A dependence of the cost of fast controls for the heat equation on the support of initial datum
aa r X i v : . [ m a t h . O C ] F e b A DEPENDENCE OF THE COST OF FAST CONTROLS FOR THE HEATEQUATION ON THE SUPPORT OF INITIAL DATUM
HOAI-MINH NGUYEN
Abstract.
The controllability cost for the heat equation as the control time T goes to 0 is well-known of the order e C/T for some positive constant C , depending on the controlled domain andfor all initial datum. In this paper, we prove that the constant C can be chosen to be arbitrarilysmall if the support of the initial data is sufficiently close to the controlled domain, but notnecessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano’sapproach in which a new spectral inequality is established. The main ingredient of the proof ofthe new spectral inequality is three-sphere inequalities with partial data. Key words: heat equations, fast controls, controllability cost, spectral inequalities, three-sphereinequalities.
Mathematics Subject Classification:
Introduction
We are interested in the dependence of the cost of fast controls for the heat equation on thesupport (location) of the initial data. Let ω ( Ω be a bounded, open subset of R d ( d ≥ T > u ∈ L (Ω), and f ∈ L ((0 , T ) × ω ). Let A be a Lipschitz, symmetric, uniformly elliptic, matrix-valued function defined in Ω. Consider the unique solution u ∈ L ((0 , T ); H (Ω)) ∩ C ([0 , T ]; L (Ω))of the system(1.1) ∂ t u − div (cid:0) A ( x ) ∇ u (cid:1) = f ω in (0 , T ) × Ω ,u = 0 on (0 , T ) × ∂ Ω ,u ( t = 0 , · ) = u in Ω . Here and in what follows, D denotes the characteristic function of a set D of R d . It is well-knownfrom the work of Gilles Lebeau and Luc Robbiano [8], via spectral inequalities and the work ofAndrei Fursikov and Yu Imanuvilov [5], via Carleman’s estimates that one can act on ω using f to bring u from the initial state u (arbitrary) at time 0 to the final state 0 at time T (arbitrarilypositive).For D ⊂ Ω, set(1.2) c ( T, ω, D ) = sup k u k L =1supp u ⊂ D inf f ∈ L ((0 ,T ) × ω ) u ( T, · )=0 where u satisfies (1.1) k f k L ((0 ,T ) × ω ) . For T ∈ (0 , c e c /T ≤ c ( T, ω, Ω) ≤ C e C /T , for some positive constants c , c , C , and C independent of T . The second inequality followsfrom the observability inequality [5, 8], and the first inequality was obtained by Luc Miller [13] and others [3, 22]. There is significant literature covering other aspects of the cost of the controlfor heat equations [2, 4], the transport equation with small viscosities [1, 6, 10, 11], and the waveequation [7, 19, 20]. The cost of fast controls were also considered for linear thermoelasticity [9],Schr¨odinger equations [4, 11, 12, 18], and plate vibrations [12]. Similar questions were previouslyaddressed in finite dimensions by Thomas Seidman [21].The goal of this paper is to show a dependence of c ( T, ω, D ) on D . More precisely, we prove Theorem 1.1.
Let T ∈ (0 , and ε > . Assume that ω ⋐ Ω is of class C , and set, for r > , (1.4) ω r = n x ∈ R d ; dist( x, ω ) < r o . There exist two constants δ ∈ (0 , and C ε > , depending only on ε , ω , Ω , and the elliptic andLipschitz constants of A , such that (1.5) c ( T, ω, ω δ ) ≤ C ε e ε/T . Remark 1.1.
The constants δ and C ε in Theorem 1.1 are independent of T .When ω = Ω, the dependence of c and C on Ω has been studied extensively, see e.g. [3,13] andthe references therein. Nevertheless, to our knowledge, the dependence of the cost on the supportof initial datum for the heat equation has not been considered in the literature. Theorem 1.1 isnew even in one dimensional case.Theorem 1.1 is expected in the sense that if the support of the initial data is not too far from thecontrolled region, then it is easier to control. Even in this regard, this intuition is not completelytransparent since the propagation speed is infinite and hence the support of the solution at anypositive time is generally the whole domain Ω. Known examples used in the moment method forthe heat equations (mainly for one dimensional space) and other equations give the same size ofthe control cost for initial datum formed by eigenfunctions of the corresponding operator. Fromthis aspect, Theorem 1.1 is thus unexpected.The proof of Theorem 1.1 is in the spirit of Gilles Lebeau and Luc Robbiano’s approach [8]in which we establish a new spectral inequality. Let 0 < λ ≤ λ ≤ . . . be the sequence ofthe eigenvalues of the operator − div( A ∇· ) with the zero Dirichlet boundary condition, and let e , e , . . . be the corresponding eigenfunctions, i.e.,(1.6) ( − div( A ∇ e i ) = λ i e i in Ω ,e i = 0 on ∂ Ω . Assume that { e i , i ≥ } forms an orthogonal basis in L (Ω). Set, for λ > E ≤ λ = X λ i ≤ λ a i e i ( x ); a i ∈ R . One of the key elements of Gilles Lebeau and Luc Robbiano’s approach is the following spectralinequality(1.8) k v k H (Ω) ≤ Ce C √ λ k v k L ( ω ) ∀ v ∈ E λ , where C is a positive constant independent of λ .In this paper, we also follow this approach. Nevertheless, to capture the dependence on thesupport of the initial datum, we use and establish the following new spectral inequality (comparewith (1.8)). OST OF FAST CONTROLS FOR THE HEAT EQUATION 3
Proposition 1.1.
