Null controllability of the heat equation on the Heinsenberg group

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Null controllability of the heat equation on the Heinsenberg group Liviu Ignat Institute of Mathematics of the Romanian Academy Paris, November 2010 Joint work with Enrique Zuazua Liviu Ignat (IMAR) Heat equation on the Heinsenberg group IHP, Paris, November 2010 1 / 37

  • romanian academy

  • liviu ignat

  • heinsenberg group

  • squares vector

  • null controllability

  • cd ?

  • heat equation


Published : Tuesday, June 19, 2012
Reading/s : 24
Origin : univ-orleans.fr
Number of pages: 42
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Liviu Ignat
Paris, November 2010 Joint work with Enrique Zuazua
Institute of Mathematics of the Romanian Academy
Null controllability of the heat equation on the Heinsenberg group
73/10
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3
Heat Equation
4
Controlability
5
Carleman inequality
1
Heisenberg Group
/273
Outline
2
Sum of squares vector fields
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HeR)eqattiuaononHehtsnieebnerggriLivIungtaI(AM
Heisenberg Group
1
Carleman inequality
5
Controlability
4
Heat Equation
3
2
N,vorasiPHP,uoIp7
Outline
Sum of squares vector fields
03/3r201emberoupreGgesbnHie
= (x1+y1, . . . , x2d+y2d, s+u+12dX(xkyd+kyd+kyk). k=1
(x1, . . . , x2d, s)(y1, . . . , y2d, u)
04/37
or
The Heisenbergh groupHdcan be viewed asR2d+1endowed with the group multiplication
uorGpisenbergHeC×d,RviuI).Liz0)2=(z0s1+,0+sz(z+0s=)0,(z),s(zawplourgehthtiwdewodneember201aris,NovuoIpPHP,neebgrrgeHthnseitiuaononeH)RqetatangAMI(
(x1, . . . , x2d, s)(y1, . . . , y2d, u)
The Heisenbergh groupHdcan be viewed asR2d+1endowed with the group multiplication
or Cd×R, endowed with the group law
(z, s)(z0, s0) = (z+z0, s+s01+2=(zz0)).
= (x1+y1, . . . , x2d+y2d, s+u+12dX(xkyd+kyd+kyk). k=1
/37HeisrgGrenbepuothonontiennseieH)RAMI(taauqetaeHovemis,N0104ber2rguoebgrP,raIpPHLiviuIgn
The Heisenberg Laplacian - a second order operator
Fork∈ {1, . . . , d}consider the vector fields
2d ΔH=XXk2. k=1
Xk=xkxd2+kX,sd+k=xd+k+x2k.s
ovember20105/37uproreGgesbnHiegroupIHP,Paris,NnohtHeiesnneebgrR)MAatHeuaeqontiiviLngIuI(ta
Heisenberg Group In divergence form ΔHu= div(Aru) whereAis the following(2d+ 1)×(2d+ 1)matrix: 0II01221((xxd+1,1...,x,..,.dx)2Td)T, 21(xd+1, . . . , x2d)12(x1, . . . , xd)|x4|2 Liviu Ignat (IMAR)Heat equation on the Heinsenberg group / 37IHP, Paris, November 2010 6
rGgrpuoisHebeen3/70102rebmevoN,
Remark 1.ΔHis no-where elliptic.
Remark 2. It a basic example of a hypoelliptic operator
Ford= 1
A(x1, x2, s) =1210x21201x1x1221412+xxx1222, Observe thatAis a semi-positive matrix withdet(A(x, s)) = 0for all (x, s)H1andrank(A) = 2 .
7onontheHeinsenbegrrguoIpPHP,rasiLiuIviatgnAMI(eH)Rqetaitau
equaHeatonthtionIungiLivAM)RtaI(ar,PHPpIemov,NisnesnieHeuorggrebrleceotervsqsauumofSds7/3
2
3
Outline
Sum of squares vector fields
r2be0801
5
Controlability
4
Heat Equation
Heisenberg Group
1
Carleman inequality
xaEor:Flempattnocsnicneocerevets,Tethes(seesarqufsmoSusdlerotcev73/
Definition: A differential operatorLishypoellipticif wheneveruandf are distributions satisfyingLu=f,umust beCon any open set where fisC
is,November20109ebgrrguoIpPHP,rathonontiennseieH)RAMI(taauqetaeH|0.P(ξ)uIgnLiviseP|pmil1||+(α)ξξ|,di|0=6|Rξ,thch|αatleupsusαrolanlt-iditnoF:owingconsthefollevig)stneiceoctanstonhcitswontiqeauitlarenedlirtiaarpaLinebook
rgbeougreieHennsN,simevoPHIpraP,at(IMAR)LiviuIgnitnonohteHtaqeau
Definition: A differential operatorLishypoellipticif wheneveruandf are distributions satisfyingLu=f,umust beCon any open set where fisCExample: For constant coefficients, Treves (see the bookLinear partial differential equations with constant coefficients) gives the following condition: For alln-tuplesαsuch that|α| 6= 0,
ξRd,|ξ| → ∞implies1|+P|αP(ξ)(ξ|)| →0.
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