J. Math. Anal. Appl. 295 (2004) 527–538 www.elsevier.com/locate/jmaa Note on the cost of the approximate controllability for the heat equation with potential K.-D. Phung Received 2 September 2003 Available online 28 May 2004 Submitted by R. Triggiani Abstract We prove the approximate controllability for the heat equation with potential with a cost of order ec/ε when the target is in H01 (Ω) with a precision in L2 (Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2 (Ω) of the heat equation with potential is established. 2004 Elsevier Inc. All rights reserved. 1. Introduction and main results Throughout this paper, Ω is a bounded domain in Rn , n 1, with a boundary ∂Ω of class C 2 , ω is a non-empty open subset of Ω and T > 0 is a real number. Further, we denote · ∞ the usual norm in L∞ (Ω × (0, T )) and we consider a = a(x, t) a function in L∞ (Ω × (0, T )). In this paper we study the following heat equation with a potential a in L∞ (Ω × (0, T )): ∂t u − ∆u + au = f · 1|ω in Ω × (0, T ), u = 0 on ∂Ω × (0, T ), (1.1) u(·, 0) = u0 in Ω, where f ∈ L2 (ω × (0, T )), u0 ∈ H01 (Ω) and 1|ω denotes the characteristic function of the set ω. It is well known from [6] or [2] that we can act through f ∈ L2 (ω × (0, T )) when u0 ∈ L2 (Ω) in order to get the null controllability result u(·, T ) = 0 and furthermore, the following estimate holds [3]: there exists a constant c0 > 0 such that E-mail address: kim-dang.phung@cmla.ens-cachan.fr. 0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.03.059 528 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 1 2/3 f L2 (ω×(0,T )) exp c0 1 + + T a∞ + a∞ u0 L2 (Ω) . T (1.2) (1.2) is an explicit estimate with respect to both quantities T > 0 and a∞ 0, of the control function f and may be viewed as a measure of the cost of the null controllability for the heat equation with potential. Here we ask whether the following steering property for the heat equation with potential holds when u0 = 0: there exist two constants D > 1 and c = c(T , a∞ ) > 1 depending on both quantities T > 0 and a∞ 0 such that for all ε > 0, for all ud ∈ H01 (Ω), there exists a suitable approximate control function f depending on ε such that Du 1 d H (Ω) 0 (1.3) f L2 (ω×(0,T )) c exp ud L2 (Ω) ε and u(·, T ) − ud 2 ε. L (Ω) (1.4) Our goal is to measure the cost of the approximate control function f and furthermore to give an explicit estimate with respect to ε, T and a∞ 0. This problem has received a particular attention from [3] where it is proved that the cost of the approximate controllability for the heat equation with potential is of order ec/ε if 2 ud ∈ H 2 (Ω) ∩ H01 (Ω), and of order ec/ε if ud ∈ H01 (Ω). But when a is a constant or more generally√of the form a(x, t) = a1 (x) + a2 (t), where a1 ∈ L∞ (Ω) and a2 ∈ L∞ (0, T ), the order ec/ ε if ud ∈ H 2 (Ω) ∩ H01 (Ω) is optimal [3, Theorem 