EA David L Russell
email: luc.miller@math.polytechnique.fr

A DIRECT LEBEAU-ROBBIANO STRATEGY FOR THE OBSERVABILITY OF HEAT-LIKE SEMIGROUPS Dedicated to

Keywords: 2000 Mathematics Subject Classification. 93B07, 35B37 observability, null-controllability, fast control cost, heat semigroup

This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces final-observability of an evolution semigroup from a spectral inequality, i.e. some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. Contrary to the original Lebeau-Robbiano strategy, it does not have recourse to null-controllability and it yields the optimal bound of the cost when applied to the heat equation, i.e. c 0 exp(c/T ), or to the heat diffusion in potential wells observed from cones, i.e. c 0 exp(c/T β ) with optimal β. It also yields simple upper bounds for the cost rate c in terms of the spectral rate.

This paper also gives geometric lower bounds on the spectral and cost rates for heat, diffusion and Ginzburg-Landau semigroups, including on non-compact Riemannian manifolds, based on L 2 Gaussian estimates.

1.

Introduction. This paper concerns the so-called "Lebeau-Robbiano strategy" for the null-controllability of linear evolutions systems like the heat equation. The Lebeau-Robbiano strategy was originally devised for the heat flux on a bounded domain of R d observed from some open subset of this domain. It originally starts from the interior observability estimate for sums of eigenfunctions of the Dirichlet Laplacian proved by some Carleman estimates at the end of the nineties in joint papers of Lebeau with Jerison, Robbiano and Zuazua, cf. § 2.4.

In the last decade, many people have contributed applications, e.g. to nodal sets of sums of Laplacian eigenfunctions in [START_REF] Jerison | Nodal sets of sums of eigenfunctions, Harmonic analysis and partial differential equations[END_REF], to coupled wave and heat equations in the same domain in [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF], to the heat equation in unbounded domains in [START_REF]Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds[END_REF], to anomalous diffusions in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF], cf. § 4.1, to structural damping, e.g. the plate equation with square root damping, in [START_REF]The cost of fast non-structural controls for a linear elastic system with structural damping[END_REF][START_REF] Avalos | Boundary and localized null controllability of structurally damped elastic systems, Control methods in PDE-dynamical systems[END_REF], cf. § 4.2, to thermoelastic plates without rotatory inertia in [START_REF] Benabdallah | Null controllability of a thermoelastic plate[END_REF][START_REF]On the cost of fast controls for thermoelastic plates[END_REF][START_REF] Cokeley | Localized null controllability and corresponding minimal norm control blowup rates of thermoelastic systems[END_REF][START_REF]How violent are fast controls. III[END_REF], to the heat transmission problem in [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF], to diffusions in a potential well of R d in [START_REF]Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones[END_REF], cf. § 4.3, to the heat equation discretized in time or space in [START_REF] Zheng | Controllability of the time discrete heat equation[END_REF][START_REF] Boyer | Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations[END_REF], to semigroups generated by non-selfadjoint elliptic operators in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]. We also refer to the survey [START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF].

The Lebeau-Robbiano strategy was already stated in abstract settings with bounds on the cost of fast control of the form c 0 exp(c/T β ) in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF][START_REF]How violent are fast controls. III[END_REF]. Our goal is to retain the most general features of both papers while simplifying the proof to improve the estimate of the cost.

The paper [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] concerns semigroups generated by negative self-adjoint operators, introduces some notion of observability on spectral subspaces, cf. § 3.6. It links the exponent β in the fast control cost estimate to some exponent in this notion, but falls just short of the optimal exponent. It combines final-observability and null-controllability as in the original setting, but does not use Weyl's eigenvalues asymptotics, not even the discreteness of the spectrum of A. The assumptions brought out in [START_REF]On the cost of fast controls for thermoelastic plates[END_REF] and introduced as an abstract framework in [START_REF]How violent are fast controls. III[END_REF] allow generators which are not self-adjoint, but do not apply to the semigroups considered in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF][START_REF]Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones[END_REF]. Thus the notion of relative observability on growth spaces adopted in § 2.2 is a little more general. The paper [START_REF]How violent are fast controls. III[END_REF] achieves the breakthrough of reaching the exponent β = 1 which is optimal for the heat equation, but it adds approximate null-controllability as another layer to the strategy.

Here, the strategy goes directly from relative observability on growth spaces to the estimate of fast final-observability cost, and reaches the optimal exponents β for the observation from cones of heat diffusion in potential wells V (x) = |x| 2k , k ∈ N * , cf. § 4.3. Its sheer simplicity yields straightforward upper bounds of the rate c in the fast control cost estimate. Since it leaves null-controllability out, it can be seen as a shortcut to the original Lebeau-Robbiano strategy.

Section 2 introduces the abstract setting, states and proves the direct Lebeau-Robbiano strategy. The abstract result is connected to the original Lebeau-Robbiano setting in § 2.4. Section 3 gives further background, four lemmas which may be of independent interest and some open problems. Section 4 describes the application of the main result to the P.D.E. problems considered in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF][START_REF]The cost of fast non-structural controls for a linear elastic system with structural damping[END_REF][START_REF]Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones[END_REF]. Section 5 gives lower bounds on the rates in the cost of fast control and in the observability on spectral subspaces (e.g. the estimate for sums of eigenfunctions in § 2.4).

