {(Am+1)

1�+1 u ��; T2(m+1)�

� C1z�C2

Z T2m

T2m+1

u (�; t)j! dtNext, take z2� 1� =

1� + 1 and sum the telescoping series from

m = 0 to +1 to complete the proof with C� =2(1+�)

�(1+�)

12� �1

�� .

Page 25: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Lebeau-Robbiano sum of eigenfunctions

���ei = �iei in ,ei 2 H1

0 () , 0 < �1 < �2 � �3 � � � �, �i ! +1 .

P�i��

jaij2 � CeCp�

Z!

����� P�i�� aiei (x)�����2

dx

Reference: Jerison-Lebeau (1996)

Page 26: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Observation at one point in time implies LR

ku (�; T )kL2() ��C1e

C2T� ku (�; T )kL2(!)

�� �ku (�; 0)kL2()

�1��

=)

P�i��

jaij2 � C21e4C1

1+�2 ( 1��� )

�1+� �

�1+�

Z!

����� P�i�� aiei (x)�����2

dx

Example : � = 1=3 for the bilaplacian.

Page 27: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 28: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 29: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 30: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 31: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 32: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 = ku (�; T )k2 �

��C21e

2C2T�

u (�; T )j! 2�� �ku (�; 0)k2�1��=

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� P�i��

��aie�jT ��2!1��

0@C21e 2C2T�

Z!

����� P�i�� aiei (x)�����2

dx

1A� e2�T (1��) P�i��

jaij2!1��

Page 33: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 � C21e2C2T� e2

1����T

Z!

����� P�i�� aiei (x)�����2

dx

Next, optimize w.r.t. T by choosing T =�

�1��

C2�

� 11+� to conclude

P�i��

jaij2 � C21e4C1

1+�2 ( 1��� )

�1+� �

�1+�

Z!

����� P�i�� aiei (x)�����2

dx .

Page 34: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 � C21e2C2T� e2

1����T

Z!

����� P�i�� aiei (x)�����2

dx

Next, optimize w.r.t. T by choosing T =�

�1��

C2�

� 11+� to conclude

P�i��

jaij2 � C21e4C1

1+�2 ( 1��� )

�1+� �

�1+�

Z!

����� P�i�� aiei (x)�����2

dx .

Page 35: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of Lebeau-Robbiano

ku (�; T )k ��C1e

C2T�

u (�; T )j! �� ku (�; 0)k1��u (�; t) =

P�i��

aie�j(T�t)ei (x)

=)P�i��

jaij2 � C21e2C2T� e2

1����T

Z!

����� P�i�� aiei (x)�����2

dx

Next, optimize w.r.t. T by choosing T =�

�1��

C2�

� 11+� to conclude

P�i��

jaij2 � C21e4C1

1+�2 ( 1��� )

�1+� �

�1+�

Z!

����� P�i�� aiei (x)�����2

dx .

Page 36: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Conversely

P�j�� jaj j

2 � eC2(1+��)Z!

���P�j�� ajei (x)���2 dx

=)8� 2 (0; 1)

ku (�; T )k � eC�1+( 1

�T )�

1���ku (�; 0)k�

u (�; T )j! 1��=)

ku (�; T )k � eC4�1+( 1T )

�1��

�e

�C4Tln�ku(�;0)kku(�;T )k

��� u (�; T )j! See Yubiao Zhang (CRAS 2016) for the Kolmogorov equation.

Page 37: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Estimate with weights implies desired observationLet 0 � t1 < t2 < t3 � T . Denote M = �ln(T�t3+~)+ln(T�t2+~)

�ln(T�t2+~)+ln(T�t1+~)

and � (x; t) = �jx�x0j24(T�t+~) �

n2 ln(T � t+ ~)

�Zju (x; t2)j2 e�(x;t2)dx

�1+M�Zju (x; t3)j2 e�(x;t3)dx

�Zju (x; t1)j2 e�(x;t1)dx

�M

=)

ku (�; T )k � eC(1+1T ) u (�; T )j! � ku (�; 0)k1��

Page 38: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Let 0 � t1 < t2 < t3 � T . Denote M = �ln(T�t3+~)+ln(T�t2+~)�ln(T�t2+~)+ln(T�t1+~)

and � (x; t) = �jx�x0j24(T�t+~) �

n2 ln(T � t+ ~)�Z

ju (x; t2)j2 e�(x;t2)dx

�1+M�Zju (x; t3)j2 e�(x;t3)dx

�Zju (x; t1)j2 e�(x;t1)dx

�MChoose t3 = T , t2 = T � `~, t1 = T � 2`~, then M` =

ln(`+1)ln( 2`+1`+1 )

and�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

Page 39: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Let 0 � t1 < t2 < t3 � T . Denote M = �ln(T�t3+~)+ln(T�t2+~)�ln(T�t2+~)+ln(T�t1+~)

and � (x; t) = �jx�x0j24(T�t+~) �

n2 ln(T � t+ ~)�Z

ju (x; t2)j2 e�(x;t2)dx

�1+M�Zju (x; t3)j2 e�(x;t3)dx

�Zju (x; t1)j2 e�(x;t1)dx

�MChoose t3 = T , t2 = T � `~, t1 = T � 2`~, then M` =

ln(`+1)ln( 2`+1`+1 )

and�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

Page 40: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Let 0 � t1 < t2 < t3 � T . Denote M = �ln(T�t3+~)+ln(T�t2+~)�ln(T�t2+~)+ln(T�t1+~)

and � (x; t) = �jx�x0j24(T�t+~) �

n2 ln(T � t+ ~)�Z

ju (x; t2)j2 e�(x;t2)dx

�1+M�Zju (x; t3)j2 e�(x;t3)dx

�Zju (x; t1)j2 e�(x;t1)dx

�MChoose t3 = T , t2 = T � `~, t1 = T � 2`~, then M` =

ln(`+1)ln( 2`+1`+1 )

and�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

Page 41: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and� (x; t) = �jx�x0j2

4(T�t+~) �n2 ln(T � t+ ~)and D = max

x2jx� x0j

�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

=)�Zju (x; T � `~)j2 e

�jx�x0j2

4(`+1)~ dx

�1+M`

1

(`+1)n=2

�Zju (x; T )j2 e

�jx�x0j2

4~ dx

�Zju (x; T � 2`~)j2 e

�jx�x0j2

4(2`+1)~ dx

�M`

Page 42: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and� (x; t) = �jx�x0j2

4(T�t+~) �n2 ln(T � t+ ~) and D = max

x2jx� x0j

�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

=)�Zju (x; T � `~)j2 e

�jx�x0j2

4(`+1)~ dx

�1+M`

1

(`+1)n=2

�Zju (x; T )j2 e

�jx�x0j2

4~ dx

�Zju (x; T � 2`~)j2 e

�jx�x0j2

4(2`+1)~ dx

�M`

Page 43: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and� (x; t) = �jx�x0j2

4(T�t+~) �n2 ln(T � t+ ~) and D = max

x2jx� x0j

�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

=)�Zju (x; T � `~)j2 e

�jx�x0j2

4(`+1)~ dx

�1+M`

1

(`+1)n=2

�Zju (x; T )j2 e

�jx�x0j2

4~ dx

�Zju (x; 0)j2 dx

�M`

Page 44: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and� (x; t) = �jx�x0j2

4(T�t+~) �n2 ln(T � t+ ~) and D = max

x2jx� x0j

�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

=)�Zju (x; T � `~)j2 dx

�1+M`

e�D

4(`+1)~ (1+M`) 1

(`+1)n=2

�Zju (x; T )j2 e

�jx�x0j2

4~ dx

�Zju (x; 0)j2 dx

�M`

Page 45: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and� (x; t) = �jx�x0j2

4(T�t+~) �n2 ln(T � t+ ~) and D = max

x2jx� x0j

�Zju (x; T � `~)j2 e�(x;T�`~)dx

�1+M`

�Zju (x; T )j2 e�(x;T )dx

�Zju (x; T � 2`~)j2 e�(x;T�2`~)dx

�M`

=)�Zju (x; T )j2 dx

�1+M`

e�D

4(`+1)~ (1+M`) 1

(`+1)n=2

�Zju (x; T )j2 e

�jx�x0j2

4~ dx

�Zju (x; 0)j2 dx

�M`

Page 46: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and D = maxx2

jx� x0j

Zju (x; T )j2 e

�jx�x0j2

4~ dx

=

Zjx�x0j2�r

+

Zjx�x0j2�r

ju (x; T )j2 e�jx�x0j

2

4~ dx

=)�Zju (x; T )j2 dx

�1+M`

e�D

4(`+1)~ (1+M`) 1

(`+1)n=2

� Z

jx�x0j2�rju (x; T )j2 dx+ e

�r4~

Zju (x; 0)j2 dx

!

��Z

ju (x; 0)j2 dx

�M`

Page 47: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and D = maxx2

jx� x0j

Zju (x; T )j2 e

�jx�x0j2

4~ dx

=

Zjx�x0j2�r

+

Zjx�x0j2�r

ju (x; T )j2 e�jx�x0j

2

4~ dx

=)�Zju (x; T )j2 dx

�1+M`

e�D

4(`+1)~ (1+M`) 1

(`+1)n=2

� Z

jx�x0j2�rju (x; T )j2 dx+ e

�r4~

Zju (x; 0)j2 dx

!