Let ε ∈ (0 , . There exist two constants δ ∈ (0 , and C ε > , dependingonly on ε , ω , Ω , and the elliptic and Lipschitz constants of A , such that, for λ > , k v k L ( ω δ ) ≤ C ε e ε √ λ k v k L ( ω ) ∀ v ∈ E λ . Remark 1.2.
It is important to emphasize here that the constants δ and C ε in Proposition 1.1are independent of λ .The proof of Proposition 1.1 is in the spirit of [8]. Nevertheless, we use three-sphere inequalitieswith partial data, which was recently established by the author, to quantitatively capture thedependence of the support. These inequalities have been derived and applied to the study ofcloaking using negative-index materials [15,16]. A typical example of these inequalities is, see [16,Theorem 2.1], Theorem 1.2.
Let d ≥ , Λ ≥ , < R < R , and let Γ = n x = ( x ′ , x d ) ∈ ∂B R ; x d = 0 o .Denote O r = n x ∈ R d ; dist( x, Γ) < r o , D r = B R \ ( B R ∪ O r ) , and Σ r = ∂B R \ ¯ O r for r > .For every α ∈ (0 , , there exists r ∈ (0 , R − R ) , depending only on α , Λ , Γ , R , and R , suchthat for every r ∈ (0 , r ) , there exists r ∈ (0 , r ) , depending only on r , α , Λ , R , and R , suchthat for ( d × d ) Lipschitz, uniformly elliptic, symmetric, matrix-valued function M defined in D r verifying, in D r , (1.9) Λ − | ξ | ≤ hM ( x ) ξ, ξ i ≤ Λ | ξ | ∀ ξ ∈ R d and |∇M ( x ) | ≤ Λ , and for V ∈ [ H ( D r )] m satisfying (1.10) | div( M∇ V ) | ≤ Λ (cid:0) |∇ V | + | V | (cid:1) in D r for some Λ ≥ , we have (1.11) k V k H ( B R r \ B R r ) ≤ C (cid:16) k V k H / (Σ r ) + kM∇ V · x/ | x |k H − / (Σ r ) (cid:17) α k V k − αH ( D r ) , for some positive constant C , depending only on α , Λ , Λ , R , R , and d . The geometry of Theorem 1.2 is given in Figure 1.We will use a variant of Theorem 1.2, given in Proposition 1.2, to derive Theorem 1.1. Neverthe-less, we present Theorem 1.2 here to highlight the difference between the three-sphere inequalitiesused in this paper and the standard three-sphere ones. In (1.11), one only uses the information ofΣ r (a portion of ∂B R , see Figure 1) in the first interpolation term. The terminology partial data comes from this. The constants r , r , and r are independent of Λ , but the constant C doesdepend on Λ . If instead of Σ r , one uses ∂B R , inequality (1.11) is then known. Using knownthree-sphere inequalities and the arguments of the propagation of smallness, one can prove (1.2)for some α ∈ (0 , α ∈ (0 , r , r , r . Even if v is a solutionof the Laplace equation in two dimensions, using Hadamard three-sphere (circles) inequalitiesand the arguments of propagation of smallness, as far as we know, one can only obtain (1.11) forsome small α , even though one replaces Σ r by ∂B R \ { x } for some x ∈ ∂B R . The possibilityto take α close to 1 is crucial for the proof of Theorem 1.1 where ε can be arbitrarily small.This point is also crucial for the cloaking applications considered in [15, 16]. Several applicationsof Theorem 1.2 concerning variants of Hadamard’s three-circle inequalities with partial data aregiven in [16]. HOAI-MINH NGUYEN Σ r O r Γ R r r R D r B R + r \ B R + r Figure 1.