2. Setting and main result.

2.1. Observability cost. We consider the abstract differential equation φ(t) = Aφ(t), φ(0) = x ∈ E, t ≥ 0,

where A : D(A) ⊂ E → E is the generator of a strongly continuous semigroup (e tA ) t≥0 on a Hilbert space E. The solution is φ(t) = e tA x. Although we may think of A as a nonpositive self-adjoint operator with an orthonormal basis of eigenfunctions for example, cf. § 3.6, our setting has applications where A has no eigenvalues (e.g. in § 4.1 when M = R d ) or A is not a self-adjoint operator bounded from above (e.g. A does not even generate an analytic semigroup in § 4.2 for γ < 1/2, cf. [START_REF] Chen | Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1 2[END_REF][START_REF] Haraux | Analyticity and regularity for a class of second order evolution equations[END_REF]). We also consider an observation operator C ∈ L(D(A), F) admissible for this semigroup, cf. § 3.1, i.e. C is a continuous operator from D(A) with the graph norm to another Hilbert space F and satisfies (norms in E and F are both denoted • )

T 0 Ce tA x 2 dt ≤ Adm T x 2 , x ∈ D(A), T > 0.
(

N.b. the admissibility constant T → Adm T > 0 is nondecreasing. We may think of C as a bounded operator from E to itself for example.

We say that (A, C, T ) is observable at cost κ T > 0 if

e T A x 2 ≤ κ T T 0 Ce tA x 2 dt, x ∈ D(A). (3)
N.b. as T → +∞, √ κ T does not grow more than the semigroup and, e.g., when

A is nonpositive self-adjoint T → κ T is nonincreasing and decays at least like 1/T , cf. § 3.2. This final-observability of (1) through C in time T > 0 is equivalent to a controllability property for which κ T is the ratio of the size of the input annihilating the disturbance to the size of this disturbance, cf. § 3.2. We are interested in the asymptotic growth of κ T as T → 0 and think of κ T as the cost of fast control.

f (t) e tA x 2 -f (qt) x 2 ≤ t 0 Ce τ A x 2 dτ, x ∈ D(A), t ∈ (0, T ], (4)
holds with f (t) → 0 as t → 0 + , q ∈ (0, 1) and T > 0, then κ T ≤ 1/f ((1 -q)T ) for T ∈ (0, T ], i.e. the fast control cost does not grow more than the inverse of f .

Proof. Let T ≤ T . Let T 0 = T , T k+1 = T k -τ k , τ k = q k (1 -q)T , k ∈ N. The series τ k = T defines a disjoint partition ∪(T k+1 , T k ] = (0, T ]. Applying (4) to x = e T k+1 A y and t = τ k yields f (τ k ) e T k A y 2 -f (τ k+1 ) e T k+1 A y 2 ≤ T k T k+1 Ce tA y 2 dt, y ∈ D(A), k ∈ N.
Adding these inequalities yields, since the left hand side is a telescoping series,

f (τ 0 ) e T A y 2 -f (τ k ) e T k A y 2 ≤ T T k Ce tA y 2 dt, y ∈ D(A), k ∈ N.
Taking the limit k → ∞ completes the proof since f (τ k ) → 0 and the continuous function t → e tA y is bounded on the compact set [0, T ].

2.2.

Relative observability on growth subspaces. We assume that there is a nondecreasing family of semigroup invariant spaces E λ ⊂ E, λ > 0 (i.e. e tA E λ ⊂ E λ ⊂ E λ , t > 0, λ > λ) satisfying the semigroup growth property (namely some time-decay) with exponent ν ∈ (0, 1) and rate m ≥ 0

e tA x ≤ m 0 e mλ ν e -λt x , x ⊥ E λ , t ∈ (0, T 0 ), λ > 0. ( 5
)
We call them growth spaces. We think of them as spectral subspaces of A, i.e. σ(A E ⊥ λ ) ⊂ {z ∈ σ(A) | Re z ≤ -λ}, and we think of (5) as a spectrally determined growth property, cf. § 3.6.

We also assume that there is an observation operator C 0 ∈ L(D(A), F) satisfying the bound relative to C on growth spaces with exponent α ∈ (0, 1) and rate a > 0

C 0 x 2 ≤ a 0 e 2aλ α Cx 2 , x ∈ E λ , λ > 0. ( 6
)
We call C 0 a reference operator and the property (6) of C: observability on growth subspaces relatively to C 0 . We think of C 0 as a simple operator with a good estimate of fast control like the identity operator, cf. § 3.7.

Main result.

When the reference operator C 0 satisfies the observability cost estimate with exponent β > 0 and rate b > 0

e T A x 2 ≤ b 0 e 2b T β T 0 C 0 e tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ), (7)
we claim that C satisfies a similar estimate with exponent1 max β, α 1-α , ν 1-ν :

Theorem 2.2. Under the assumptions (5), ( 6) and ( 7)

with β = α 1-α = ν 1-ν , the system (A, C, T ) is observable at a cost κ T such that 2c = lim sup T →0 T β ln κ T < ∞.
More precisely, this rate c is bounded in terms of an implicitly defined s > 0:

c ≤ c * := (β + 1)b a + m β+1 β β β s (β+1) 2 β
, with s(s

+ β + 1) β = (β + 1)β β 2 β+1 b 1 β+1 a + m . (8)
Moreover, if the admissibility constant in (2) satisfies Adm T → 0 as T → 0, then there exists T > 0 such that κ T ≤ 4a 0 b 0 exp( 2c * T β ) for T ∈ (0, T ]. Since c > 0 for some "meaningful" example, cf. § 4.3, there are no lower β such that lim sup T →0 T β ln κ T < ∞ under these assumptions.

