Sharp observability estimates for heat equations

Transcription

1 Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor Sylvain Ervedoza Enrique Zuazua Sharp observability estimates for heat equations Abstract he goal of this article is to derive new estimates for the cost of observability of heat equations. We develop a new method allowing to show that, when the corresponding wave equation is observable, the heat equation is also observable. his method allows to describe the licit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain. We show that our estimate is sharp in some cases, and in particular in one space dimension and in the multi-dimensional radially symmetric case. Our result extends the ones in [] to the multi-dimensional setting and improves the ones available in the literature, namely those by Miller [8,3,3] and enenbaum and ucsnak [39]. his paper has mainly been developed while the first author was a Visiting Fellow of the Basque Center for Applied Mathematics (BCAM. Also partially supported by the Agence Nationale de la Recherche (ANR, France, Project C-QUID number BLAN and Project CISIFS number N Second author partially supported by the Grant MM8-354 of the MICINN (Spain, project PI-4 of the Basque Government, the ERC Advanced Grant FP NUMERIWAVES, and the ESF Research Networking Program OP- PDE. S. Ervedoza CNRS ; Institut de Mathématiques de oulouse UMR 59 ; F-36 oulouse, France, Université de oulouse ; UPS, INSA, INP, ISAE, U, UM ; IM ; F-36 oulouse, France ervedoza@math.univ-toulouse.fr. E. Zuazua IKERBASQUE, Basque Foundation for Science, E-48 Bilbao - Basque Country - Spain Basque Center for Applied Mathematics (BCAM, Bizkaia echnology Park, Building 5, E-486 Derio - Basque Country - Spain zuazua@bcamath.org

2 Sylvain Ervedoza, Enrique Zuazua Our approach is based on an licit representation formula of some solutions of the wave equation in terms of those of the heat equation, contrarily to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also lain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment the applications of our techniques to controllability properties of heat-type equations. Keywords Heat equations Wave equations ransmutation Observability cost Controllability Cost Introduction. Setting he goal of this article is to study the problem of the cost of observability for heat type equations. o fix the ideas, we will mainly consider the classical constant coefficient heat equation although our methods and results apply to a large class of parabolic abstract problems. Let Ω be a bounded domain and consider the heat equation with state z, solution of t z x z =, (t, x R + Ω, z(t, x =, (t, x R + Ω, (. z(, x = z (x, x Ω. We analyze the problem of observability, which is dual to the controllability one, as we shall lain in Section 3 (see also [3], and that consists in getting global estimates on the solutions in terms of the energy concentrated on some subdomain of the domain Ω where the equation evolves. here is an extensive literature on the subject. In particular, using Carleman inequalities as in [5], one can prove that for any subdomain ω Ω, there exist constants C, γ > and γ > such that any solution z of the heat equation (. satisfies Ω Ω ( γ t z(t, x dtdx C ω z(t, x dtdx, (. and for all >, ( γ z(, x dtdx C z(t, x dtdx. (.3 ω hese are so called observability inequalities that assert that the energy of solutions concentrated in ω yields an upper bound of the energy everywhere in Ω. For that to happen, because of the strong irreversibility of the heat semigroup, an onentially vanishing weight is needed at t = in (. and, similarly, the constant in (.3 grows onentially as. he constants C, γ and γ on the observability inequality (. depend on the geometric properties of ω and Ω.

3 Sharp observability estimates for heat equations 3 his paper is mainly devoted to the analysis of the constant γ. Our goal is to prove a new upper bound on the best constant γ in (. that, all along this article, will be referred to as being the onential observability cost. Moreover, this bound will be shown to be sharp in some geometric configurations and in particular in -d, a fact that was unknown until now. As we shall lain later in Section.4, this constant γ characterizes the reachability set for (.. he constant γ is called the finite-time onential observability cost. Estimates like (.3 are particularly relevant in small time. Note that, according to Lions [3], estimate (.3 is equivalent to estimating the cost of null-controllability in time t =, i.e. the norm of the map that, to an initial data z L (Ω, associates the control u of minimal L ((, ω-norm such that the solution of t z x z = u(t, xχ ω (x, (t, x R + Ω, z(t, x =, (t, x R + Ω, (.4 z(, x = z (x, x Ω. satisfies z( =. here are several previous results on this subject yielding various lower and upper bounds on γ and γ that we briefly present below. he first remark is that, obviously, γ γ. (.5 Lower bounds: he following lower bound on the constant γ fulfilling (. was obtained by comparison with the Green function of the heat equation (see [8]: γ d, with d = sup d(x, ω. (.6 x Ω Indeed, the Green function centered at a point x in Ω \ ω at a geodesic distance d(x, ω of the observation region ω, which decays as Ct N/ ( x x /4t away from ω, shows that, necessarily, γ d(x, ω / is needed for all x Ω in order to ensure (.3. On the other hand, in [3,4], using the functions z ρ (t, x = sin (4πt n/ ( ρ x t ( 4t (ρ x solutions of the heat equation in R N, it is shown that, ρ >, (.7 γ d, with d = sup{ρ, such that B(x, ρ Ω\ω}. (.8 Note that, always, d d, but in some geometrical situations, d = d. his is the case in particular when Ω\ω is a ball and ω is a neighborhood of Ω. Upper bounds: On the other hand, as mentioned above, Carleman inequalities guarantee that (.3 holds with a finite constant γ >, hence also (. for some constant γ. But this technique does not provide any licit