Geometry of Theorem 1.2 in two dimensions
We now introduce some notations to state the local version of Theorem 1.2, which is used inthe proof of Proposition 1.1. For d ≥ x = ( x , x , e x ) ∈ R × R × R d − , we use the polarcoordinate (ˆ r, θ ) with θ ∈ ( − π, π ] for the pair ( x , x ); the variable e x is irrelevant for d = 2. For0 < γ < γ < R >
0, we denote(1.12) Y γ ,γ ,R = n x ∈ R d ; θ ∈ ( − π/ , π/ , γ R < ˆ r < γ R, and | e x | < R o , (see also Figure 2). The following variant of Theorem 1.2 in a half plane, see [16, Theorem 3.1],is the key ingredient of the proof of Proposition 1.1. Proposition 1.2.
Let d ≥ , Λ ≥ , and R ∗ < R < R ∗ . Then, for any α ∈ (0 , , there exists aconstant ˆ γ ∈ (0 , , depending only on α , Λ , R ∗ , R ∗ , and d such that for every ˆ γ ∈ (0 , ˆ γ ) , thereexists ˆ γ ∈ (0 , ˆ γ ) , depending only on α , ˆ γ , Λ , R ∗ , R ∗ , and d such that, for real, symmetric,uniformly elliptic, Lipschitz matrix-valued functions M defined in D ˆ γ := Y ˆ γ , ,R verifying, in D ˆ γ , (1.13) Λ − | ξ | ≤ hM ( x ) ξ, ξ i ≤ Λ | ξ | ∀ ξ ∈ R d and |∇M ( x ) | ≤ Λ , and for V ∈ [ H ( D ˆ γ )] m satisfying (1.14) | div( M∇ V ) | ≤ Λ (cid:0) |∇ V | + | V | (cid:1) in D ˆ γ for some Λ ≥ , we have, with Σ ˆ γ = ∂D ˆ γ ∩ (cid:8) x = 0 (cid:9) , (1.15) k V k H ( Y ˆ γ , ˆ γ , R ) ≤ C (cid:16) k V k H / (Σ ˆ γ ) + kM∇ V · η k H − / (Σ ˆ γ ) (cid:17) α k V k − αH ( D ˆ γ ) , for some positive constant C , depending only on α, ˆ γ , Λ , Λ , R ∗ , R ∗ , and d . Here and in what follows, η , · · · , η d denotes the standard basis of R d , i.e., η = (1 , , . . . , η d = (0 , . . . , , OST OF FAST CONTROLS FOR THE HEAT EQUATION 5ˆ γ R γ R Σ γ ˆ γ R Y ˆ γ , ˆ γ , R D γ Figure 2.
Geometry of Y γ ,γ , R , Σ γ , and D γ in two dimensions. proof. The proof is much simpler for the case A = I and d = 2, but already contains several keyideas [15].The paper is organized as follows. Section 2 is devoted to the proof of Proposition 1.1. Theproof of Theorem 1.1 is given in Section 3.2. Spectral inequality
This section is devoted to the proof of Proposition 1.1. The key ingredient of the proof is:
Lemma 2.1.
Let M be a Lipschitz, symmetric, uniformly elliptic, matrix-valued defined in Ω × ( − , and let ϕ ∈ H (Ω × ( − , be such that | div( M ∇ ϕ ) | ≤ Λ( |∇ ϕ | + | ϕ | ) in Ω × ( − , . Set D r = n X = ( x, x d +1 ) ∈ R d +1 ; dist( X, ω × { } ) < r o for r > .Given α ∈ (0 , , there exist two constants δ ∈ (0 , and C α > , depending only on α , ω , Ω , andthe Lipschitz and elliptic constants of M , such that (2.1) k ϕ k H ( D δ ) ≤ C α (cid:16) k ϕ k H / ( ω ×{ } ) + k M ∇ ϕ · η d +1 k H − / ( ω ×{ } ) (cid:17) α k ϕ k − αH (Ω × ( − , . Remark 2.1.
The constant δ and C α are independent of ϕ . Proof of Lemma 2.1.