N.b. the condition Adm T → 0 as T → 0 for the better bound in theorem 2.2 holds for example when C is bounded from E to F, cf. § 3.1.

Corollary 1. Under the same assumptions as theorem 2.2, the cost rate c is bounded more explicitly in the following cases, with the abbreviation a m = a + m:

i. If (6) holds with α = 1 2 (i.e. β = 1) then c * = 4b 2 a m + 2 √ b - √ a m -4
.

ii. If [START_REF] Boyer | Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations[END_REF] holds for any b then c ≤ a β+1 m (β + 1) β(β+1) β -β 2 . iii. If (6) holds for any a then c ≤ b.

iv. If b > a β+1 m (β + 1) β(β+1) β -β 2 then c * ≤ b a m 1 α b (1-α) 2 a 1-α m - (β + 1) α β α 2 -1
.

N.b. (ii) applies to the identity operator as reference operator C 0 , cf. § 3.7. N.b. if [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] holds with m = 0 or for any m > 0 then a m can be replaced by a. Theorem 2.2 for α = 1 2 (i.e. β = 1) and m = 0 is due to Seidman with some less precise and less simple cost rate bound than (8); e.g., in the case (ii) with β = 1 and m = 0 which applies to the original setting in § 2.4, [46, theorem 2.4] proves 2 c ≤ 8a 2 instead of c ≤ 4a 2 , and does not state (i).

With the exponential bound b 0 e bT -β in (7) replaced by a polynomial bound b0 T b

(so that (ii) applies), the papers [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF][START_REF]On the cost of fast controls for thermoelastic plates[END_REF] only prove lim sup T →0 T β ln κ T < ∞ for β > α 1-α , hence fall short of the optimal exponent. ] and [24, theorem 14.6] prove [START_REF] Benabdallah | Null controllability of a thermoelastic plate[END_REF] with exponent α = 1 2 using the semiclassical local elliptic Carleman estimates of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. In this case [START_REF] Benabdallah | Null controllability of a thermoelastic plate[END_REF] writes as

M |v(x)| 2 dx ≤ a 0 e 2a √ λ Ω |v(x)| 2 dx, λ > 0,
for any sum of eigenfunctions v = µ≤λ ϕ µ , -∆ϕ µ = µϕ µ . Since [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] holds with m = 0 (cf. § 3.6) and ( 7) holds with any b > 0 (cf. § 3.7), the corollary 1(ii) of theorem 2.2 proves that this estimate on sums of eigenfunctions implies the bound on fast control for any c > 4a 2 :

M |φ(T, x)| 2 dx ≤ κ T T 0 Ω |φ(t, x)| 2 dx dt, κ T = c 0 e 2c T , T ∈ [0, T 0 ],
for any solution of the Cauchy problem ∂ t φ -∆φ = 0, φ(0, •) ∈ L 2 (M ).

The exponent α = 1 2 is always sharp in this setting as proved in [24, proposition 14.9] (cf. also [30, proposition 5.5]), i.e. the above estimate on sums of eigenfunctions implies a > 0. Theorem 5.3 improves this into a ≥ sup y∈M dist(y, Ω)/2. N.b. the cost estimate in [33, theorem 2.1] (cf. also theorem 5.1) combined with c > 4a 2 given by theorem 2.2 only proves a lower bound on a which is worse by a factor 2. This could mean that "something is lost" in the proof of theorem 2.2.

In this general setting, the cost upper bound lim sup T →0 T ln κ T < ∞ is due to Seidman (it is deduced in [START_REF]How violent are fast controls. III[END_REF] from the above estimate on sums of eigenfunctions, and the first such upper bound was proved in [START_REF]Two results on exact boundary control of parabolic equations[END_REF]) and the cost lower bound lim inf T →0 T ln κ T ≥ sup y∈M dist(y, Ω) 2 /2 is due to [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] (the first lower bound was proved in dimension one in [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF]). In the Euclidean case, this upper bound was proved in [START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF] by global Carleman estimates with singular weights of the Èmanuilov type (with a less precise lower bound). Under the geometrical optics condition on Ω, a more precise upper bound is deduced in [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] by the control transmutation method from the observability of the wave group in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]: lim sup T →0 T ln κ T ≤ c * L 2 Ω , where L Ω is the length of the longest generalized geodesic in M which does not intersect Ω, and c * is determined by a one-dimensional observability estimate for which c * ≤ (2

f (T ) e T A x 2 -g(T ) x 2 ≤ T 0 Ce tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ], (9)
holds with f (T ) = f 0 exp(-2/(d 2 T ) β ) and g(T ) = g 0 exp(-2/(d 1 T ) β ), where f 0 , g 0 , d 1 < d 2 are positive, then for all d ∈ (0, d 2 -d 1 ) there exists T ∈ (0, T 0 ] such that κ T ≤ f -1 0 exp(2/(dT ) β ) for T ∈ (0, T ]. Moreover, if g 0 ≤ f 0 then we may take d = d 2 -d 1 and T = T 0 .
Proof. To apply lemma 2.1, we compute the least q such that g(T ) ≤ f (qT ) for all T ∈ (0, T ]. We find q = d1 d2 h(T ) with h(T