4 4 Sylvain Ervedoza, Enrique Zuazua ression on how the onential observability constants γ, γ depend on the geometry of the problem under consideration. he existing upper bounds refer mainly to the case where the Geometric Control Condition (GCC is satisfied. GCC asserts that all the rays of Geometric Optics in Ω, reflected according to Descartes law on the boundary, enter the domain ω in some finite uniform time S (see [4] for a more precise description of the GCC. his imposes, of course, important constraints on the geometry of the control subdomain ω. his condition is sharp in the context of the observability of the wave equation but, as mentioned above, is not needed to establish the observability inequality for the heat equation. According to [4, 5], the GCC is equivalent to the following observability property for the corresponding wave equation: here exists C > such that any solution y of the wave system { ss y x y =, (s, x R Ω, (.9 y =, (s, x R Ω, satisfies Ω y(, x dx + y s (, H (Ω C ω y(s, x dsdx. (. In (.9, s stands for the time variable of the wave equation, since it is convenient to distinguish it from the time t for the heat process. he time S needed for the GCC to hold, in view of the finite velocity of propagation of waves ( in the present model, is necessarily such that S d. his is so since, roughly, in time d one can only guarantee that the ray along the geodesic path reaches the observation set, after evolving along a back and forth trajectory, while the GCC requires the same to hold for all the rays. However, there are many cases in which S >> d or even S is infinite. his is precisely the case when ω fails to satisfy the GCC in any finite time. his happens, for instance, when Ω is the unit ball and ω is a ball centered at the origin and of radius r <. However, there are non-trivial situations in which we can guarantee that S = d = d, and in particular in the -d setting, as we will lain in Section 4. Under the GCC, it has been shown that the observability inequality (.3 holds for the heat equation for all γ > γ S /, (. with γ = 8(36/37 in [8,3]. his upper bound on γ was later improved to γ = 3 in [39]. As a consequence of this, according to (.5, the observability inequality (. holds for any γ satisfying γ > 3S /. (. But, even when S = d, this upper bound (. on the best observability constant is larger (by a multiplicative factor 3 than the lower bound (.8.

5 Sharp observability estimates for heat equations 5 A sharp result on γ in -d. he results in [] imply that, for onedimensional parabolic equations on an interval of length L controlled from one of the points on the boundary, γ in (. can be chosen to be any constant strictly larger than L /, whereas (. does not hold if γ < L /. his is the unique existing result in the literature on the optimality of γ in (. as far as we know. Note that the results in [] are stated from the point of view of the reachability set of the heat equation but this set is fully determined by the constant γ in (., see Section.4. he techniques used in [] are based on a precise study of the biorthogonal family of (( n π t n in L (,, which is not available in higher dimensions. Also note that this seems to indicate that the lower bound (.8 for γ should be d / instead of d /. So far, this is only a conjecture. As we have said, the fact that the observability property of waves implies the observability of the heat equation is well-known. But this has not been proved directly so far, but rather in the context of the dual equivalent controllability problem. o be more precise, Russell in his pioneer work [37] observed that the exact controllability property of the wave equation implies the null controllability of the corresponding heat process. his, by duality, allows also showing the link between the observability properties of these two models. he original approach of Russell was based on the method of moments ([37], and has been more recently modified and replaced by the so-called transmutation method ([8,3,3], that has been employed to give the quantitative results on the onential observability cost mentioned above. ransmutation is easier to apply: it is inspired in Kannai s transform, which allows writing the solutions of the controlled abstract heat equation in terms of the corresponding controlled solutions of the wave model. his approach has been also recently used in [33] to derive an efficient method for numerically computing the control for heat equations. he main result of this paper ensures that, under the GCC, the observability inequality (. holds for γ = S / (or very close variants; see Section 4 for more details. his significantly improves the known estimates (.. According to the lower bound (.8, we conclude that our result is sharp when S = d. he later is true, as we mentioned above, in one space dimension and in some simple multidimensional geometries: for instance, for any domain Ω when the control set ω is a neighborhood of its boundary such that Ω \ ω is a ball, see Section 4. Note that even in the dimensional case, it also improves the results in [] up to the critical case γ = L /. Our approach is also based on a transmutation method, but applied directly on the observability context rather than from the control point of view. he main novelty is that we write solutions of the wave equation as a function of that of the heat equation, in the opposite sense to the classical Kannai transform. his might seem counterintuitive since solutions of the heat equation propagate at an infinite speed, and this could be an obstruction to get the solutions of the wave equation, with a finite velocity of propaga-

6 6 Sylvain Ervedoza, Enrique Zuazua tion. But, in fact, this may be done since our transform maps solutions of the heat equation into a class of analytic solutions of the wave one. Once solutions of the wave equation have been written in terms of those of the heat equation, applying the well known observability properties of the wave equation under the GCC, one recovers observability inequalities for the heat equation with sharp onential observation cost. his method will be formulated and presented in an abstract setting containing the heat equation but also other parabolic problems as, for instance, the fourth order diffusion operator.. he main result Let X be a Hilbert space and A be a self-adjoint positive definite unbounded operator on X with dense domain D(A and compact resolvent. We then introduce the following abstract heat equation: { t z + Az =, t R +, (.3 z( = z, and its corresponding wave equation: { ss y + Ay =, s R, y( = y, s y( = y. (.4 he observation is done through an operator B L(D(A, U, where U is a Hilbert space. As we mentioned above, our approach applies under the assumption that the observability property holds for this abstract wave equation as made precise below. Assumption here exist a time S > and a constant C w = C wave such that any solution y of the wave equation (.4 with initial data (y, y D(A D(A / satisfies A / y X + y X C w By(s U ds. (.5 Our main result is the following one: heorem. Let A be a self-adjoint unbounded positive definite operator with dense domain and compact resolvent and B be an observation operator B L(D(A, U such that Assumption holds. hen there exists C > such that for any z solution of (.3 with initial data z D(A, the following estimate holds ( S z(t X t dt C Bz(t U dt. (.6

