Quantitative uniqueness for time periodic heat equation with potential and its applications

-

Documents
42 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

Niveau: Supérieur, Licence, Bac+2
Di?erential and Integral Equations Volume 19, Number 6 (2006), 627–668 QUANTITATIVE UNIQUENESS FOR TIME-PERIODIC HEAT EQUATION WITH POTENTIAL AND ITS APPLICATIONS K-D. Phung Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China G. Wang Department of Mathematics, Wuhan University, Wuhan 430072, China The Center for Optimal Control and Discrete Mathematics Huazhong Normal University, Wuhan 730079, China (Submitted by: Viorel Barbu) Abstract. In this paper, we establish a quantitative unique contin- uation property for some time-periodic linear parabolic equations in a bounded domain Ω. We prove that for a time-periodic heat equa- tion with particular time-periodic potential, its solution u(x, t) satis- fies ?u(·, 0)?L2(Ω) ≤ C ?u(·, 0)?L2(?) where ? ? Ω. Also we deduce the asymptotic controllability for the heat equation with an even, time- periodic potential. Moreover, the controller belongs to a finite dimen- sional subspace and is explicitly computed. 1. Introduction and main results Throughout this paper, Ω is a connected bounded domain in Rd, d ≥ 1, with a boundary ∂Ω of class C2, ? is a non-empty open subset of Ω. Let 1 > 0 be the first eigenvalue of the operator ?? with the Dirichlet boundary condition (i.