) = (1 + inf t∈(0,T ) t β d β 1 ln f0 g0 ) -1 β
where the parenthesis is 1 when g 0 ≤ f 0 and positive when T is small enough. Now

κ T ≤ 1 f ((1-q)T ) = 1 f0 exp( 1 (d3T ) β ) with d 3 = d 2 -d 1 h(T ) → d 2 -d 1 as T → 0.
We proceed with the proof of theorem 2.2. For ease of exposition, we start with the case m = 0 in (5) and complete the general case at the very end of § 2.5.

Plugging ( 6) in ( 7) yields

e τ A φ 2 ≤ a 0 b 0 e 2aλ α + 2b τ β τ 0 Ce tA φ 2 dt, φ ∈ E λ , τ ∈ (0, T 0 ). ( 10
)
Given x ∈ D(A) and T ∈ (0, T 0 ), we introduce an observation time τ = εT with ε ∈ (0, 1), a spectral threshold λ defined by (rλ) α = 1 τ β with r > 0, the orthogonal projection of x on E λ denoted x λ , and x ⊥ λ = x -x λ . Since E λ is semigroup invariant, we may apply [START_REF] Chen | A mathematical model for linear elastic systems with structural damping[END_REF] to φ = e (1-ε)T A x λ and obtain:

e T A x λ 2 ≤ 1 4f (T ) T (1-ε)T Ce tA x λ 2 dt, f (T ) = 1 4a 0 b 0 exp - 2 T β a + br α r α ε β . (11)
We put the factor 4 in the definition of f because we shall use twice the inequality:

y + z 2 ≤ 2( y 2 + z 2 ), y ∈ E, z ∈ E. (12)
Using ( 12) then (2) yields

T (1-ε)T Ce tA x λ 2 dt ≤ 2 T (1-ε)T Ce tA x 2 dt + 2 Adm εT e (1-ε)T A x ⊥ λ 2 . ( 13
)
Using ( 12) again, then [START_REF] Chen | Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1 2[END_REF] and finally (13) yields

f (T ) e T A x 2 ≤ T (1-ε)T Ce tA x 2 dt + Adm εT e (1-ε)T A x ⊥ λ 2 + 2f (T ) e T A x ⊥ λ 2 .
Applying ( 5) with m = 0 to x ⊥ λ yields

f (T ) e T A x 2 -m 2 0 Adm εT e -2(1-ε)T λ + 2f (T )e -2T λ x ⊥ λ 2 ≤ T 0 Ce tA x 2 dt.
Since x ⊥ λ ≤ x , Adm εT ≤ Adm T0 and f (T ) ≤ f (T 0 ), we deduce the approximate observability estimate

f (T ) e T A x 2 -m 2 0 (Adm T0 +2f (T 0 )) e -2(1-ε)T λ x 2 ≤ T 0 Ce tA x 2 dt. ( 14
)
Recalling that here β = α 1-α so that T λ = 1/(rε β/α T β ), this proves [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF] with [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF] still holds with g 0 = f 0 with a smaller T 0 . Therefore lemma 2.3 proves the theorem for c * = (d 2 -d 1 ) -β , for any ε ∈ (0, 1) and r > 0. Now, introducing for convenience γ = 1 β and s = ε 1-ε , we are left with maximizing with respect to r > 0 and s > 0:

f 0 = 1 4a 0 b 0 , g 0 = m 2 0 (Adm T0 +2f (T 0 )) , d 2 = ε (ar -α + b) 1 β and d 1 = ε 1 α r 1 -ε 1 β . As T 0 → 0, if Adm T0 → 0 then g 0 → 0, hence
d a,b (r, s) = d 2 -d 1 = ε (ar -α + b) γ - r 1 -ε γ ε 1 α = s s + 1 r γ h -γ (r) -s γ ,
where h(r) = ar

γ γ+1 + br, since 1 α = 1 + γ, ε = s s+1 and 1 -α = 1 β+1 = γ γ+1 . N.b. d a,b (r, s) > 0 for r small enough already proves c < ∞.
The optimality condition ∇d a,b = 0 writes successively, abbreviating h = h(r),

γr γ-1 (h -γ -s γ ) = r γ γh h γ+1 , 1 (s+1) 2 (h -γ -s γ ) = γs γ s+1 , h γ+1 (h -γ -s γ ) = rh = γ γ+1 h + 1 -γ γ+1 br, h = (γs γ (s + 1) + s γ ) -1 γ = 1 s (γs + γ + 1) -1 γ .
Plugging the last equation (h in terms of s) in the former yields r in terms of s:

br γ + 1 = h γ+1 h -γ -s γ - γ γ + 1 h = h γ+1 h -γ γ + 1 -s γ = γ γ + 1 (sh) γ+1 , hence r = γb -1 (γs + γ + 1) -γ+1 γ .
Plugging this once in h(r) in terms of s yields