7 Sharp observability estimates for heat equations 7 Besides, for all > there exists C( > such that for any z solution of (.3 with initial data z D(A, the following is satisfied: ( S z(t X t dt C( Bz(t U dt. (.7 he following comments are in order: In view of this result, one can take any γ > S / in (. when the GCC is satisfied, see Section 4 for more details, and even the critical case γ = S / provided the norm in the left-hand side of (. is weakened. he finite time estimate (.7 should be made precise further in the sense that it would be interesting to get licit bounds on how the constant C( grows as tends to zero. his issue is discussed in Section 3. and in Section 5. If B is assumed to be in L(D(A /, U, since A is positive definite, the right hand side of (.6 is finite for any solution of (.3 with initial data in X. his is so because solutions z of (.3 with initial data in X belong to L (R +, (νtdt; D(A /, for some ν > smaller than the first eigenvalue of A. Accordingly, when B L(D(A /, U, by density, estimate (.6 can be extended to any z X. If B does not belong to L(D(A /, U but only to L(D(A, U, one cannot guarantee a priori that the integrals in (.6 are finite for any initial data z X but the inequalities (.6 (.7 make sense for initial data in D(A. In Section 3, we will lain how our transmutation technique developed for heorem. can be applied directly in a finite-time horizon, using different transmutation kernels that are compactly supported in time t (,. In particular, our transmutation method can be used to get a bound on the cost of controllability γ in (.3 (see Section 3., though the bound we obtain is worst than the ones in [8,39] when. We shall lain why our method fails to improve the bounds in [39]. Our method also identifies an observed quantity for which not only the observability inequality holds but the reverse is also true (see Section 3.3. In other words, we will give an licit norm on the initial data which is equivalent to some norm of the observation. Note for instance that, although (. holds, the reverse is not true. his issue is of course of particular interest with respect to the control problem, as we lain in Section 3.4. In particular, this can be used to determine a Hilbert Uniqueness Method algorithm to compute smooth controls. his partially lains why transmutation allows to avoid the ill-posedness of the problem of numerically computing the controls (see [33]. We also list a number of examples in which our approach applies. In particular, we focus on the -dimensional heat equation. We then consider the case Ω = (,, the unit square, with observation on the boundary, first when GCC holds, and then when the observation is done only on one side of the unit square. In that later case, though GCC does not hold, transmutation can be applied and also yields in that particular case estimates on the onential observability cost.

8 8 Sylvain Ervedoza, Enrique Zuazua he outline of the article is as follows. First, in Section, we prove heorem.. We also briefly comment in Section 3 how the techniques we have developed for heorem. can be adapted to deal with a finite time horizon, and comment their control theoretical consequences. In Section 4, we discuss applications of heorem. on some examples. Finally, in Section 5, we give some further comments and open problems. Proof of the main result We proceed in several steps that will be presented in different paragraphs.. ransmutation : from heat processes to waves As we have lained above, transmutation has been so far used to transform results on the control of the wave equation into results on the control of the heat one. For that to be done one has to write the solutions of the heat equation in terms of those of the wave equation in the spirit of the classical Kannai transform (see, for instance, [8]. But here we apply the transmutation method at the level of the observability property. More precisely, we want to derive observability inequalities for the heat equation as a consequence of the existing observability inequalities for the wave equation. For this to be done one has to write the solutions of the wave equation in terms of those of the heat equation. Such transform is rather counterintuitive since, in view of the finite velocity of propagation underlying the wave operator, it might seem unnatural to try to ress its solutions in terms of the heat kernel which diffuses at an infinite speed. But this can be done, indeed, for a suitable class of initial data and this suffices to our purposes. he key observation of the present article is as follows: heorem. Let z X and z = z(t be the solution of the abstract heat equation (.3 with initial datum z. For any finite S >, the solution of the abstract wave equation (.4 with initial data y, y = S 4 πt3/ ( S z(t dt, (. 4t in the time interval < s < S can be represented as ( ( ss s S y(s = sin z(t dt. (. R + (4πt / t 4t Proof (heorem. Let us consider z the solution of the abstract heat equation (.3 with initial data z X. One can check directly the statement of heorem., showing that y given by (. is a solution of (.4. However, for giving a better insight to the reader, we rather lain how we got this

9 Sharp observability estimates for heat equations 9 result, linking the trajectory z(t to one of the solutions of the abstract wave equation (.4. o do this, we look for a solution y of (.4 in the form y(s = k(t, sz(t dt, (.3 R + where k = k(t, s is a suitable kernel to be made precise below, describing how the wave and heat semigroups are related. In order to identify the kernel k we formally apply the abstract wave operator to y: ss y(s + Ay(s = ss k(t, sz(t dt + R + k(t, saz(t dt R + = ss k(t, sz(t dt k(t, s t z(t dt R + R + = ss k(t, sz(t dt + R + t k(t, sz(t dt lim (k(t, sz(t + k(, sz. R t + his shows that y is a solution of the wave equation (.4 if k satisfies t k + ss k =, t R +, s R, k(, s =, s R, (.4 lim t k(t, s =, s R. Note that in this system s plays the role of the space variable and that we are dealing with the adjoint heat equation that can be easily transformed into the standard forward one by the change of variables t t. he existence of such non-trivial kernels k is well known (see, e.g., [8], even if, of course, problem (.4 is severely ill-posed. In particular, according to the uniqueness results in [7], if we assume that, for some constant M, k(t, s M (Ms, t R +, s R, then k. herefore, the solution we are looking for, k, has to violate this growth condition. Note that, formally, for any k satisfying (.4, we automatically get that y given by (.3 is a solution of the abstract wave equation (.4. But for the estimates we will derive later to obtain heorem., we will need precise estimates on one such non-trivial kernel k. A key further observation with respect to the constructions in [7, 8] is that, in the present context, we only need the solution k to be defined for s (, S. We can then look for k satisfying, instead of (.4, the following restricted system: t k(t, s + ss k(t, s =, t R +, s (, S, k(, s =, s (, S, (.5 lim t k(t, s =, s (, S.

10 Sylvain Ervedoza, Enrique Zuazua Such k satisfying (.5 can be given licitly: ( ( ss s S k(t, s = sin. (.6 (4πt / t 4t Furthermore, k satisfies the following identities: S k(t, =, t R +, s k(t, = 4 ( S, t R πt3/ +. 4t Summarizing, if z is the solution of the abstract heat equation (.3, then the function y given by (.3 with this kernel k is precisely a solution of the wave equation (.4 for s (, S with initial data (y, y as in (.. his concludes the proof of heorem.. Remark. Observe also that the function k in (.6 can be obtained from the Appell transform (see [4] out of the separated variable solution v(t, s = sin(ss/ (S t/4 of the adjoint heat equation t v + ss v =. Also note that this is the same kernel as the one constructed in [3,4] (see (.7 but with t replaced by t to switch from the heat operator to the present adjoint one. here, it was used to prove estimates from below for γ.. Observability by transmutation Using the transmutation formula of the previous paragraph we can derive a first observability inequality for the heat equation as a consequence of the corresponding one for the wave equation. he following holds: heorem. Let B be an observation operator B L(D(A, U. If Assumption holds and A is self-adjoint, positive definite and with compact resolvent, for any z solution of (.3 with initial data z D(A, S 4 πt C w 3/ ( S 4t sin (4πt / z(t dt ( ss t where C w is the constant in (.5. X ( s S Bz(t dt 4t U ds, (.7 Proof (heorem. Let z D(A and consider z(t the corresponding solution to the abstract heat equation (.3. hen heorem. yields a solution y of the wave equation on (, S, licitly given through identity (.. Using (.5, we immediately obtain A / S y + y X C w k(t, sbz(tdt ds, (.8 X R + for those initial data (y, y given by (.. his is exactly (.7. U

