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

Reza Aftabizadeh

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

• Wuhan University
• Heat equation

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 12
• Langue: English

Extrait

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- fiesu(·,0) L 2 (Ω) ≤Cu(·,0) L 2 (ω) whereω⊂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 inR d ,d≥1, with a boundary∂Ω of classC 2 ,ω Letis a non-empty open subset of Ω.  1 >0 be the first eigenvalue of the operator−Δ with the Dirichlet boundary condition (i.e., the smallest strictly positive eigenvalue of−Δ inH 01 (Ω)). Let T >0 be a real number; we denote by. ∞ the usual norm inL ∞ (Ω×(0, T)). In this paper, we study an infinite-dimensional system generated by the following parabolic equation  ∂ t y− y Δy =0 +ay= on f ∂Ω× Ωin 0(,+∞ × ) (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

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K-D. Phung and G. Wang

with an even,T-periodic, bounded potentiala=a(x, t)∈L ∞ (Ω×R) satis- fying

 1 ≤ a ∞ anda(x, t+T) =a(x, t) =a(x,−t) in Ω×R.(1.2) We assume thaty(·,0) =y o ∈L 2 (Ω) andf=f(x, t)∈L l 1 oc (0,+∞;L 2 (Ω)), so that (1.1) admits a unique solutiony=y(x, t)∈C([0,+∞);L 2 (Ω)). Let {G(t, s)} 0≤s≤t<+∞ be theT-periodic evolutionary process onL 2 (Ω), such that y(·, t) =G(t, s)y(·, s) +  s t G(t, r)f(·, r)dr,(1.3) for all 0≤s≤t, where theT-periodicity ofG(t, s) meansG(t+T , s+T) = G(t, s). Notice that any bounded function inL ∞ (Ω×(0, T /2)) can be extended to be an even,T-periodic potentiala∈L ∞ (Ω×R). Here, we are only interested in the case where (a ∞ − 1 )≥0 andais not positive, because if (a ∞ − 1 )<0 ora≥the 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) :L y( 2 · ( , ) ) Ω 0−→ −→ y L ( 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 inL 2 (Ω) formed of eigen- functions (z jo ) j≥1 ofG(T ,0) corresponding to eigenvalues (λ j ) j≥1 , where λ j =λ j (T) depends onT,λ j ∈R,λ j →0 asj→+∞ we may. Thus, arrange the eigenfunctions so that the sequence{|λ j |} j≥1 is non-increasing, · · · ≤ |λ m+1 | ≤ |λ m | ≤ · · · ≤ |λ 1 |. Clearly,λ j = 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 eigenvalueλ j (T) ofG(T ,0) satisfies  1 + ln| T λ j (T)   | ≤ a ∞ .(1.5) The first result of the paper is as follows. Theorem 1.There exist two constantsc o ∈(0,1)andC >0,both only depending onωandΩ, such that if we choose the time-periodicityT∈(0, c o ]

Quantitative uniqueness for time-periodic heat equation

and an even,T-periodic potentiala∈L ∞ (Ω×R)such that

1/4  1 ≤ a ∞ ≤ 1  T c o  ,

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then any eigenfunctionz jo ofG(T ,0)inL 2 (Ω),corresponding to the eigen- valueλ j (T)with|λ j (T)| ≥1,satisfies

 Ω  G(t,0)z jo (x)  2 dx≤e Ca 4 ∞ /3   G(t,0)z jo (x) 2 dx  ω

for allt≥0.

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 eigenfunctionsz jo meopatofcerhroPniac´rgtotheespondin eigenvaluesλ j with|λ j | ≥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 constantsc o ∈(0,1)andC >0,both only depending onΩandωsuch that if we choose the time-periodicity, T∈(0, c o ] and an even,T-periodic potentiala∈L ∞ (Ω×R)such that

 1 ≤ a ∞ ≤ 1  c o  1/4 T ,

and if there existsm >0such that

· · · ≤ |λ m+1 (T)|<1≤ |λ m (T)|=· · ·=|λ 2 (T)|=|λ 1 (T)|,

630K-D. Phung and G. Wang then for each initial datay o ∈L 2 (Ω),the control functionf c ∈C([0, T];L 2 (Ω)) satisfying ⎧ (i)f c (x, T−t) =  σ j (t)G(t,0)z jo (x), j=1,··,m ⎛ σ 1 (t) ⎨ ⎪ ⎝ ⎜ σ m . (t) ⎠ ⎞ T   ω G(t,0)z κo (x)G(t,0)z jo (x)dx  1≤κ,j≤m  −1 (ii) ⎟ =− 1 ⎛  Ω y o (x)G(T ,0)z 1 o (x)dx ⎞ . × ⎜ o (x)G(T ,0)z m o (x)dx ⎠ ⎟ , ⎝  Ω y ⎪ ⎩ (iii)  0 T f c (·, s) L 2 (ω) ds≤e Ca 4 ∞ /3 e ( a ∞ − 1 ) T y o  L 2 (Ω) , implies that the solutiony=y(x, t)∈C([0,+∞);L 2 (Ω))of the following heat equation with potentialaand control functionf c , ⎨ ⎩ ⎧ ∂ t y−Δy+ay y = =0 f c o · n 1 | ∂ ω Ω ×(0 × ,T ( ) 0,+ in ∞ Ω ), ×(0,+∞), y(,0) =y o inΩ, · satisfies, for allt≥T, y(·, t) L 2 (Ω) ≤C m e −γt y o  L 2 (Ω) , where C ( a 4/3 +aT )

γ= −ln| T λ m+1 (T)| >0andC m = 2e |λ m+ ∞ 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)z jo = (λ j ) n z jo (1.6) for alln∈N, it is clear that if all eigenvaluesλ j satisfy|λ j |<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