Since ω is of class C , by using local charts and a change of variables,it suffices to prove the following result: Let ˆ M be a Lipschitz, symmetric, uniformly elliptic,matrix-valued function defined in Q := ( − , d +1 , and let ˆ ϕ ∈ H ( Q ) be such that(2.2) | div( ˆ M ∇ ˆ ϕ ) | ≤ ˆΛ( |∇ ˆ ϕ | + | ˆ ϕ | ) in Q. HOAI-MINH NGUYEN
Q x x d +1 δ Σ a ) Q x x d +1 b ) Figure 3. a ): Geometry of inequality 2.3 in two dimensions with Σ := Q ∩ { x d +1 =0; x < } . b ) The way to obtain (2.1) from (2.3) for d = 1; ω is the orange interval, Ω isthe blue interval, D δ is the region whose boundary is violet. Given α ∈ (0 , δ >
0, depending only on α , and the Lipschitz and elliptic constantsof ˆ M , such that(2.3) k ˆ ϕ k H ( B δ ) ≤ C (cid:16) k ˆ ϕ k H / (Σ) + k ˆ M ∇ ˆ ϕ · η d +1 k H − / (Σ) (cid:17) α k ˆ ϕ k − αH ( Q ) , where Σ := Q ∩ { x d +1 = 0; x < } for some positive constant C depending only on α , theLipschitz and elliptic constants of ˆ M , and ˆΛ.Here and in what follows in this proof, B r denotes the open ball centered at 0 and of radius r > R d +1 .It is important to note that in (2.3), the norms in the RHS are considered in the set Σ which isdefined by Q ∩ { x d +1 = 0; x < } and is not given by the set Q ∩ { x d +1 = 0 } . See a) of Figure 3for the geometry of (2.3) and b) of Figure 3 for the ideas behind using local charts and coveringarguments to obtain (2.1) from (2.3).We will make a change of variables in order to apply Proposition 1.2. To this end, for X =( x , · · · , x d +1 ) ∈ Q \ (cid:8) x d +1 = 0; x ≤ (cid:9) , define R ( X ) = ( y , x , · · · , x d , y d +1 ) , with ( y , y d +1 ) = ˆ re iθ/ if ( x , x d +1 ) = ˆ re iθ for ˆ r >
0, and θ ∈ ( − π, π ).Set ˆ ϕ = ˆ ϕ ◦ R − in ˆ Q := R (cid:16) Q \ (cid:8) x d +1 = 0; x ≤ (cid:9)(cid:17) . Set f ( x ) = div( ˆ M ∇ ˆ ϕ )( x ) in Q, f ( x ) = f | det( ∇R ) | ◦ R − ( x ) in Q , and(2.4) ˆ M = ∇R ˆ M ∇R T | det( ∇R ) | ◦ R − in ˆ Q . OST OF FAST CONTROLS FOR THE HEAT EQUATION 7
It is clear from (2.4) that the elliptic and Lipschitz constants of ˆ M are bounded by the ellipticand Lipschitz constants of ˆ M , up to a constant C , depending only on d .Since div( ˆ M ∇ ˆ ϕ ) = f in Q , it follows from a change of variables that(2.5) div( ˆ M ∇ ˆ ϕ ) = f in Q . We have det( ∇R )( x ) = 1 / x ∈ Q, |∇ ˆ ϕ ( x ) | ≤ |∇R ( x ) ||∇ ˆ ϕ ◦ R ( x ) | ≤ C |∇ ˆ ϕ ◦ R ( x ) | for x ∈ Q. Since | f | ≤ C (cid:0) |∇ ϕ | + | ϕ | (cid:1) in Q by (2.2), we derive from (2.5) that | div( ˆ M ∇ ˆ ϕ ) | ≤ ˆΛ (cid:0) |∇ ϕ | + | ϕ | (cid:1) in Q , for some ˆΛ >
0, depending only on Λ and d .Set(2.6) Γ , + = n ( x , . . . , x d +1 ); x = 0 , x d +1 > , x j ∈ ( − ,
1) for 2 ≤ j ≤ d o and(2.7) Γ , − = n ( x , . . . , x d +1 ); x = 0 , x d +1 < , x j ∈ ( − ,
1) for 2 ≤ j ≤ d o . Apply Proposition 1.2 to ˆ ϕ with R = 1, ˆ γ = ˆ γ /