11 Sharp observability estimates for heat equations Note that, in (.7, the left hand-side term constitutes a norm on z whereas the right hand-side one should be estimated in terms of the norm of the observation Bz(t. his will be done in the next paragraph..3 Further estimates In view of the estimate (.7 in heorem., in order to get the main result in heorem. it is sufficient to estimate the integrals on both sides of (.7. For this to be done, it will be convenient to use the spectral decomposition of the functional space X on the basis of the eigenfunctions of A : Since A is a self-adjoint positive definite operator with compact resolvent, its spectrum consists in a sequence of positive eigenvalues < µ µ j µ j+ and an orthonormal (in X basis of corresponding eigenvectors Φ j satisfying AΦ j = µ j Φ j. We now prove classical estimates from below for the left hand side of (.7 and from above for its right hand side. Estimates on the left hand side of (.7. Lemma. here exists a constant C > such that ( S z(t X t dt C ( S z(t dt t3/ 4t X. (.9 o be more precise, if z = a j Φ j, solutions z of (.3 with initial data z satisfy: a j ( µ j C ( + µ j j / ( S z(t X t dt C j t3/ ( S 4t z(t dt X, (. a j ( µ j ( + µ j /. (. Proof (Lemma. Expanding z on the basis (Φ j as z = j a jφ j, the corresponding solution z of (.3 is z(t = j a j Φ j ( µ j t. (. his implies in particular that S e 4t t3/ z(t dt = X j a j ( t S e 3/ 4t µjt dt. (.3 We need to determine a lower bound for F (µ = ( S t3/ 4t µt dt.

12 Sylvain Ervedoza, Enrique Zuazua For this, set µ = S/( µ, and remark that F (µ But, for α [,, µ t ( S 3/ ( S µ ( S µ 4t µt dt µ t µ / 3/ ( µt dt ( t dt. µ µ t3/ and 3 α α ( t dt = t3/ α ( α 3 ( t dt 3/ α t5/ ( t dt t5/ 3 α 5/ which implies in particular, for α, that α ( t dt 3 ( α, 4 α3/ α t ( t dt ( α. 3/ 4 α3/ Hence, for µ 6/S, F (µ ( µ S ( µ Sµ /4 S S( + µ. /4 For µ [, 6/S ], one easily checks that F is continuous and does not vanish. hus it is bounded from below by some positive constant. We conclude that there exists c > such that for all µ R +, F (µ c ( µ, (.4 ( + µ /4 which implies (. by (.3. Now, using the same notations as in (., let us remark that ( S z(t X t dt = j a j ( S t µ jt dt. (.5

13 Sharp observability estimates for heat equations 3 Estimate (. then follows from the following one: for µ µ >, (recall that µ is the smallest eigenvalue of A ( S t µt dt = S ( ( µ t + dt µ t S µ S µ 3 ( µ dt + S µ ( 3 ( µ + S µ ( 3S µ 3 ( µt dt C ( µ µ C ( µ + µ. (.6 Hence, from (.5-(.6, there exists a constant C such that (. holds. Estimate (.9 immediately follows from (.-(.. his concludes the proof of Lemma.. Estimates on the right hand side of (.7. Lemma. For all >, there exists a constant C ( such that Moreover, k(t, sbz(t dt U ds C ( t S e 3/ 4t z(t dt X. (.7 lim C ( =. (.8 Proof (Lemma. Let >. Using that, for some constant C independent of >, dt t log (t + dt t log (t + dt + dt min{,} t log (t + ( + C, for each s (, S, we get k(t, sbz(tdt U ( k(t, s Bz(t U t dt log (t + dt t log (t + ( + ( ss C 4πt sin e s t Bz(t U t t log (t + dt ( + C log (t + Bz(t U dt. (.9

14 4 Sylvain Ervedoza, Enrique Zuazua But B belongs to L(D(A, U. hus, log (t + Bz(t U dt C ( + t z(t D(A dt. (. Using the same ansion of the heat solutions as in (., we obtain ( + t z(t D(A dt j a j ( + t µ 4 j ( tµ j dt C j a j ( µ j µ 3 j( +. (. his shows that, for some C > independent of, k(t, sbz(t dt ( + 3 ds C a j ( µ j µ 3 j. hus, for all >, setting C ( = sup {( µ + S µ( + µ / µ µ µ we have a j ( µ j µ 3 ( + 3 j C ( j j U j 3 ( + 3 }, (. a j ( µ j ( + µ j /, (.3 and, obviously, lim C ( = because µ, the smallest eigenvalue of A, is strictly positive. Estimate (.7 and the limit (.8 then follow immediately from estimates (.,(. and (.3. Lemma.3 For all >, there exists a constant C( such that k(t, sbz(t dt ds C Bz(t U dt. (.4 Proof (Lemma.3 For each s (, S, k(t, sbz(tdt C C U k(t, s Bz(t U 4πt sin U ( ss t t dt ( Bz(t s t U S t dt t ( s S Bz(t U t dt t dt. (.5

15 Sharp observability estimates for heat equations 5 Besides, for t (, S, ( s S ( s S ds = ds t t S t ( s S ( s S = ds + t S ds t t ( S S + + t C t, t and, obviously, whatever t > is, ( s S ds S. t Combining these two estimates, we deduce that for all t >, ( s S ds min{s, C t}. t hus, integrating (.5 in s (, S, we obtain the desired estimate (.4. We are now in position to prove heorem.. Proof (heorem. Combining heorem., Lemmas. and.3, we obtain ( S z(t dt t3/ 4t C Bz(t U dt X + CC ( ( S z(t dt t3/ 4t. (.6 aking large enough so that which can be done by Lemma., we obtain ( S z(t dt t3/ 4t C CC ( /, (.7 X Bz(t U X dt. (.8 his implies (.7 for from (.9. Estimates (.7 for and (.6 are then straightforward. o prove (.7 in any time > (and smaller than, we use a compactness argument to show that for all >, there exists a constant C such that for any z solution of (.3 with initial data z = a j Φ j D(A, ( S z(t dt t3/ 4t X C Bz(t U dt. (.9