6.2]. Let us make the first observation: by simple changes of variables, we are reduced to check that for all ε > 0, for all T > 0, for all wd ∈ H01 (Ω) there exists a function ϑ such that ϑL2 (ω×(0,T )) ceD/ε wd L2 (Ω) , and such that the following steering property holds: w(·, 0) − wd 2 εwd H 1 (Ω) , L (Ω) 0 where w is the solution of the following backward heat equation with potential: ∂t w + ∆w − aw = ϑ · 1|ω in Ω × (0, T ), w = 0 on ∂Ω × (0, T ), w(·, T ) = 0 in Ω. (1.5) (1.6) (1.7) Now let us consider ϕ the solution of the heat equation without control when the initial data ϕ0 ∈ L2 (Ω): ∂t ϕ − ∆ϕ + aϕ = 0 in Ω × (0, T ), (1.8) ϕ = 0 on ∂Ω × (0, T ), ϕ(·, 0) = ϕ0 in Ω. Then by classical integrations by parts, we get K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 529 T ϑ(x, t)ϕ(x, t) dx dt = − 0 ω w(x, 0)ϕ(x, 0) dx Ω (−w(·, 0) + wd )ϕ0 dx − = Ω wd ϕ0 dx. (1.9) Ω Also suppose that the solution w of (1.7) exists with (1.6) and (1.5) then for all ε > 0, for all wd ∈ H01 (Ω), we have using Cauchy–Schwarz inequality wd ϕ0 dx εwd H 1 (Ω) ϕ0 L2 (Ω) + ceD/ε wd L2 (Ω) ϕL2 (ω×(0,T )) . (1.10) 0 Ω Consequently, we obtain choosing wd = (−∆)−1 ϕ0 that ϕ0 H −1 (Ω) εϕ0 L2 (Ω) + ceD/ε ϕL2 (ω×(0,T )) , ∀ε > 0, (1.11) if ϕ0 = 0, (1.12) or equivalently ϕ0 L2 (Ω) ϕ0 L2 (Ω) c exp D ϕL2 (ω×(0,T )) ϕ0 H −1 (Ω) or equivalently ϕ0 H −1 (Ω) ln 2 + D 1 ϕ0 L2 (Ω) c ϕL2 (ω×(0,T )) ϕ0 L2 (Ω) , (1.13) where the values of the constants c = c(T , a∞ ) > 1 and D > 1 may changed from line (1.11) to line (1.13) but not their dependence. (1.13) is a quantitative estimate for unique continuation for initial data in L2 (Ω) of the heat equation with potential from an interior observation. This kind of logarithmic estimate already appears in the context of the cost of approximate control and stabilization for hyperbolic equation [5,7]. Here we will follow the strategy in [7] (see also [4]) to prove that (1.13) implies an approximate result with an estimate (1.3) of the cost function. The first main result of this paper is as follows. Theorem 1. There exist two constants c1 , c2 > 1 such that for all ε > 0, for all T > 0, for all a ∈ L∞ (Ω × (0, T )), for all ud ∈ H01 (Ω), there exists a control function fε ∈ L2 (ω × (0, T )) such that D∇ud L2 (Ω) fε L2 (ω×(0,T )) E exp ud L2 (Ω) , ε u(·, T ) − ud 2 ε, (1.14) L (Ω) where u ∈ C([0, T ]; L2 (Ω)) is the unique solution of the heat equation with potential and control function ∂t u − ∆u + au = fε · 1|ω in Ω × (0, T ), u = 0 on ∂Ω × (0, T ), (1.15) u(·, 0) = 0 in Ω, 530 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 and E > 1, D > 1 are given by 2 D = D(T , a∞ ) = c1 T ec1 T a∞ + T1 , 2 2/3 E = E(T , a∞ ) = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )). (1.16) Of course we will also need to prove the estimate (1.13). Our second main result is as follows. Theorem 