2, and ˆ γ = 0, and in R d +1 with ( x , x , e x )being replaced by (cid:0) x , x d +1 , ( x , · · · , x d ) (cid:1) . There exists ˆ γ > k ˆ ϕ k H (cid:0) ( B ˆ γ \ B ˆ γ / ) ∩{ x > } (cid:1) ≤ C k ˆ ϕ k − αH ( ˆ Q ) × (cid:16) k ˆ ϕ k H / (cid:0) Γ , + (cid:1) + k ˆ M ∇ ˆ ϕ · η k H / (cid:0) Γ , + (cid:1) + k ˆ ϕ k H / (cid:0) Γ , − (cid:1) + k ˆ M ∇ ˆ ϕ · η k H / (cid:0) Γ , − (cid:1)(cid:17) α . Since ˆ ϕ = ˆ ϕ ◦ R in Q , it follows from a change of variables, see e.g. [14, Lemma 2], that k ˆ ϕ k H (cid:0) ( B ˆ γ \ B ˆ γ / ) \{ x d +1 =0; x < } (cid:1) ≤ C k ˆ ϕ k − αH ( Q ) (cid:16) k ˆ ϕ k H / (Σ) + k ˆ M ∇ ˆ ϕ · η d +1 k H − / (Σ) (cid:17) α . Since ˆ ϕ ∈ H ( Q ), and hence in particular ˆ ϕ ∈ H ( B ˆ γ ), we obtain(2.8) k ˆ ϕ k H ( B ˆ γ \ B ˆ γ / ) ≤ C k ˆ ϕ k − αH ( Q ) (cid:16) k ˆ ϕ k H / (Σ) + k ˆ M ∇ ˆ ϕ · η d +1 k H − / (Σ) (cid:17) α . Using the fact | div( ˆ M ∇ ˆ ϕ ) | ≤ ˆΛ( |∇ ϕ | + | ϕ | ) in Q, and ˆ M is symmetric, uniformly elliptic and Lipschitz, one has (2.9) k ˆ ϕ k H ( B γ ) ≤ C k ˆ ϕ k H ( B γ \ B γ / ) . Assertion (2.3) now follows from (2.8) and (2.9) with δ = γ . The proof is complete. (cid:3) Remark 2.2.
One of the key points of the proof is the assertion (2.3). This assertion is known ifone replaces the set Q ∩ { x d +1 = 0; x < } by Q ∩ { x d +1 = 0 } and the proof in this case can bedone as in [8]. However, this does not imply (2.1). The proof of (2.3) follows from Proposition 1.2,which is non-trivial.We are ready to give One can prove (2.9) using a contradiction argument and the unique continuation principle.
HOAI-MINH NGUYEN
Proof of Proposition 1.1.
Since v ∈ E ≤ λ , there exists a i ∈ R with λ i ≤ λ , such that v ( x ) = X λ i ≤ λ a i e i ( x ) in Ω . As in the spirit of [8], set, with X = ( x, x d +1 ) ∈ Ω × R , V ( X ) = X λ i ≤ λ λ − / i a i sinh( λ / i x d +1 ) e i ( x ) , where sinh t = ( e t − e − t ) for t ∈ R . Since − div x ( A ( x ) ∇ e i ( x )) = λ i e i ( x ) in Ω, it follows that(2.10) ∂ x d +1 V + div x (cid:0) A ( x ) ∇ x V (cid:1) = 0 in Ω × R ,V ( X ) = 0 for X ∈ Ω × { } ,∂ x d +1 V ( X ) = v ( x ) for X ∈ Ω × { } . Given α ∈ (0 , V , there exist two constants δ = δ ( α ) ∈ (0 ,
1) and C α >
0, depending only on α , ω , Ω, and the elliptic and Lipschitz constants of A , such that(2.11) k V k H ( D δ ) ≤ C α (cid:16) k V k H / ( ω ×{ } ) + k ∂ x d +1 V k H − / ( ω ×{ } ) (cid:17) α k V k − αH (Ω × ( − , . Using (2.10), we derive from (2.11) that(2.12) k V k H ( D δ ) ≤ C α k v k αL ( ω ) k V k − αH (Ω × ( − , . Since A is Lipschitz, by the regularity theory of elliptic equations , one has k ∂ x d +1 V k L ( ω δ ) ≤ C α k V k H ( D δ ) , which yields(2.13) k v k L ( ω δ ) ≤ C α k V k H ( D δ ) . On the other hand, by the standard spectral inequality (1.8), one gets(2.14) k V k H (Ω × ( − , ≤ Ce C √ λ k v k L ( ω ) , for some positive constant C , depending only on ω , Ω, and the elliptic and Lipschitz constants of A .Combining (2.12), (2.13) and (2.14) yields(2.15) k v k L ( ω δ ) (2.13) ≤ C α k V k H ( D δ ) (2.12) ≤ C α k v k αL ( ω ) k V k − αH (Ω × ( − , ≤ C α e C (1 − α ) √ λ k v k L ( ω ) . By choosing α such that C (1 − α ) = ε , we derive from (2.15) that(2.16) k v k L ( ω δ ) ≤ C ε e ε √ λ k v k L ( ω δ ) . The proof is complete. (cid:3) One can directly apply the quotient method due to Louis Nirenberg [17].
OST OF FAST CONTROLS FOR THE HEAT EQUATION 9 Proof of Theorem 1.1
The proof of Theorem 1.1 is based on the following lemma, which will be derived from thespectral inequality stated in Proposition 1.1.