16 6 Sylvain Ervedoza, Enrique Zuazua We argue by contradiction. Fix (, and assume that there is no constant C such that (.9 holds. It would then exist a sequence z n of solutions of (.3 with initial data z,n = j a j,nφ j D(A such that ( S z t3/ n (t dt 4t X =, lim Bz n (t n U =. (.3 Note that, using the ansion of z on the basis (Φ j, for all n N, ( S z t3/ n (t dt 4t where β j = X = j a j,n β j, and that, according to estimate (.4, for some C >, ( S t3/ 4t µ jt dt, (.3 β j C ( µ j. (.3 ( + µ j /4 hus, (a j,n β j is bounded in l (N and, extracting a sequence if necessary, (a j,n β j weakly converges to some sequence (b j β j in l (N. But, due to (.3, for all ε >, there exists a constant c ε such that for all (a j l (N, t a j ( µ j t c ε a j βj. j j L (ε, his implies in particular that, setting z(t = j b j ( µ j tφ j for all t > ε, z n weakly converges to z in L (ε, ; X weak-. Due to the regularizing effects of the abstract heat equation under consideration, this implies that z n strongly converges to z in L (ε, ; D(A and z n (ε strongly converges to z(ε in D(A. herefore, choosing ε < /3, z( +ε solves (.3 with initial data z(ε and, due to (.3 and the strong convergence of z n to z in L (ε, ; D(A, B z(t = for t (ε,. But solutions of (.3 are analytic in positive time with values in D(A. Hence B z(t = for all t > ε and in particular on (ε, + ε. Applying (.7 with to z( + ε, we deduce that z(t for all t > ε. Hence the limit sequence (b j is identically zero. But z n strongly converges to z in L (, ; D(A. Since B L(D(A; U, we deduce that Bz n strongly converges to B z in L (, ; U. Consequently, due to (.3, Bz n strongly converges to zero in L (, ; U. But then, according to (.8, (a j,n β j strongly converges to zero in l (N, which is in contradiction with (.3. Hence we have proved (.9 for any positive time >. Estimate (.9 then yields (.7 in any time > and concludes the proof of heorem..

17 Sharp observability estimates for heat equations 7 Remark. he regularizing effect of the abstract heat semigroup allows also showing that for all p > and γ > S /, any solution z of (.3 satisfies ( γ z(t D(A t p dt C(γ, p Bz(t U dt (.33 with a constant C = C(γ, p >. Indeed, writing z(t = j a jφ j ( µ j t and using (.6, we get ( γ z(t D(A t p dt C j a j µ p ( γ µ j j, (.34 ( + µ j / which easily yields the claimed result (.33 by (.38 and the estimates (.3. Remark.3 For convenience, we have assumed that B is bounded from D(A to U, but our arguments apply similarly when the operator B is unbounded from D(A p to U, whatever p N is. he proofs are the same, except for Lemma. and the compactness argument used in the proof of (.9, where straightforward modifications need to be applied. his allows to deal with weaker observability properties, such as pointwise observations, as we will lain in Section 4. Remark.4 It would be interesting to know if the following observability inequality holds: For all >, there exists a constant C( such that solutions z of (.3 satisfy ( S z(t dt t3/ 4t X C( k(t, sbz(t dt (.35 where k is the function given by (.6. Using heorem. and Lemma., we immediately get that this is true for > for large enough. However, for >, the compactness argument used in the proof of heorem. cannot be applied directly and requires the following unique continuation property: If z denotes a solution of the abstract heat equation (.3, ( s (, S, k(t, sbz(t dt = = t (,, Bz(t =. (.36 Whether or not this unique continuation property holds for any time > is an open problem. Of course, using that k solves (.5, this is equivalent to prove that solutions y of ss y + Ay = k(, sz(, s (, S (.37 with initial data as in (. satisfying By(s = for all s (, S vanishes identically. Of course, the source term in (.37 makes the classical unique continuation results of no use for that particular problem. U ds,

18 8 Sylvain Ervedoza, Enrique Zuazua.4 A first application to control Let us remark that, under the assumptions of heorem., the proof of heorem. yields (.9. Hence, for any time >, there exists a constant C such that for all z solution of (.3 with initial data z = a j Φ j, a j βj C j Bz(t U dt, (.38 with β j as in (.3. his can be used to show that the reachability space R, which is the set of all functions that can be obtained as z( for z solution of the abstract control system z + Az = B u(t, t, z( =, (.39 with u L (, ; U, contains the set of all data z = j a jφ j satisfying j a j β j <. (.4 Of course, from the estimates (.3, this is implied by a j ( + µ j / (S µ j <. (.4 j In a more concise form, this means that A /4 ( AX R. Indeed, following [3, 3], let us introduce the functional J defined for ϕ X by J(ϕ = Bϕ(t U dt ϕ, z X, where ϕ is the solution of the adjoint heat equation t ϕ + Aϕ =, t (,, ϕ( = ϕ. hen define the completion X of {ϕ X} with respect to the norm ϕ obs = Bϕ(t U dt. Due to estimate (.38, if z = a j Φ j satisfies (.4, the functional J is well-defined, continuous, convex and coercive in X. It therefore has a unique minimizer ψ X which defines a control function u(t = Bψ(t (or, more precisely, u(t = Bψ, where B is the unique continuous extension of the map ϕ Bϕ(t on X. As one can check by writing the Euler- Lagrange equation satisfied by ψ, the corresponding solution z of (.39 satisfies z( = z. Note that, in [38] (see also [3], it is proved that the reachability set is independent of >, which is consistent with the fact that the subspace of the reachability set we have found does not depend on time.