γs + γ + 1 = s(γs + γ + 1) γ+1 γ h = s γ br h = sγ a b r -1 γ+1 + 1 .
Simplifying γs and plugging r in terms of s again yields the equation for s in ( 8):

s γ (γs + γ + 1) = γ + 1 a γ b γ γ 2 γ+1
, which has a unique solution since the L.H.S. increases from 0 to +∞ as s does. We still denote s this solution. The corresponding r = γb -1 (γs + γ + 1)

-γ+1 γ satisfies r 1 γ+1 = γ b 1 γ+1 (γs + γ + 1) -1 γ = s a b γ γ+1 .
The second equation of the first system traducing the optimality condition ∇d a,b = 0 yields:

d a,b (r, s) = s s + 1 r γ h -γ (r) -s γ = γs γ+1 r γ = γs (γ+1) 2 a b γ γ + 1 γ(γ+1) . Now c * = d -β a,b (r, s) is (8) with m = 0 since 1 + γ = β+1 β , γ+1 γ = β + 1 and (γ+1) 2 γ = (β+1) 2 β
. Corollary 1 in the case a m = a is deduced by the following arguments.

i. The positive solution of the quadratic equation in [START_REF] Bragg | Related problems in partial differential equations[END_REF]

is s = 1 + 2 √ b a -1. ii. Eliminating b from (8) yields c * = (a/(β + 1)) β+1 β -β 2 (s + β + 1) (β+1) 2
, and the implicit equation yields s → 0 as b → 0. iii. Eliminating a from (8) yields c * = b(s + β + 1) β+1 /s β+1 , and the implicit equation yields s → ∞ as a → 0.

iv. The easiest lower bound for s is s + β + 1 ≥ (β + 1)

1 β β ( β β+1 ) 2 b 1 (β+1) 2 a 1 β+1
, obtained by plugging s + β + 1 ≥ s in its implicit equation. We now complete the general case m = 0 in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. The proof uses (5) only once: in the equation before [START_REF] Davies | Heat kernel bounds, conservation of probability and the Feller property[END_REF]. We may divide this equation by e mλ α and keep the same right hand side since e -mλ α ≤ 1. This yields [START_REF] Davies | Heat kernel bounds, conservation of probability and the Feller property[END_REF] with f (T ) replaced by f (T )e -mλ α . This amounts to replacing a by a + m in the definition of d 2 after [START_REF] Davies | Heat kernel bounds, conservation of probability and the Feller property[END_REF] and therefore in the conclusion (8).

Comments.

3.1. Admissibility. Any C ∈ L(E, F) satisfies the admissibility condition (2) with Adm T = T C 2 . The more general setting in § 2.1 is canonical (cf. [START_REF] Weiss | Admissible observation operators for linear semigroups[END_REF]) and required in many P.D.E. problems, e.g. when the heat flux is observed on the boundary rather than an open subset of the domain. Although it should be sufficient for any P.D.E. problems, it might be useful to circumvent the admissibility assumption: Proof. In the proof of theorem 2.2, the admissibility condition ( 2) is only used once, for x = x ⊥ λ orthogonal to the growth space E λ , in this manner: [START_REF] Edward | Internal null-controllability for a structurally damped beam equation[END_REF] and this only affects the definition of the function g(T ) = g 0 exp(-2/(d 1 T ) β ) used in [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF]. Recall that ε ∈ (0, 1) and r > 0 have been fixed (in order to maximize d a,b ). We shall prove, for any δ ∈ (0, 1) small enough, any g 1 > 0 and some smaller T 0 ,

T (1-ε)T Ce tA x 2 dt ≤ Adm T0 m 2 0 e 2mλ ν e -2(1-ε)T λ x 2 , x ⊥ E λ , T ∈ (0, T 0 ),
T (1-ε)T Ce τ A x 2 dτ ≤ g 1 e 2mλ ν e -2(1-(1+δ)ε)T λ x 2 , x ⊥ E λ , T ∈ (0, T 0 ). ( 17
)
Indeed replacing ( 16) by ( 17), ( 9) still holds with g 0 and d 1 replaced by g 1 and

d 1,δ = ε 1 α r 1-(1+δ)ε 1 β . Since d 1,δ → d 1
as δ → 0 this will not affect the range of d obtained by applying lemma 2.3, nor the conclusion of theorem 2.2.

With the graph norm on D(A), C ∈ L(D(A), F) means Cx ≤ C ( x + Ax ). We only need to prove [START_REF] Fattorini | Ordinary differential equations in linear topological spaces I[END_REF] with C replaced by A since the proof of ( 17) with C replaced by the identity is the same, only shorter. We use the small parameter δ ∈ (0, 1) to decompose the lower integration bound in [START_REF] Fattorini | Ordinary differential equations in linear topological spaces I[END_REF] in this geometric way: [START_REF] Dolecki | A general theory of observation and control[END_REF], τ β ln Ae τ A ≤ δ 2β+2 , τ ∈ (0, T 0 ), for a smaller T 0 . This with τ = τ 3 and (5) yield