2. There exist two constants c1 , c2 > 1 such that for all T > 0, for all a ∈ L∞ (Ω × (0, T )), for all initial data u0 ∈ L2 (Ω) such that u0 = 0, the solution u of the homogeneous heat equation with potential ∂t u − ∆u + au = 0 in Ω × (0, T ), u = 0 on ∂Ω × (0, T ), (1.17) u(·, 0) = u0 in Ω, satisfies u0 H −1 (Ω) ln 2 + D 1 u0 L2 (Ω) E uL2 (ω×(0,T )) u0 L2 (Ω) , (1.18) where E > 1, D > 1 are given by 2 D = c1 T ec1 T a∞ + T1 , 2 2/3 E = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )). Remarks. (1) Note that E(T , 0) is a constant not depending on T > 0. (2) An application of Theorem 1 to get a space of exact controllable target data may be possible following [7] based on properties of Riesz basis (see [8]). The plan of the paper is as follows. In Section 2 we prove Theorem 1 as an application of Theorem 2. Section 3 contains the proof of Theorem 2. Finally in the last section some comments are added. 2. Proof of Theorem 1 Theorem 1 is easily deduced from the following result. Theorem 3. There exist two constants c1 , c2 > 1 such that for all ε > 0, for all T > 0, for all a ∈ L∞ (Ω × (0, T )), for all wd ∈ H01 (Ω), there exists a control function ϑε ∈ L2 (ω × (0, T )) such that D ϑε L2 (ω×(0,T )) exp wd L2 (Ω) , ε w(·, 0) − wd 2 εwd H 1 (Ω) , (2.1) L (Ω) 0 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 531 where w ∈ C([0, T ]; L2 (Ω)) is the unique solution of the backward heat equation with potential and control function −∂t w − ∆w + aw = Eϑε · 1|ω in Ω × (0, T ), w = 0 on ∂Ω × (0, T ), (2.2) w(·, T ) = 0 in Ω, and E > 1, D > 1 are given by 2 D = c1 T ec1 T a∞ + T1 , 2 2/3 E = exp(c2 (1 + T a∞ (1 + ec2 T a∞ ) + a∞ )). Proof. We will use the logarithmic estimate (1.18) from Theorem 2. Let us introduce the operator C : ϑ ∈ L2 ω × (0, T ) → w(·, 0) ∈ L2 (Ω), where w is the solution of ∂t w + ∆w − aw = −Eϑ · 1|ω w = 0 on ∂Ω × (0, T ), w(·, T ) = 0 in Ω. (2.3) in Ω × (0, T ), (2.4) It is well known that if ϑ ∈ L2 (ω × (0, T )), then w ∈ C([0, T ]; H01(Ω)) and in particular w(·, 0) ∈ H01 (Ω) ⊂ L2 (Ω) with compact injection. Thus, the operator C is linear, continuous and compact from L2 (ω × (0, T )) to L2 (Ω). We define F = Im C the space of exact controllability initial data with the following norm: (2.5) w0 F = inf ϑL2 (ω×(0,T )) | Cϑ = w0 . We will need to construct the dual operator of C. Let u0 ∈ L2 (Ω), we consider the unique solution u ∈ C([0, T ]; L2(Ω)) of ∂t u − ∆u + au = 0 in Ω × (0, T ), (2.6) u = 0 on ∂Ω × (0, T ), u(·, 0) = u0 in Ω, and we know from Theorem 2 that D u0 H −1 (Ω) u0 L2 (Ω) ln 2 + Eu 2 u0 L2 (Ω) . (2.7) L (ω×(0,T )) Let us introduce the operator K : u0 ∈ L2 (Ω) → Eu|ω ∈ L2 ω × (0, T ) , (2.8) is linear, continuous, compact from L2 (Ω) where u is the solution of (2.6). The operator K to L2 (ω × (0, T )). Remark that the operator K is the adjoint of C and