19 Sharp observability estimates for heat equations 9 Remark also that our results improve the ones in [], where it was proved using biorthogonals that ( (S + ε AX R for any > and ε > for the case of d heat equation observed from one boundary. Indeed, using the estimates in Section.3, one can rewrite the results in [] as follows: one can take any γ > S / in (. in -d when controlling from one boundary. However, the techniques used in [] are restricted to the -d case controlled from one boundary, in which case the control problem can be formulated licitly as a moment problem. herefore other situations (distributed controls in -d or any case in higher dimension do not seem to be handled by the techniques in []. Our result also improves some other existing ones in higher dimension, as for instance those in [3, Appendix A], stating that ( α AX R for any > for any α > 4 (36/37S. 3 Observability and Controllability in finite time So far our approach has been presented in an infinite time horizon, in the sense that the transmutation kernel k in (.6 is not compactly supported in time t R +. Below, we lain that there are many possible choices of transmutation kernels, and among them, many that are compactly supported in time t (,. However, as we shall lain below, they are less licit as before and therefore the estimates we obtain that way are worst than the ones in the literature. Despite of this, the use of these finite time horizon kernels yields new results for a broad class of abstract heat equations. 3. ransmutation in finite time horizon Here, our goal is to show that there are many kernel functions k(t, s, vanishing after some time >, that can be used to transmute from heat to waves. Following the proof of heorem., given >, one should then construct a nontrivial solution k of t k (t, s + ss k (t, s =, t (,, s (, S, k (, s =, s (, S, (3. k (, s =, s (, S. Such k can be constructed following the classical method of ychonoff (see [7, p.] and [8]. he idea is to look for a solution k as a power series ansion in s of the form k (t, s = n s n n! g n(t, (3. where the functions g n are smooth and supported on [, ]. A necessary condition for such ansion to solve (3. is to have g n = ( n g (n, g n+ = ( n g (n, n N. (3.3

20 Sylvain Ervedoza, Enrique Zuazua Such a function k can be constructed by taking g (t and g (t of the form ( ( α g (t = t +, t (,, t (3.4 t R \ (,, where α > is some positive parameter. It is well-known that g is a smooth function, but to guarantee the convergence of the power series ansion (3., we need more precise estimates, that can be derived using Cauchy s formula (see [7, Pb.3 p.73]: Lemma 3. For each δ (,, for all n N and t (,, ( g (n n! (t (δ min{t, t} n α. (3.5 ( + δ min{t, t} Proof (Lemma 3. Note that, due to the fact that g is symmetric in /, we can restrict ourselves to prove (3.5 only for t (, /. Fix t (, /. Note that g is real analytic in a neighborhood of t and can then be extended to an holomorphic function in a neighborhood of t, for instance in the ball B(t, δt of center t and radius δt, δ (,. hus, the Cauchy formula yields g (t = g (τ dτ, (3.6 iπ Γ (t,δt τ t where Γ (t, δt denotes the circle of center t and radius δt. We then obtain that g (n (t = n! g (τ dτ. (3.7 iπ (τ t n+ Γ (t,δt Now, licit computations easily yields that, for t (, / and τ Γ (t, δt ( ( g (τ = αre τ + ( α, τ t( + δ where Re(τ denotes the real part of τ C and estimate (3.5 follows immediately. Lemma 3. allows to prove the convergence of the series (3. and to obtain the estimate (similarly as in [7, p.] ( ( s k (t, s s min{t, t} δ α. (3.8 ( + δ For (3.8 to be well-defined on (, S for t and t and for k to solve the time boundary conditions in (3., we need that, for some δ (,, α S ( + /δ, that is α > S. We thus have the following:

21 Sharp observability estimates for heat equations Proposition 3. For any finite S >, for any α > S, there exists a function k satisfying (3. with k (t, = and s k (t, = g (t given by (3.4 such that for any δ (, satisfying α > S ( + /δ, for any (t, s (, (, S, estimate (3.8 holds and, for all p N, ( ( p p! s t k (t, s (δ min{t, t} p s min{t, t} δ α. ( + δ (3.9 Only (3.9 has not been proved, but it follows from Lemma 3. and identity (3. immediately. Details are left to the reader. Of course, such k can be used for transmutation, similarly as in heorem.. o be more precise, if z X and z = z(t is the solution of the abstract heat equation (.3 with initial datum z, the function y = y(s given by y(s = k (t, sz(t dt, s (, S, (3. is a solution of the abstract wave equation (.4 on (, S with initial data y, y = ( ( α t + z(t dt. (3. t Let us finally emphasize that any kernel k solution of (3. can be used for transmutation, which illustrates the flexibility of this approach. 3. Exponential observability cost in finite time As we recalled in the introduction, estimates on the cost γ of controllability in small time in (.3 for heat like equations are available in the literature (see [3,39]. he goal of this paragraph is to lain that our approach also applies to that particular issue, using for instance the function k given by Proposition 3. but, so far, yields a weaker result (but with an easier proof than the ones in the articles [8, 39]. More precisely, we claim that for all solutions of the abstract heat equation (.3, the finite time observability inequality (.3 holds with γ > 6S for all > with a constant C independent of >. his of course follows from the estimate (3.8 and similar estimates as the ones in Section.3. he proof is left to the reader. Let us now lain why this result is so far from the bounds obtained in [3,39]. his is due to the fact that we have very rough estimates on the function k, which is ected to be highly oscillatory, similarly as k in (.6.

22 Sylvain Ervedoza, Enrique Zuazua In particular, one could look for a solution k α of (.5 of the form (3. with g (t = and α ( g (t = α, t >, 4πt 3/ t, t, α >. hough such function k α corresponds to the licit solution k α (t, s = 4πt ( s /4 α t sin ( s α t, (3. estimates on g (t and its derivatives will only yield that, for all δ (,, ( ( s k α (t, s s t δ α. (3.3 + δ Of course, estimate (3.3 only guarantees the existence of k α for t R + and s ( α/, α/ whereas on the licit formula (3., one immediately sees that k α is well-defined on (t, s R + ( α, α. his indicates that the above estimates do not take into account in a satisfactory way the strong oscillating behavior of the function k α and the conjectured ones of the functions k. his also lains why our technique fails to provide sharp estimates on the finite time onential observability cost γ in ( wo-sided inequalities When dealing with the wave equation, one often obtains two sided inequalities of the following form: here exist some strictly positive constants c w, C w, such that any y solution of (.4 satisfies c w By(s A U ds / y + y X C w X By(s U ds. (3.4 his states, in addition to (.5, an admissibility result, always true when B L(D(A /, U, but consequence of a more subtle hidden regularity property when this is not the case (and in particular when considering boundary observation through the normal derivative of solutions for the Dirichlet Laplacian, see e.g. [3]. Inequality (3.4 can be combined with any kernel k solution of (3. (such kernel exists, see Proposition 3. to obtain a two-sided observability inequality for the heat equation. o simplify the presentation, we further assume that k is odd in the variable s. (Otherwise, replace k by k (t, s k (t, s.