��Z

ju (x; 0)j2 dx

�M`

Page 48: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and D = max

x2jx� x0j

�Zju (x; T )j2 dx

�1+M`

�(`+ 1)n=2

�e

D4(`+1)~ (1+M`)

u (�; T )j! 2 + e D4(`+1)~ (1+M`)e

�r4~ ku (�; 0)k2

���Z

ju (x; 0)j2 dx

�M`

Choose ` > 1 such that D4(`+1)~ (1 +M`) ' D

4(`+1)~ (1 + ln`) =r8~ ,�Z

ju (x; T )j2 dx

�1+M`

(`+ 1)n=2�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��Zju (x; 0)j2 dx

�M`

Page 49: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and D = max

x2jx� x0j

�Zju (x; T )j2 dx

�1+M`

�(`+ 1)n=2

�e

D4(`+1)~ (1+M`)

u (�; T )j! 2 + e D4(`+1)~ (1+M`)e

�r4~ ku (�; 0)k2

���Z

ju (x; 0)j2 dx

�M`

Choose ` > 1 such that D4(`+1)~ (1 +M`) ' D

4(`+1)~ (1 + ln`) =r8~ ,�Z

ju (x; T )j2 dx

�1+M`

(`+ 1)n=2�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��Zju (x; 0)j2 dx

�M`

Page 50: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�Here ~ < 1, ` > 1, 2`~ < T , M` 'ln` and D = max

x2jx� x0j

�Zju (x; T )j2 dx

�1+M`

�(`+ 1)n=2

�e

D4(`+1)~ (1+M`)

u (�; T )j! 2 + e D4(`+1)~ (1+M`)e

�r4~ ku (�; 0)k2

���Z

ju (x; 0)j2 dx

�M`

Choose ` > 1 such that D4(`+1)~ (1 +M`) ' D

4(`+1)~ (1 + ln`) =r8~ ,�Z

ju (x; T )j2 dx

�1+M`

(`+ 1)n=2�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��Zju (x; 0)j2 dx

�M`

Page 51: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, 2`~ < T ,�ku (�; T )k2

�1+K� C

�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��ku (�; 0)k2�KAlso true for ~ �min(1; T= (2`)) with C = KeK=T .

Optimize w.r.t ~ > 0 such that�ku (�; T )k2

�1+K� KeK=T

�2 u (�; T )j! ku (�; 0)k��ku (�; 0)k2�K

This completes the proof.

Page 52: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, 2`~ < T ,�ku (�; T )k2

�1+K� C

�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��ku (�; 0)k2�KAlso true for ~ �min(1; T= (2`)) with C = KeK=T .

Optimize w.r.t ~ > 0 such that�ku (�; T )k2

�1+K� KeK=T

�2 u (�; T )j! ku (�; 0)k��ku (�; 0)k2�K

This completes the proof.

Page 53: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Proof of �weights imply observation�

Here ~ < 1, 2`~ < T ,�ku (�; T )k2

�1+K� C

�er8~

u (�; T )j! 2 + e�r8~ ku (�; 0)k2��ku (�; 0)k2�KAlso true for ~ �min(1; T= (2`)) with C = KeK=T .

Optimize w.r.t ~ > 0 such that�ku (�; T )k2

�1+K� KeK=T

�2 u (�; T )j! ku (�; 0)k��ku (�; 0)k2�K

This completes the proof.

Page 54: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

ODE implies Estimate with weights

8><>:1

2y0 (t) +N (t) y (t) = 0

N 0 (t) � 1

T � t+ ~N (t)

[y (t2)]1+M � y (t3) [y (t1)]M with M = �ln(T�t3+~)+ln(T�t2+~)

�ln(T�t2+~)+ln(T�t1+~)

TRUE with y (t) =Zju (x; t)j2 e�(x;t)dx

and N (t) =

Zjru (x; t)j2 e�(x;t)dxZju (x; t)j2 e�(x;t)dx

when convex

Page 55: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Resume

When convex, take e�(x;t) = 1

(T�t+~)n=2e�jx�x0j

2

4(T�t+~)

then y (t) =Zju (x; t)j2 e�(x;t)dx

and N (t) =

Zjru (x; t)j2 e�(x;t)dxZju (x; t)j2 e�(x;t)dx

solve the ODE

8><>:1

2y0 (t) +N (t) y (t) = 0

N 0 (t) � 1

T � t+ ~N (t)

which implies the �estimate with weights�which implies observation in one point in time

then, Lebeau-Robbiano inequality and observability

Page 56: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

General caseWhen 2 C2, localize with � 2 C10 (Bx0;R), �u solves aparabolic equation with a second member g .Replace jx� x0j by the geodesic distance d (x; x0) .Then y (t) =

Zj�u (x; t)j2 e�(x;t)dx

and N (t) =

Zjr (�u) (x; t)j2 e�(x;t)dxZj�u (x; t)j2 e�(x;t)dx

solve an ODE

8>><>>:����12y0 (t) +N (t) y (t)

���� � � C

T � t+ ~ + C�y (t) +

��R (�u) ge

�dx��

N 0 (t) ��1 + CR

T � t+ ~ + C�N (t) +

R jgj

2 e�dx

y (t)

which implies the �estimate with weights� (ok if CR < 1).Next propagate the balls to conclude the observation at one pointin time.