Lemma 3.1.
Let < T < , λ > , and let v ∈ E λ . Let v ∈ L ((0 , T ) , H (Ω)) ∩ C ([0 , T ] , L (Ω)) be the unique solution of the system (3.1) ∂ t v − div( A ∇ v ) = 0 in (0 , T ) × Ω ,v = 0 on (0 , T ) × ∂ Ω ,v ( t = 0 , · ) = v in Ω . For ε > , there exist two constants δ ∈ (0 , and C ε > , depending only on ε , ω , and Ω , andthe elliptic and Lipschitz constants of A , such that k v ( T, · ) k L ( ω δ ) ≤ C ε δ − T − / e ε √ λ k v k L ((0 ,T ) × ω ) . Recall that ω r is defined in (1.4). Remark 3.1.
The constants δ and C ε in Lemma 3.1 are independent of λ and T . Proof.
By Proposition 1.1, there exist δ ∈ (0 ,
1) and C ε >
0, such that(3.2) k ξ k L ( ω δ ) ≤ C ε e ε √ λ k ξ k L ( ω ) for ξ ∈ E ≤ λ . Since v ∈ E ≤ λ , it follows that v ( t, · ) ∈ E ≤ λ for t ∈ (0 , T ). We derive from (3.2) that(3.3) k v ( t, · ) k L ( ω δ ) ≤ C ε e ε √ λ k v ( t, · ) k L ( ω ) for t ∈ (0 , T ) . Fix ϕ ∈ C ∞ c ( R d ) such that 0 ≤ ϕ ≤ ϕ = 1 in ω δ , supp ϕ ⊂ ω δ , and |∇ αx ϕ | ≤ C/δ | α | forall multi-indices α with | α | ≤
2. Here and in what follows in this proof, C denotes a positiveconstant, depending only on ω , Ω, and the elliptic and Lipschitz constants of A .Set(3.4) u ( t, x ) = ϕ ( x ) v ( t, x ) in (0 , T ) × Ω , and denote(3.5) g ( t, x ) = − (cid:16) h A ( x ) ∇ v ( t, x ) , ∇ ϕ ( x ) i + v ( t, x ) div( A ( x ) ∇ ϕ ( x )) (cid:17) in (0 , T ) × Ω , where h· , ·i denotes the standard scalar product in R d .We derive from (3.1) and the symmetry of A that ( ∂ t u − div( A ∇ u ) = g in (0 , T ) × Ω ,u = 0 on (0 , T ) × ∂ Ω . Multiplying the equation of u by u and integrating by parts in ( t, T ) × Ω, we obtain, for 0 ≤ t ≤ T ,(3.6) 12 ˆ Ω | u ( T, x ) | dx + ˆ Tt ˆ Ω h A ( x ) ∇ u ( s, x ) , ∇ u ( s, x ) i dx ds = 12 ˆ Ω | u ( t, x ) | dx + ˆ Tt ˆ Ω g ( s, x ) u ( s, x ) dx ds. We next estimate the last term of (3.6). Since, for x ∈ Ω and s ∈ (0 , T ), h A ( x ) ∇ v ( s, x ) , ∇ ϕ ( x ) i u ( s, x ) (3.4) = h A ( x ) ϕ ( x ) ∇ v ( s, x ) , ∇ ϕ ( x ) i v ( s, x ) (3.4) = h A ( x )( ∇ u ( s, x ) − v ( s, x ) ∇ ϕ ( x )) , ∇ ϕ ( x ) i v ( s, x ) , it follows that, for s ∈ (0 , T ),(3.7) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω h A ( x ) ∇ v ( s, x ) , ∇ ϕ ( x ) i u ( s, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ Ω δ − |∇ u ( s, x ) || v ( s, x ) | dx + ˆ Ω δ − | v ( s, x ) | dx. We also have, for x ∈ Ω and s ∈ (0 , T ),(3.8) | v ( s, x ) div( A ( x ) ∇ ϕ ( x )) || u ( s, x ) | ≤ Cδ − | v ( s, x ) | , since A is Lipschitz and |∇ αx ϕ | ≤ C/δ | α | for all multi-indices α with | α | ≤
2. Combining (3.7)and (3.8) yields(3.9) (cid:12)(cid:12)(cid:12)(cid:12) ˆ Tt ˆ Ω g ( s, x ) u ( s, x ) dx ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ˆ Tt ˆ Ω δ − |∇ u ( s, x ) || v ( s, x ) | dx ds + ˆ Tt ˆ Ω δ − | v ( s, x ) | dx ds. Using (3.9) and the ellipticity of A , and applying Young’s inequality, we derive from (3.6) that,for t ∈ (0 , T ), ˆ Ω | u ( T, x ) | dx ≤ ˆ Ω | u ( t, x ) | dx + Cδ − ˆ Tt ˆ Ω | v ( s, x ) | dx ds. Integrating the above inequality with respect to t from 0 to T , we derive that ˆ Ω | u ( T, x ) | dx ≤ Cδ − T − ˆ T ˆ Ω | v ( s, x ) | dx ds. Since v = ϕu , 0 ≤ ϕ ≤ ϕ = 1 in ω δ , and supp ϕ ⊂ ω δ , it follows that ˆ ω δ | v ( T, x ) | dx ≤ Cδ − T − ˆ T ˆ ω δ | v ( t, x ) | dx dt. We derive from (3.3) that ˆ ω δ | v ( T, x ) | dx ≤ C ε δ − T − e ε √ λ ˆ T ˆ ω | v ( t, x ) | dx dt, which is the conclusion. The proof is complete. (cid:3) We are ready to give
Proof of Theorem 1.1.