23 Sharp observability estimates for heat equations 3 hen the transmutation technique applies and yields: c w k (t, sbz(t dt ds U s k (t, z(t dt C w X k (t, sbz(t dt U ds. (3.5 Concerning the observed quantity on the initial datum, observe that for z = j a jφ j, we have s k (t, z(t dt = X j where β j (k = a j (β j (k, s k (t, ( µ j t dt. (3.6 We thus define the set of observable states with k as the Hilbert space given by { } O(k = z = j a j Φ j, z O(k = j a j (β j (k <. (3.7 Let us emphasize that this space depends on the kernel transmutation function k. Rewriting (3.5 using this norm, we deduce that there exist two strictly positive constants c, c such that c z O(k k (t, sbz(t dt ds c z O(k. (3.8 Remark 3. he same can be done with the kernel k as in (.6, β j as in (.3 and O(k as { } O(k = z = j a j Φ j, z O(k = j U a j β j <, (3.9 if is large enough. Indeed, according to Remark.4 and Lemma., for large enough, for any, for any z solution of the abstract heat equation (.3, it holds c z O(k k(t, sbz(t dt ds c z O(k. (3. Note that in (3.9, the space O(k is independent of the time >. But whether or not estimate (3. holds in arbitrarily small values of > is an open problem, see Remark.4. U

24 4 Sylvain Ervedoza, Enrique Zuazua Remark 3. Let us remark that this is not the first time that one derives such equivalence of norms between an observation and the solutions. Indeed, the by-now classical Fursikov-Imanuvilov s Carleman estimate derived in [5] also yields, for some weights η = η(t, x (whose definition is given in an intricate way that reflects the geometrical setting, see [5] for the detailed definition of η, that, given >, Ω and ω, there exists a constant C such all solutions z of (. satisfies Ω η(t, x z(t, x dt dx C η(t, x z(t, x dtdx, (3. ω and of course, the reverse inequality also holds true. 3.4 Application to control In the sequel, we assume that (3.4 holds for the abstract wave equation, a fact that is well known to be true in many relevant situations. For the solutions of the corresponding heat equation it then follows that the twosided inequalities (3.8 are true. hese inequalities can be used to deal precisely with the dual control problem. A technical assumption. For what follows, it is interesting to further assume that there exists a constant C such that for any z solution of the abstract heat equation (.3, z( X C z O(k. (3. his is automatically fulfilled in most applications because of the strong regularizing effect of heat-like equations. Estimate (3. means that the map z z( is continuous from the set of observable states with k to X. In particular, (3. and (3.8 imply z( X C k (t, sbz(t dt Writing (3. on the basis of eigenfunctions of A and recalling the definition (3.7 of the space O(k and of the coefficients β j (k, one easily checks that (3. holds if and only if there exists a constant C such that for all µ >, ( µ C s k (t, ( µt dt. (3.3 Note that the kernel function k given in Proposition 3. satisfies (3.3 (or equivalently (3. or (3.5 below: Indeed, when t (, s k (t, is non-trivial and non-negative U ds. s k (t, e µt dt e µ s k (t, dt,

25 Sharp observability estimates for heat equations 5 and then C in (3.3 can be taken as C = / sk (t, dt. We emphasize that many of the non-trivial kernels k solutions of (3. satisfy assumption (3.. his is the case for instance for the kernels k given by Proposition 3.. Namely, for any non-trivial non-negative g such that the ansion (3. converges, (3. holds using the same arguments as above. In the following, the transmutation kernel k solution of (3. is fixed and assumed to satisfy (3.. he reachability set. Define the reachability set (its name will be justified hereafter { R(k = z = a j Φ j, z R(k = } a j (β j (k <, (3.4 j j which is the dual space of O(k. Note that, using this spectral representation of solutions of the heat equation (.3, one immediately sees that estimate (3. (equivalently (3.3 is equivalent to the existence of a constant C such that for any z solution of the abstract heat equation (.3, z( R(k C z X. (3.5 In particular, this implies that, if z X, then z( belongs to the reachability set R(k, meaning that all free trajectories of the heat semigroup belong to R(k. Let us then consider the following control problem: For z X, z R(k, to find a control u so that the solution z of satisfies t z + Az = B u, t (,, z( = z, (3.6 z( = z. (3.7 o deal with this problem, in view of the previous two-sided observability inequalities, following the ideas in Subsection.4, we introduce the functional J on O(k as J(ϕ = k ( t, sbϕ(t dt ds where ϕ is the solution of the adjoint heat equation U + ϕ(, z X ϕ, z O(k R(k, (3.8 t ϕ + Aϕ =, t (,, ϕ( = ϕ. (3.9 For convenience, we introduce the free heat equation t z + A z =, t (,, z( = z. (3.3

26 6 Sylvain Ervedoza, Enrique Zuazua Using this function z, multiplying (3.3 by ϕ solution of (3.9, we immediately get z, ϕ( X = z(, ϕ O(k R(k. (3.3 Besides, estimate (3.5 implies that Setting the functional J can be rewritten as J(ϕ = k ( t, sbϕ(t dt z( R(k C z X. (3.3 Z = z z(, (3.33 U ds ϕ, Z O(k R(k. (3.34 Since Z R(k (see (3.3, using (3.8, we deduce that the functional J is continuous and coercive in the space O(k. Since it is strictly convex, it has a unique minimum ψ O(k which satisfies ( ψ O(k C Z R(k C z R(k + z X. (3.35 Writing the Euler-Lagrange equation satisfied by ψ, setting, for s (, S, ( v(s = k ( t, sbψ(t dt = B k ( t, sψ(t dt, (3.36 where ψ is the solution of the abstract heat equation (3.9 corresponding to ψ, we obtain that for all ϕ O(k, v(s, or, equivalently, k ( t, sbϕ(t dt U ds ϕ, Z O(k R(k =, (3.37 k ( t, sv(s ds, Bϕ(t U dt + ϕ(, z X ϕ, z O(k R(k =. (3.38 his implies that the function u(t = k ( t, sv(s ds, where v is as in (3.36, (3.39 is an admissible control function for (3.6: Indeed, multiplying (3.6 by ϕ solution of (3.9, we obtain that, for all ϕ O(k, u(t, Bϕ(t U dt + ϕ(, z X ϕ, z( O(k R(k =,