we denote C ∗ = K. Indeed, by multiplying (2.6) by w the solution of (2.4) where ϑ ∈ L2 (ω × (0, T )) and by applying the Green formula, we obtain the following duality relation: T ϑ(x, t)Eu(x, t) dx dt = 0 ω w(x, 0)u(x, 0) dx, Ω 532 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 i.e., for all ϑ ∈ L2 (ω × (0, T )), for all u0 ∈ L2 (Ω), we have T ϑK(u0 ) dx dt = 0 ω u0 C(ϑ) dx. (2.9) Ω We define F = (Im C) with the norm T 1/2 2 Eu(x, t) dx dt u0 F = K(u0 )L2 (ω×]0,T [) = . (2.10) 0 ω Note that (2.9) and Holmgren theorem imply that F = Im C is dense in L2 (Ω). From the duality relation (2.9), we have by choosing ϑ = K(u0 ) ∈ L2 (ω × (0, T )) that K(u0 )2 2 = u0 C K(u0 ) dx. (2.11) L (ω×(0,T )) Ω We use the notation (·, ·) to describe the scalar product on L2 (Ω). Let B = C ◦ K. Then (2.11) becomes: for all u0 ∈ L2 (Ω), K(u0 )2L2 (ω×(0,T )) = (B(u0 ), u0 ). The operator B is non-negative, compact from L2 (Ω) to L2 (Ω) and self-adjoint on 2 L (Ω). We deduce that it has a discrete spectrum and we associate in L2 (Ω) the Hilbert basis with eigenfunctions ξn of B and eigenvalues µn , where µn > 0, is non-increasing and tends to zero. element g ∈ L2 (Ω) we have the Fourier expansion g = Consequently, for2 every 2 (g, ξ )ξ , g = n n n>0 n>0 |(g, ξn )| < +∞ and also L2 (Ω) 2 K(g)2 2 = (Bg, g) = µn (g, ξn ) . (2.12) L (ω×(0,T )) n>0 Note that from (2.10), (2.12) and by duality, we deduce that 1 (g, ξn )2 . g2F = µn n>0 Let us introduce the sets Sn = {m > 0 | αn+1 < µm αn }, where αn = eµ1 +e e−e for 2 all n > 0. Then each function g ∈ L (Ω) can be represented in the form g = n>0 gn , where gn = (g, ξm )ξm . (2.13) n m∈Sn Also, we have g2L2 (Ω) = n>0 gn 2L2 (Ω) , where gn 2L2 (Ω) = m∈Sn |(g, ξm )|2 . Further, g2F = n>0 K(gn )2L2 (ω×(0,T )) , where 2 K(gn )2 2 = µm (g, ξm ) . (2.14) L (ω×(0,T )) m∈Sn K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 533 From (2.14), we have, if gn = 0, that gn 2L2 (Ω) 1 1 < . αn K(gn )2 2 αn+1 L (ω×(0,T )) (2.15) From (2.7) and (2.15), we obtain, if gn = 0, that gn H −1 (Ω) D gn L2 (Ω) . √ [ln(2 + 1/ αn )] (2.16) Let us introduce, with gn given by (2.13), 2 2 gn gI = αn+1 gn L2 (Ω) < +∞ . I = g= n>0 (2.17) n>0 Then the dual space of I is defined by 1 2 2 I = g= gn gI = gn L2 (Ω) < +∞ . αn+1 n>0 (2.18) n>0 From (2.15) and more exactly F ⊂ I , we have that I ⊂ F. (2.19) z ∈ H01 (Ω). Now, let N > 0. Let zn + zn z= nN Then we can write, in with zn = (z, ξm )ξm . n>N L2 (Ω), (2.20) m∈Sn The properties of the sequence (αn )n>0 and (2.19), (2.16) imply that there exists a constant c3 > 0 such that the following relations hold: 2 2 1 zn zn c3 F I nN nN 2 1 = z , ξ p m αn+1 n>0 m∈Sn pN 2 1 = (z, ξ )(ξ , ξ ) q q m αn+1 n>0 = nN m∈Sn pN q∈Sp 1 αn+1 1 αN+1 1 αN+1 m∈Sn q∈S1 ∪···∪SN 2 (z, ξq )(ξq , ξm ) m∈S1 ∪···∪SN q∈S1 ∪···∪SN m∈S1 ∪···∪SN 2 (z, ξq )(ξq , ξm ) (z, ξm )2 1 αN+1 z2L2 (Ω) , (2.21) 534 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 2 zn − z 2 nN = L (Ω) zn 2L2 (Ω) = n>N n>N zH 1 (Ω) zn H −1 (Ω) 0 n>N zH 1 (Ω) 0 n>N D zn L2 (Ω) √ [ln(2 + 1/ αn )] zH 1 (Ω) 0 (z, zn ) n>N D2 √ [ln(2 + 1/ αn )]2 zn 2L2 (Ω) n>N 1 D zH 1 (Ω) zn − z √ 2 2 0 [ln(2 + 1/ αn )] L (Ω) n>N nN D zH 1 (Ω) c3 zn − z (2.22) √ 2 . 