  • periodic potential

  • stable eigenvalues

  • strong unique

  • unique solution

  • heat equation

  • boun- ded domains

  • continuation property

  • linear time-periodic


Sujets

Informations

Publié par
Nombre de visites sur la page 12
Langue English
Signaler un problème
Differential and Integral Equations
Volume 19, Number 6 (2006), 627–668
QUANTITATIVE UNIQUENESS FOR TIME-PERIODIC HEAT EQUATION WITH POTENTIAL AND ITS APPLICATIONS
K-D. Phung Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
G. Wang Department of Mathematics, Wuhan University, Wuhan 430072, China The Center for Optimal Control and Discrete Mathematics Huazhong Normal University, Wuhan 730079, China
(Submitted by: Viorel Barbu)
Abstract.In this paper, we establish a quantitative unique contin-uation property for some time-periodic linear parabolic equations in a bounded domain Ω. We prove that for a time-periodic heat equa-tion with particular time-periodic potential, its solutionu(x, t) satis-fiesu(·,0)L2(Ω)Cu(·,0)L2(ω)whereωΩ. Also we deduce the asymptotic controllability for the heat equation with an even, time-periodic potential. Moreover, the controller belongs to a finite dimen-sional subspace and is explicitly computed.
1.Introduction and main results Throughout this paper, Ω is a connected bounded domain inRd,d1, with a boundaryΩ of classC2,ω Letis a non-empty open subset of Ω. 1>0 be the first eigenvalue of the operatorΔ with the Dirichlet boundary condition (i.e., the smallest strictly positive eigenvalue ofΔ inH01(Ω)). Let T >0 be a real number; we denote by.the usual norm inL×(0, T)). In this paper, we study an infinite-dimensional system generated by the following parabolic equation tyyΔy=0+ay=onfΩ×Ωin0(,+×)(0,,+),(1.1)
Accepted for publication: February 2006. AMS Subject Classifications: 35B10, 93B07, 93B60. The first author thanks the support of the NCET od China under grant NCET-04-0882, and the NSF of China under grants 10371084 and 10525105. 627
628
K-D. Phung and G. Wang
with an even,T-periodic, bounded potentiala=a(x, t)L×R) satis-fying
1≤ aanda(x, t+T) =a(x, t) =a(x,t) in Ω×R.(1.2) We assume thaty(·,0) =yoL2(Ω) andf=f(x, t)Ll1oc(0,+;L2(Ω)), so that (1.1) admits a unique solutiony=y(x, t)C([0,+);L2(Ω)). Let {G(t, s)}0st<+be theT-periodic evolutionary process onL2(Ω), such that y(·, t) =G(t, s)y(·, s) +stG(t, r)f(·, r)dr,(1.3) for all 0st, where theT-periodicity ofG(t, s) meansG(t+T , s+T) = G(t, s). Notice that any bounded function inL×(0, T /2)) can be extended to be an even,T-periodic potentialaL×R). Here, we are only interested in the case where (a1)0 andais not positive, because if (a1)<0 orathe system (1.1) is clearly stable when0, then f= 0. The Poincar´e map is usually defined byG(T+t, t restrict our at-). We tention on the operatorG(T ,0) G(T ,0) :Ly(2·(,))Ω0yL(2·,(T))Ω.(1.4)
It is well known thatG(T ,0) is compact by the smoothing action of the diffusion. Sinceais even andT-periodic, we see thatG(T ,0) is self-adjoint. Consequently, there is a complete orthonormal set inL2(Ω) formed of eigen-functions (zjo)j1ofG(T ,0) corresponding to eigenvalues (λj)j1, where λj=λj(T) depends onT,λjR,λj0 asj+ we may. Thus, arrange the eigenfunctions so that the sequence{|λj|}j1is non-increasing, · · · ≤ |λm+1| ≤ |λm| ≤ · · · ≤ |λ1|. Clearly,λj= 0 from the backward uniqueness property for the linear parabolic equation. Furthermore, we will see that for allT >0 and for all even,T-periodic, bounded potentialsa, any eigenvalueλj(T) ofG(T ,0) satisfies 1+ ln|Tλj(T) | ≤a.(1.5) The first result of the paper is as follows. Theorem 1.There exist two constantsco(0,1)andC >0,both only depending onωandΩ, such that if we choose the time-periodicityT(0, co]
Quantitative uniqueness for time-periodic heat equation
and an even,T-periodic potentialaL×R)such that
1/4 1≤ a1Tco,
629
then any eigenfunctionzjoofG(T ,0)inL2(Ω),corresponding to the eigen-valueλj(T)with|λj(T)| ≥1,satisfies
ΩG(t,0)zjo(x)2dxeCa4/3G(t,0)zjo(x)2dx ω
for allt0.
The knowledge of the eigencouple of the Poincar´e map plays a key role in the study of periodic parabolic systems (see [14], [15]). Theorem 1 implies clearly that the eigenfunctionszjomeopatofcerhroPniac´rgtotheespondin eigenvaluesλjwith|λj| ≥1 have the unique continuation property. It is classical in control theory for linear partial differential equations that the unique continuation property is linked with the approximate controlla-bility and, more precisely, a quantitative uniqueness result yields an estimate of the cost of the approximate control (see e.g. [12]). Here, our second the-orem establishes an asymptotic controllability (or open loop stabilizability) for the heat equation with time-periodic potential. Theorem 2.There exist two constantsco(0,1)andC >0,both only depending onΩandωsuch that if we choose the time-periodicity, T(0, co] and an even,T-periodic potentialaL×R)such that
1≤ a1co1/4 T ,
and if there existsm >0such that
· · · ≤ |λm+1(T)|<1≤ |λm(T)|=· · ·=|λ2(T)|=|λ1(T)|,
630K-D. Phung and G. Wang then for each initial datayoL2(Ω),the control functionfcC([0, T];L2(Ω)) satisfying (i)fc(x, Tt) =σj(t)G(t,0)zjo(x), j=1,··,m σ1(t) σm.(t)TωG(t,0)zκo(x)G(t,0)zjo(x)dx1κ,jm1 (ii)=1 Ωyo(x)G(T ,0)z1o(x)dx. ×o(x)G(T ,0)zmo(x)dx, Ωy (iii)0Tfc(·, s)L2(ω)dseCa4/3e(a1)TyoL2(Ω), implies that the solutiony=y(x, t)C([0,+);L2(Ω))of the following heat equation with potentialaand control functionfc, tyΔy+ayy==0fco·n1|ωΩ×(0×,T()0,+inΩ),×(0,+), y(,0) =yoinΩ, · satisfies, for alltT, y(·, t)L2(Ω)CmeγtyoL2(Ω), where C(a4/3+aT)
γ=ln|Tλm+1(T)|>0andCm= 2e|λm+1(T)|2. The notion of asymptotic controllability is standard in the nonlinear con-trol theory of finite-dimensional systems (see e.g. [7]). Here, (i) says that our control has a finite-dimensional structure (see also [4], [5]). (ii) implies that the operator associating the initial data with the control function is linear which gives a kind of robustness property of our control. (iii) gives us an explicit expression of the cost of the control, from which we see easily that we can act a control on the equation in a very short timeT, but as a payment, evenness andT-periodicity for the potentialaare required. From the following property of the Poincar´e map, G(nT ,0)zjo= (λj)nzjo(1.6) for allnN, it is clear that if all eigenvaluesλjsatisfy|λj|<1, then the system (1.1) is stable without control. On the other hand, if the modu-lus of some eigenvalues ofG(T ,0) are bigger than one, then the equation