Fix λ = c /T where c is a large positive constant determined later. Set(3.10) H := X λ i ≤ λ a i e − λ i ( T/ − t ) e i ( x ); a i ∈ R , x ∈ Ω , t ∈ (0 , T / ⊂ L (cid:0) (0 , T / × Ω (cid:1) . Equip H with the standard scalar product in L (cid:0) (0 , T / × Ω (cid:1) . Then, H is a Hilbert space (offinite dimensions). OST OF FAST CONTROLS FOR THE HEAT EQUATION 11
Let ϕ ∈ H , and set(3.11) v ( t, x ) = ϕ ( T / − t, x ) for ( t, x ) ∈ (0 , T / × Ω . It follows from the definition of H in (3.10) that ( ∂ t v − div( A ∇ v ) = 0 in (0 , T / × Ω ,v = 0 on (0 , T / × ∂ Ω , and moreover, v ( t = 0 , · ) ∈ E ≤ λ .By Lemma 3.1, there exist two constants δ ∈ (0 ,
1) and C ε >
0, depending only on ε , ω , Ω, c ,and the Lipschitz and elliptic constants of A , such that k v ( T / , · ) k L ( ω δ ) ≤ C ε δ − T − / k v k L (cid:0) (0 ,T/ × Ω (cid:1) . This implies, by (3.11),(3.12) k ϕ (0 , · ) k L ( ω δ ) ≤ C ε δ − T − / e ε/T k ϕ k L (cid:0) (0 ,T/ × ω (cid:1) . Fix such constants δ and C ε .Fix u ∈ L (Ω) with supp u ⊂ ω δ . We will construct a control with support in (0 , T ) × ω ,which steers u from time 0 to 0 at time T for which the cost is bounded by C ε e ε/T k u k L (Ω) .Since u ∈ L (Ω) with supp u ⊂ ω δ , using the Riesz representation theorem, we derive from(3.12) that there exists f ∈ H , such that(3.13) ˆ Ω u ( x ) ϕ (0 , x ) dx = ˆ T/ ˆ ω f ( s, x ) ϕ ( s, x ) dx ds for ϕ ∈ H, and(3.14) k f k L (cid:0) (0 ,T/ × ω (cid:1) ≤ C ε δ − T − / e ε/T k u k L (Ω) . Let u ∈ L ((0 , T / H (Ω)) ∩ C ([0 , T / L (Ω)) be the unique solution of the system(3.15) ∂ t u − div (cid:0) A ( x ) ∇ u (cid:1) = f ω in (0 , T / × Ω ,u = 0 on (0 , T / × ∂ Ω ,u ( t = 0 , · ) = u in Ω . Since ( ∂ t ϕ + div( A ∇ ϕ ) = 0 in (0 , T / × Ω ,ϕ = 0 on (0 , T / × ∂ Ω , for ϕ ∈ H, multiplying the equation of u by ϕ ( ∈ H ) and integrating by parts in (0 , T / × Ω, we obtain ˆ Ω u ( T / , x ) ϕ ( T / , x ) dx − ˆ Ω u (0 , x ) ϕ (0 , x ) dx = ˆ T/ ˆ ω f ( s, x ) ϕ ( s, x ) dx ds for ϕ ∈ H. Using (3.13), we derive that(3.16) ˆ Ω u ( T / , x ) ϕ ( T / , x ) dx = 0 for ϕ ∈ H. In other words, the projection of u ( T / , · ) into E ≤ λ is 0. Thus,(3.17) u ( T / , x ) = X λ i >λ h u ( T / , · ) , e i i L (Ω) e i ( x ) in Ω , where h· , ·i L (Ω) denotes the standard scalar product in L (Ω).On the other hand, by the standard energy estimate, we have ˆ Ω | u ( T / , x ) | dx ≤ ˆ Ω | u (0 , x ) | dx + C ˆ T/ ˆ ω | f ( s, x ) | ds dx. We derive from (3.14) that(3.18) k u ( T / , · ) k L (Ω) ≤ C ε δ − T − / e ε/T k u k L (Ω) . Let u ∈ L (( T / , T / H (Ω)) ∩ C ([ T / , T / L (Ω)) be the unique solution of the system(3.19) ∂ t u − div (cid:0) A ( x ) ∇ u (cid:1) = 0 in (0 , T / × Ω ,u = 0 on (0 , T / × ∂ Ω ,u ( t = T / , · ) = u ( T / , · ) in Ω . It follows from (3.17) and (3.18) that k u (2 T / , · ) k L (Ω) ≤ e − λT/ k u ( T / , · ) k L (Ω) ≤ C ε δ − T − / e ε/T − λT/ k u k L (Ω) , which yields, since λ = c /T ,(3.20) k u (2 T / , · ) k L (Ω) ≤ C ε δ − T − / e ε/T − c / (3 T ) k u k L (Ω) . On the other hand, there exists f ∈ L ((2 T / , T ) × Ω) with support in [2