0 ln(2 + 1/ αN+1 ) L (Ω) nN We conclude the proof of Theorem 3 by choosing N > 0 such that 1 D ≈ ln 2 + √ ε αN+1 in order to have, with z = wd , Cϑ − wd L2 (Ω) c3 εwd H 1 (Ω) , 0 D wd L2 (Ω) . ϑL2 (ω×(0,T )) c3 exp ε 2 (2.23) 3. Proof of Theorem 2 We proceed in three steps. Step 1. We apply the observability estimate [3, Theorem 1.2] with a simple change of variable in time. Then there exists a constant c0 > 0 such that for any solution u of (1.17), for any L > 0, one has L 1 2/3 u(·, L)2 2 u(x, t)2 dx dt. exp c0 1 + + La∞ + a∞ L (Ω) L 0 ω Step 2. We look for an estimate of the form 2 u(·, 0)2 −1 constantu(·, L)L2 (Ω) . H (Ω) (3.1) (3.2) K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 535 This is a backward estimate for the heat equation. To do this, we apply the ideas in [1]. Let us consider for almost all t ∈ [0, L] such that u(x, t) = 0 the following quantity: Ψ (t) = u(·, t)2L2 (Ω) u(·, t)2H −1 (Ω) . (3.3) We first prove that 2 d 1 au(·, t)H −1 (Ω) Ψ (t) c4 a2∞ Ψ (t), dt 2 u(·, t)2 −1 H (Ω) (3.4) where c4 > 0 is a constant only depending on the geometry and may change of value in the lines below. Indeed we have the two following energy equalities: posing f = −au, 2 1 d u(·, t)2 2 + u(·, t)H 1 (Ω) = f (·, t), u(·, t) , (Ω) L 0 2 dt 2 2 1 d u(·, t) −1 + u(·, t)L2 (Ω) = f (·, t), (−∆)−1 u(·, t) . H (Ω) 2 dt Then from (3.5), (3.6) and (3.3), we get 2 d Ψ (t) = −u2H 1 (Ω) + (f, u) u2H −1 (Ω) 4 dt 0 uH −1 (Ω) + u4L2 (Ω) − u2L2 (Ω) f, (−∆)−1 u . (3.5) (3.6) (3.7) Also, u4L2 (Ω) − u2L2 (Ω) f, (−∆)−1 u 2 2 f f , (−∆)−1 u − = ∆u + , (−∆)−1 u 2 2 2 2 f f −1 2 −1 (−∆) u H 1 (Ω) − , (−∆) u ∆u + 0 2 H −1 (Ω) 2 2 2 f f 2 2 −1 , (−∆) u . uH 1 (Ω) + − (f, u) uH −1 (Ω) − 2 H −1 (Ω) 2 0 Finally, (3.7) and (3.8) imply 2 2 f f 2 d 2 −1 . Ψ (t) , (−∆) u − u −1 H (Ω) dt 2 H −1 (Ω) 2 u4 −1 H (3.8) (3.9) (Ω) Replacing f = −au, we get the desired inequalities (3.4). Consequently, from (3.4) we obtain for all t ∈ (0, L), 2 Ψ (t) ec4 La∞ Ψ (0). Secondly, remark that from (3.6) and (3.10), we have (3.10) 536 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 2 1 d u(·, t)2 −1 + Ψ (t)u(·, t)H −1 (Ω) + a(·, t)u(·, t), (−∆)−1 u(·, t) (Ω) H 2 dt 2 2 1 d u(·, t)2 −1 + Ψ (t)u(·, t)H −1 (Ω) + c4 a∞ Ψ (t)u(·, t)H −1 (Ω) H (Ω) 2 dt 1 d c4 La2∞ u(·, t)2 −1 Ψ (0) + c + e a Ψ (0) u(·, t)2H −1 (Ω) . 