27 Sharp observability estimates for heat equations 7 which, according to (3.38, implies that z( = z. his control has to have some added advantages with respect to the standard ones since it has been derived using a subtle two-sided observability inequality. In particular, as we describe now, the controls obtained by this method have added regularity properties. Smoothness of controls. Choosing ϕ = ψ in (3.37, we obtain v(s U ds = ψ, Z O(k R(k. Estimates (3.35 and (3.8 then show that v L (,S;U C Z R(k C ( z R(k + z X. (3.4 In view of (3.39 and (3.4, estimates on k and its time derivatives (in t allow to recover estimates on the control u in H k (, ; U-norms. In particular, according to (3.9, for the functions k constructed in (3., for all p N, p t k L ((, (, S. herefore, the control function u in (3.39 satisfies the following: For all p N, there exist constants C p,, C p, such that u Hp (, ;U C p, v L (,S;U C p, Z R(k C p, ( z R(k + z X. (3.4 Note that this result is specific to the controls we have constructed using the kernels k. Indeed, the recent results in [6] show that the classical controls of minimal L (, ; U-norm fail to have such a property. However, remark that the controls constructed in [5] using a minimization process of a functional based on the Carleman weights also enjoy nice regularity properties. We refer to [5] for precise statements in that direction for heat equations, and to [8, Propositions and 3] for the Stokes equations. o better understand the nature of the control for the heat equation constructed by minimization of the functional J in (3.8, we analyze in more detail the function v in (3.36. For ϕ = j a jφ j, setting y(s = k ( t, sϕ(t dt, y( = and s y( = j a jφ j β j (k, identity (3.37 reads as where Y is given by v(s, By(s U ds s y(, Y X =, (3.4 Y = j z j β j (k Φ j, for Z = j z j Φ j. (3.43 Remark that Y X = Z R(k, sy( X = ϕ O(k.

28 8 Sylvain Ervedoza, Enrique Zuazua herefore, the map ϕ = j a j Φ j s y( = j a j Φ j β j (k is an isomorphism from O(k to X, and (3.4 is satisfied for any y solution of (.4 with initial data (y(, s y( = (y, y {} X. Besides, according to (3.36, v can be written as Bỹ, where ỹ(s is given by ỹ(s = k ( t, sψ(t dt. Of course, due to the properties of the kernel k, ỹ is a solution of the abstract wave equation (.4 with initial data (, s ỹ( {} X. Hence, by (3.4, ỹ = s ỹ( is a critical point of the functional J defined by J(y = for y X, where y is the solution of By(s U ds y, Y X, (3.44 ss y + Ay =, s (, S, (y(, s y( = (, y. (3.45 Due to (3.4, the functional J is continuous, coercive and strictly convex in X, and then has a unique minimizer, given by ỹ. o sum up, v, extended as an odd function on (, S, can be computed on (, S by minimization of a suitable functional J defined entirely on the wave equation. Actually, the function v(s can also be viewed as the control of minimal L (, S; U-norm such that the solution Y of { ss Y + AY = B v, s R, (3.46 Y (S =, s Y (S = satisfies the control requirement Y ( = Y, where Y is given by (3.43. (3.47 o see that, first remark that, when y =, solutions y of (.4 are odd in the time variable s. hus, J can be written as J(y = By(s U y, Y X. Writing the Euler-Lagrange equation satisfied by J at ỹ, one easily derives that Y solution of (3.46 with v = Bỹ satisfies (3.47. Once the control v of the abstract wave equation is characterized in this manner, the results obtained in [] can be easily modified to deal with this case (using in particular that, for any τ >, if y is a solution of (.4 with y =, so is y τ (s = (y(s + τ y(s + y(s τ/τ since y is odd. In particular, when B belongs to L(D(A /, U and B B p> L(D(A p

29 Sharp observability estimates for heat equations 9 (otherwise, a time-dependent smooth weight function η(s should be introduced within the functional J in (3.8, see [], it follows that for all l, there exists a constant C l such that v Hl (,S;U + Alỹ X C l A l Y X = C l A l Z R(k. (3.48 We emphasize that (3.48 concerns the regularity properties of v = v(s. he control u for the heat equation given by (3.39 is always smooth in time provided the functions p t k all belong to L ((, (, S, without these extra regularity assumptions on B (see (3.4. Also note that, as lained in [], the extra time-regularity properties of v imply extra space regularity properties. o sum up, we have proved the following: heorem 3. Let > and k L ((, (, S be a solution of (3. satisfying (3.. Assume that (3.4 holds for solutions of the abstract wave equation (.4. Let z X and z R(k (defined in (3.4. Construction of the control. he functional J in (3.8 has a unique minimizer ψ on O(k (defined in (3.7, which yields a control u solving the control problem (3.6-(3.7, given by u(t = and there exists a constant C such that k ( t, sk ( τ, sbψ(τ dτds, (3.49 ψ O(k + u L (, ;U C ( z X + z R(k. (3.5 Another way to compute the control u is the following: Find the minimizer ỹ X of the functional J defined on the waves (with Y as in (3.43-(3.33, set v(s = Bỹ(s. hen the control function u is given by (3.39. Smoothness properties of the control function. For any p N, if p t k L ((, (, S, u belongs to H p (, ; U and satisfies (3.4. If for some l N, A l Z R(k and BB p l L(D(A p, then A l ỹ X, (3.48 holds and v belongs to the space V l = l p= C p ([, S]; B D(A l p. (3.5 his automatically yields the following corollary: Corollary 3. Under the assumptions of heorem 3. and with the same notations, if for all p N, p t k L ((, (, S, BB p N L(D(A p, and for some l N, A l Z R(k, the source term B u satisfies B u C ([, ]; D(A l, (3.5 and therefore Z = z z, with z solution of the control problem (3.6-(3.7 and z as in (3.3, solution of the control problem satisfies Z + AZ = B u, t (,, Z( =, Z( = Z, (3.53 Z C ([, ]; D(A l+/. (3.54