Page 57: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Carleman commutatorAnother way to get the ODE for Log convexity :

Write y (t) =Zju (x; t)j2 e�(x;t)dx =

Zjf (x; t)j2 dx.

f = ue�=2 solves

@tf ��f �1

2f

�@t�+

1

2jr�j2

�| {z }�Sf with hSf;gi=hSg;fi+

R@ @nfgd�

+ r� � rf + 12��f| {z }

�Af with hAf;gi=�hAg;fi

= 0

Computations done by Escauriaza-Kenig-Ponce-Vega (JEMS 08)=) 8>>><>>>:

1

2y0 (t)� hSf; fi = 0

N (t) y (t) = �hSf; fi

N 0 (t) � 1

y (t)h� (S0 + [S;A]) f; fi �

R@ @nfAfd�

y (t)

Page 58: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Carleman commutatorAnother way to get the ODE for Log convexity :

Write y (t) =Zju (x; t)j2 e�(x;t)dx =

Zjf (x; t)j2 dx.

f = ue�=2 solves

@tf ��f �1

2f

�@t�+

1

2jr�j2

�| {z }�Sf with hSf;gi=hSg;fi+

R@ @nfgd�

+ r� � rf + 12��f| {z }

�Af with hAf;gi=�hAg;fi

= 0

Computations done by Escauriaza-Kenig-Ponce-Vega (JEMS 08)=) 8>>><>>>:

1

2y0 (t)� hSf; fi = 0

N (t) y (t) = �hSf; fi

N 0 (t) � 1

y (t)h� (S0 + [S;A]) f; fi �

R@ @nfAfd�

y (t)

Page 59: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Carleman commutatorAnother way to get the ODE for Log convexity :

Write y (t) =Zju (x; t)j2 e�(x;t)dx =

Zjf (x; t)j2 dx.

f = ue�=2 solves

@tf ��f �1

2f

�@t�+

1

2jr�j2

�| {z }�Sf with hSf;gi=hSg;fi+

R@ @nfgd�

+ r� � rf + 12��f| {z }

�Af with hAf;gi=�hAg;fi

= 0

Computations done by Escauriaza-Kenig-Ponce-Vega (JEMS 08)=) 8>>><>>>:

1

2y0 (t)� hSf; fi = 0

N (t) y (t) = �hSf; fi

N 0 (t) � 1

y (t)h� (S0 + [S;A]) f; fi �

R@ @nfAfd�

y (t)

Page 60: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Applications

For convex, by Log convexity, it works

with � (x; t) = �jx�x0j24(T�t+~) �

n2 ln(T � t+ ~)

in order to have � (S0 + [S;A]) f = 1

T � t+ ~ (�Sf)

and N 0 (t) � 1

T � t+ ~N (t) .

But it also works with � (x; t) = �jx�x0j24(T�t+~)

Page 61: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Lebeau-Robbiano for convex domain

���ei = �iei in ,ei 2 H1

0 () , 0 < �1 < �2 � �3 � � � �, �i ! +1 .

P�i��

jaij2 � 4eC"p� 1

r"+o(")

Zjx�x0j<r

����� P�i�� aiei (x)�����2

dx

Page 62: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Lebeau-Robbiano for Schroedinger operator

���ei + V (x) ei = �iei in ,ei 2 H1

0 () , �1 � �2 � �� � �N � 0 < �N+1 � � � � .

P�i��

jaij2 � CeC(kV k2=31 +

p�)Z!

����� P�i�� aiei (x)�����2

dx

Page 63: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

I: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Lebeau-Robbiano for Schroedinger operator

(��ei � �

jxj2 ei = �iei in , � � (n�2)24 , n � 3,

ei 2 H10 () , �1 � �2 � � � �, �i ! +1 .

P�i��

jaij2 � CeCp�

Z!

����� P�i�� aiei (x)�����2

dx

Page 64: Observation estimate at one point in time for parabolic ...trelat/GDT/confs/Kim_Dang_Phung.pdfI: Main result II: A = Id and b = 0 III: Links IV: Log convexity VI: Carleman commutator

Thank You