4 ∞ H (Ω) 2 dt (3.11) 0= Consequently, integrating (3.11) on (0, L), we get the desired estimate 2 2 u(·, 0)2 −1 exp 2ec4 La∞ Ψ (0) + c4 a∞ Ψ (0) L u(·, L)H −1 (Ω) . H (Ω) (3.12) Step 3. We choose an adequate L > 0 to conclude the proof of Theorem 2. Let us denote λ1 > 0 a constant such that λ1 Ψ (0). (3.13) Now, we choose λ1 L=T T, Ψ (0) (3.14) then we obtain from (3.1) and (3.12) that the solution u of (1.17) satisfies if u(x, t) = 0 for almost all t ∈ [0, T ], u(·, 0)2 −1 E1 exp D1 Ψ (0) H (Ω) T u(x, t)2 dx dt, (3.15) 0 ω where E1 and D1 are two constants such that √ c0 D1 = 2ec4 T a2∞ T λ1 + √ , T λ1 √ E = c exp(c λ T a ec4 T a2∞ + c (1 + T a + a2/3 )). 1 4 4 1 ∞ 0 ∞ ∞ Multiplying (3.15) by Ψ (0), we get u(·, 0)2 2 E Ψ (0) exp D1 Ψ (0) u2L2 (ω×(0,T )) . 1 L (Ω) (3.16) (3.17) So, if u0 = 0, we have 1 u0 L2 (Ω) E1 Ψ (0) exp D1 Ψ (0) uL2 (ω×(0,T )) 2 √ 2 E1 exp D1 Ψ (0) uL2 (ω×(0,T )) , D1 (3.18) and also clearly for all c1 > 1, u0 L2 (Ω) D1 √ exp c1 D1 Ψ (0) . 2 E1 uL2 (ω×(0,T )) (3.19) K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 We choose c1 > 1 large enough to get u0 L2 (Ω) D1 exp c1 D1 Ψ (0) , 2+ √ 4 E1 uL2 (ω×(0,T )) 537 (3.20) and finally we have u0 H −1 (Ω) ln 2 + c1 D1 u0 L2 (Ω) . u0 L2 (Ω) D 1 √ 4 E1 uL2 (ω×(0,T )) That completes the proof of Theorem 2. (3.21) 2 4. Further comments A quantitative estimate of unique continuation for initial data in H01 (Ω) of the heat equation with potential, solution u of (1.17), may be realized by using properties of the quantity Ψ (t) = ∇u(·, t)2L2 (Ω) u(·, t)2L2 (Ω) , (4.1) which are well described in [1]. We find that there exist two constants c1 , c2 > 1 such that for all T > 0, for all a ∈ L∞ (Ω × (0, T )), for all initial data u0 ∈ H01 (Ω) such that u0 = 0, the solution u of (1.17) satisfies D1 u0 L2 (Ω) (4.2) ∇u0 L2 (Ω) , 1 ∇u0 L2 (Ω) ln 2 + E1 u 2 L (ω×(0,T )) where E1 and D1 are two constants such that D1 = c1 1 + T1 , 2/3 E1 = exp(c2 (1 + T 2 a2∞ + T a∞ + a∞ )). (4.3) This improves the finite dimensional observability estimate of [3, Theorem 1.5] in the sense that if u0 = 0, 2/3 u0 H 1 (Ω) exp c2 1 + T 2 a2∞ + T a∞ + a∞ 0 1 u0 H01 (Ω) × exp c1 1 + uL2 (ω×(0,T )) . T u0 L2 (Ω) (4.4) Acknowledgments This work was written while author had a post-doctoral position at the University Autonoma de Madrid. He thanks the departments of mathematics of the University of Autonoma de Madrid and the University of Valencia and specially Prof. E. Zuazua and Prof. Rosa Donat for their kind hospitality and fruitful exchanges. Support by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282, is acknowledged. 538 K.-D. Phung / J. Math. Anal. Appl. 295 (2004) 527–538 References [1] C. Bardos, L. Tartar, Sur l’unicité retrograde des équations paraboliques et quelques questions voisines, Arch. Rational Mech. Anal. 50 (1973) 10–25. [2] A.V. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, in: Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. [3] E. Fernandez-Cara, E. 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