(1 -ε)T = (1 -(1 + δ)ε)T + (1 -δ)δεT + δ 2 εT = τ 1 + τ 2 + τ 3 . According to
Ae tA x ≤ e δ 2β+2 τ β 3 e (t-τ3)A x ≤ e δ 2 (εT ) β m λ e -(t-τ3)λ x ≤ m λ e δ 2 (εT ) β e -(τ1+τ2)λ x ,
for all t ∈ ((1 -ε)T, T 0 ), where m λ = m 0 e mλ ν . Recalling τ 2 λ = (1-δ)δε rε β/α T β and bounding the length of the integration interval by T 0 , the proof of [START_REF] Fattorini | Ordinary differential equations in linear topological spaces I[END_REF] with C replaced by A now reduces to

T 0 m 2 0 e 2δ 2 (εT ) β e -2τ2λ = T 0 m 2 0 e -2δc δ T β ≤ g 1 , T ∈ (0, T 0 ),
where c δ → ε/(rε β/α ) > 0 as δ → 0. This does hold for T 0 ≤ g 1 /m 2 and any δ small enough for c δ to be positive.

The idea of dispensing with the admissibility assumption is due to Marius Tucsnak and Gerald Tenenbaum in the case where A is a nonpositive self-adjoint operator with an orthonormal basis of eigenfunctions. Indeed, that A is nonpositive self-adjoint implies that A generates a bounded analytic semigroup, which is equivalent to the usual time smoothing effect, sup t>0 tAe tA < ∞, which implies the weaker effect (15) assumed in lemma 3.1. N.b. although A in § 4.2 for γ < 1/2 does not generate an analytic semigroup, it is proved in [23, theorem 4.2] that sup t>0 t 1 2γ Ae tA < ∞, which also implies (15).

Controllability cost.

From the definition of κ T in (3), we have, for T < T , κ T ≤ e (T -T )A 2 κ T . This justifies our claim in § 2.1 that √ κ T does not grow more than the semigroup as T → +∞ and does not increase when the semigroup is contractive. Moreover, if κ t ≤ g(t), t ∈ (0, T ], g nonincreasing, then κ t ≤ M 2 0 g(t), t ∈ (0, T ], with M 0 = sup s∈(0,T -T ) e sA < ∞. This justifies that we restrict to some bounded intervals (0, T ] in the statements of our results. When the semigroup is bounded by M = sup t≥0 e tA , the cost bound sup T >T κ T ≤ M 2 κ T improves into the decay: sup T >T T κ T ≤ 2M 2 T κ T . Indeed, let τ k = kT , k ∈ N, and n = T /T , so that

τ n ≤ T < τ n+1 . Since e T A x ≤ M e τ k A x for k ≤ n, n e T A x 2 ≤ M 2 κ T n k=1 τ k τ k-1 Ce tA x 2 dt = M 2 κ T τn 0 Ce tA x 2 dt. Since τ n ≤ T ≤ τ 2n , the proof of κ T ≤ M 2 κ T /n ≤ M 2 κ T (2T /T ) is completed.
The dual problem to the final-observability of ( 1) is the null-controllability of A,C,T ) is observable at cost κ T then, for all f 0 , there is a u such that f (T ) = 0 and [START_REF] Dolecki | A general theory of observation and control[END_REF]). The study of the cost of fast controls was initiated by Seidman in [START_REF]Two results on exact boundary control of parabolic equations[END_REF] with a result on the heat equation obtained by Russell's method in [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF]. We refer to the surveys [START_REF]On uniform null controllability and blowup estimates[END_REF][START_REF]On exponential observability estimates for the heat semigroup with explicit rates[END_REF] and the more recent paper [START_REF] Tenenbaum | New blow-up rates for fast controls of Schrödinger and heat equations[END_REF]. An application to reachability is given in § 3.4.

ḟ (t) = A * f (t) + Bu(t), f (0) = f 0 ∈ E, t ≥ 0, (18)
T 0 u(t) 2 dt ≤ κ T f 0 2 (cf.
3.3. Integrated observability estimate. Lemma 2.1 can be seen as the discrete version of the following lemma which has been used with f (t) = exp(-c/t) when proving observability by some parabolic global Carleman estimates (cf. e.g. [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Fernández-Cara | The cost of approximate controllability for heat equations: the linear case[END_REF]). Lemma 3.2. In the setting of § 2.1, if the integrated observability estimate

T 0 f (t) e tA x 2 dt ≤ T 0 Ce tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ), ( 19
)
holds with T 0 > 0 and f an increasing function such that f (t) → 0 as t → 0 + , then, for any ε ∈ (0, 1), κ T ≤ M εT /(εT f ((1 -ε)T )), T ∈ (0, T 0 ), with M εT ≤ M T := sup t∈[0,T ] e tA 2 ≤ M T0 < ∞, i.e. the growth of the fast control cost is almost bounded by the inverse of f . Conversely, if (3) holds for T ∈ (0, T 0 ) then (19) holds with f (t) = 1/(T 0 κ t ).

Proof. The implication results from e T A x 2 = e (T -t)A e tA x 2 ≤ M T e tA x 2 and f (t) ≤ f (T ) for t ∈ (0, T ): for ε ∈ (0, 1),

εT f ((1 -ε)T ) e T A x 2 ≤ M εT T T (1-ε) f (t) e tA 2 dt ≤ M εT T 0 Ce tA 2 dt. Writting (3) as κ -1 τ e τ A x 2 ≤ τ 0
Ce tA x 2 dt, the converse results from integrating:

T 0 τ 0 Ce tA x 2 dtdτ ≤ T 0 T 0 Ce tA x 2 dtdτ = T T 0
Ce tA x 2 dt.