T / , T ] × ω , suchthat(3.21) k f k L (cid:0) (2 T/ ,T ) × Ω (cid:1) ≤ Ce C/T k u (2 T / , · ) k L (Ω) , and(3.22) u ( T, · ) = 0 in Ω , where u ∈ L (cid:0) (2 T / , T ); H (Ω) (cid:1) ∩ C (cid:0) [2 T / , T ]; L (Ω) (cid:1) is the unique solution of the system(3.23) ∂ t u − div (cid:0) A ( x ) ∇ u (cid:1) = f in (2 T / , T ) × Ω ,u = 0 on ( T / , T ) × ∂ Ω ,u ( t = 2 T / , · ) = u (2 T / , · ) in Ω . Define f ∈ L (cid:0) (0 , T ) × Ω (cid:1) as follows(3.24) f ( t, x ) = f ω in (0 , T / × Ω , T / , T / × Ω ,f in (2 T / , T ) × Ω . Since supp f ⊂ [2 T / , T ] × ω , it follows thatsupp f ⊂ [0 , T ] × ¯ ω. OST OF FAST CONTROLS FOR THE HEAT EQUATION 13
Let u ∈ L (cid:0) (0 , T ); H (Ω) (cid:1) ∩ C (cid:0) [0 , T ]; L (Ω) (cid:1) be the unique solution of the system(3.25) ∂ t u − div (cid:0) A ( x ) ∇ u (cid:1) = f in (0 , T ) × Ω ,u = 0 on ( T / , T ) × ∂ Ω ,u ( t = 0 , · ) = u in Ω . It follows from (3.15), (3.19), (3.22), and (3.23) that u ( T, · ) = 0 in Ω . Combining (3.14) and (3.21), and using (3.20), we deduce that k f k L ((0 ,T ) × Ω) ≤ C ε δ − T − / e ε/T k u k L (Ω) (cid:16) e − c / (3 T )+ C/T (cid:17) . By fixing c such that c / ≥ C , we obtain k f k L ((0 ,T ) × Ω) ≤ C ε δ − T − / e ε/T k u k L (Ω) . The conclusion follows by replacing ε by ε/ T − / e ε/ (2 T ) ≤ C ε e ε/T ;this follows by considering the case T ≥ ε and the case 0 < T < ε . (cid:3) Remark 3.2.
The conclusion of Theorem 1.1 also holds if in the definition of c ( T, w, D ), oneadditionally requires that supp f ⋐ [0 , T ] × ω . The conclusion in this case follows by applying theestablished result for the set (cid:8) x ∈ ω ; dist( x, ∂ω ) ≥ γ (cid:9) for small γ after noting that the constant δ for such a set is independent of γ for small γ . Acknowledgement:
The author thanks Jean-Michel Coron for his interest in the problem andfor many interesting discussions. The author also thanks Kim Dang Phung for discussions onthe approach of Gilles Lebeau and Luc Robbiano. This work was completed during his visit toLaboratoire Jacques Louis Lions. The author thanks the laboratory for its hospitality.
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Ecole Polytechnique F´ed´erale de Lausanne, EPFL, CAMA, Station 8, CH-1015Lausanne, Switzerland.
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