Reachability.

As the input u varies, the final state f (T ) of ( 18) spans the set of states which are reachable from f 0 in time T , denoted R(T, f 0 ). Assuming (A, C, T ) is observable for all T > 0, the usual duality in § 3.2 implies that this reachability set R = R(T, f 0 ) does not depend on T and f 0 (by an argument due to Seidman in [START_REF] Seidman | Time-invariance of the reachable set for linear control problems[END_REF], cf. [38, footnote 7]) and satisfies e tA (E) ⊂ R, t > 0.

The following lemma provides further information on the reachability set when a cost estimate as in theorem 2.2 is available. Lemma 3.3. In the setting of § 2.1, assume A is self-adjoint and σ(A) ⊂ (-∞, λ 1 ], and consider the fractional powers

A β = -(-A + λ 1 ) β , β > 0.
For all exponents β > 0, α = β β+1 , and rates b > 0, c > b(β + 1), a > (bβ)

1
β+1 /α, for all T 0 > 0, there exists c 0 > 0 such that

e aAα x 2 ≤ c 0 e 2c T β T 0 e -2b t β e tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ). ( 20
)
If (A, C, T ) is observable at a cost κ T such that 2b 0 = lim sup T →0 T β ln κ T < ∞, then the reachability set satisfies e aAα (E) ⊂ R for α = β β+1 and a > (b 0 β)

1 β+1 /α.
Proof. We first deduce the reachability statement from the previous one. For any b > b 0 , (3) holds with κ T = exp(2b/T β ), T ∈ (0, T 0 ), for T 0 small enough. The converse in lemma 3.2 proves that the integral in ( 20) is bounded by some multiple of the integral in [START_REF] Avalos | Boundary and localized null controllability of structurally damped elastic systems, Control methods in PDE-dynamical systems[END_REF]. Plugging this in (20) yields a c 1 > 0 such that

e aAα x 2 ≤ c 1 e 2c T β T 0 Ce tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ).
The same duality argument (cf. [15, (3.22)]) deduces e aAα (E) ⊂ R(T, 0) = R. Given x ∈ D(A) and T ∈ (0, T 0 ), using the spectral measure dE x (λ) of A for x:

e aAα x 2 = σ(A) e -2a(λ1-λ) α dE x , T 0 f (t) e tA x 2 dt = T 0 σ(A)
f (t)e 2tλ dE x dt.

Hence (20) reduces to

T 0 e -2j λ (t) dt ≥ 1 c0 e -2c T β e -2a(λ1+λ) α for λ ≥ -λ 1 , and further to (by changing λ into λ -λ 1 , with c 1 = c 0 min 1, e 2T0λ1 ):

T 0 e -2j λ (t) dt ≥ 1 c 1 e -2c T β e -2aλ α , T ∈ (0, T 0 ), λ ≥ 0, (21)
where

j λ (t) = b t β + tλ satisfies j λ (t) ≥ j λ (t λ ) = t λ λ α , t λ = bβ λ 1 β+1
, λ > 0.

On the one hand, if t λ < T , then Concerning the heat semigroup in § 2.4, as a corollary to the cost upper bound in § 3.2 under the geometrical optics condition, this lemma with β = 1 proves that e -a √ -∆ φ 0 is reachable for a > √ 3L Ω , φ 0 ∈ L 2 (M ), cf. [38, corollary 10]. In dimension one a better result is due to Fattorini and Russell, cf. [18, (3.19)]: if M is a segment of length L controlled from one endpoint then e -a √ -∆ φ 0 is reachable for all a > L, φ 0 ∈ L 2 (M ) (this cannot be proved by the same method for a < L, cf. [18, (3.20)]). Whether "the optimal" rate a such that e -a √ -∆ (L 2 (M )) ⊂ R can be expressed geometrically in the general setting of § 2.4 is an open question, e.g. is it sup y∈M dist(y, Ω)? 3.5. Approximate observability. The following lemma clarifies the connection of (4) in lemma 2.1 to approximate controllability, and therefore to [START_REF]How violent are fast controls. III[END_REF]. Lemma 3.4. Given the time T > 0, the cost κ > 0 and the approximation rate ε > 0, the following two properties are equivalent.

T 0 e -2j λ (t) dt ≥ t λ δt λ e -2j λ (t) dt ≥ (1 -δ)t λ e -2j λ (δt λ ) , δ ∈ (0, 1), with j λ (δt λ ) = 1 βδ β + δ t λ λ = a δ λ α , a δ = (bβ) 1 β+1 1 βδ β + δ δ→1 ---→ (bβ) 1 β+1 α , hence (
i. Approximate observability of (A, C, T ):

e T A x 2 ≤ κ T 0 Ce tA x 2 dt + ε x 2 , x ∈ D(A).
ii. Approximate null-controllability of (18):

∀f 0 ∈ E, ∃u ∈ L 2 ([0, T ], F), 1 κ T 0 u(t) 2 dt + 1 ε f (T ) 2 ≤ f 0 2 .
Proof. Consider the strictly convex C 1 functional J defined on E by density as

J(x) = κ 2 T 0 Ce tA x 2 dt + ε 2 x 2 + e T A x, f 0 , x ∈ D(A).
Property (i) implies J(x) ≥ 1 2 e T A x 2 + e T A x, f 0 , hence J is coercive. Therefore J has a unique minimizer ψ 0 ∈ E, i.e. J(ψ 0 ) = inf x∈E J(x), and

0 = ∇J(ψ 0 ) = κ T 0 e tA * BCe tA ψ 0 dt + εψ 0 + e T A * f 0 .
This equation also says that the input u(t) = κCe tA ψ 0 in (18) yields the final state f (T ) = -εψ 0 . In terms of these u and f (T ), ∇J(ψ 0 ), ψ 0 = 0 writes

1 κ T 0 u(t) 2 dt + 1 ε f (T ) 2 = κ T 0 Ce tA ψ 0 2 dt + ε ψ 0 2 = -e T A ψ 0 , f 0 . ( 22
)
Plugging this in property (i) yields e T A ψ 0 2 ≤ -e T A ψ 0 , f 0 ≤ e T A ψ 0 f 0 . Hence e T A ψ 0 ≤ f 0 . This allows to bound [START_REF]A lower bound of the norm of the control operator for the heat equation[END_REF] as in property (ii).

Conversely, taking the duality product of x ∈ D(A) with a final state of ( 18)

f (T ) = T 0 e tA * Bu(t)dt + e T A * f 0 yields f 0 , e T A x = - T 0 u, Ce T A x dt + f (T ), x .
Using the Cauchy-Schwarz inequality in E, L 2 (0, T ) and R 2 yields

| f 0 , e T A x | 2 ≤ 1 κ T 0 u(t) 2 dt + 1 ε f (T ) 2 κ T 0 Ce tA x 2 dt + ε x 2 .
Choosing f 0 = e T A x completes the proof that property (ii) implies property (i).

3.6. Growth condition, normal semigroups and spectral spaces. If the growth spaces are closed and satisfy sup Re σ(A E ⊥ λ ) = -λ, then the growth condition (5) for a given λ > 0 as t → ∞ says that the restriction of the semigroup to E ⊥ λ satisfies the spectral bound equal growth bound condition (this condition is satisfied by any eventually norm-continuous semigroup, e.g. differentiable semigroup, e.g. A is self-adjoint and bounded from above). Yet the growth bound of this restricted semigroup for small t may get worse as λ → ∞. This justifies allowing m = 0 in the growth condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. E.g. the growth condition for some non-selfadjoint elliptic operators A stated in [27, Proposition 4.12], which comes naturally from the Laplace representation of the semigroup and resolvent estimates, is precisely of the form [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] for some m > 0. When A is only mildly non-normal as in § 4.2, the loss is only polynomial in λ, hence (5) holds for any m > 0.

On the contrary, for a normal semigroup (i.e. A is normal and the real part of its spectrum is bounded from above, e.g. A is negative self-adjoint as in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF]) the natural growth spaces are its spectral spaces and ( 5) always holds with m 0 = 1 and m = 0. Indeed, it has a spectral decomposition E (a.k.a. projection-valued measure) which commutes with any operator which commutes with A, defines spectral projections E λ = E({z ∈ σ(A) | Re z > -λ}) and spectral spaces E λ = E λ (E), and provides the integral representation e tA = σ(A) e tz dE(z) hence this growth condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]. N.b. for unitary groups (i.e. A is skew-adjoint, e.g. Schrödinger or wave equations) 5) is trivial but ( 6) is never satisfied in applications.

E λ = E λ (E) = E, λ > 0, so that (
If there is an orthonormal basis {e n } of E such that -Ae n = λ n e n , then the spectral spaces are just spanned by linear combinations of normalized eigenfunctions E λ = Span {e n } λn<λ and ( 6) is an estimate on sums of eigenfunctions of A.

For A = ∆ on E = L 2 (R d ), the spectral decomposition is the Fourier transform: 6), e.g. by complex analysis.

f (-∆)φ(ξ) = f (|ξ| 2 ) φ(ξ), φ ∈ L 2 (R d ), thus φ ∈ E λ just means φ(ξ) = 0 for |ξ| 2 > λ, i.e.

Reference operator. Any A satisfies the fast control cost estimate

e T A x 2 ≤ M T T T 0 e tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ), ( 23
)
with M T = sup t∈[0,T ] e tA 2 ≤ M T0 < ∞. Thus the cost estimate ( 7) holds for any exponent β > 0 and rate b > 0 when C 0 is the identity operator. For a system of coupled P.D.E., C 0 can be the observation of a single component as in § 4.2, e.g. the operator C M in [START_REF]On the cost of fast controls for thermoelastic plates[END_REF]: for this reference operator, ( 5) with m = 0, [START_REF] Benabdallah | Null controllability of a thermoelastic plate[END_REF], and [START_REF] Boyer | Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations[END_REF] with any b > 0, are stated in this form in [START_REF]On the cost of fast controls for thermoelastic plates[END_REF]Propositions 4,[START_REF] Avalos | Boundary and localized null controllability of structurally damped elastic systems, Control methods in PDE-dynamical systems[END_REF][START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF] respectively. The assumptions [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] with m = 0 and (6) are called [H] in the abstract framework of [START_REF]How violent are fast controls. III[END_REF].

"

Converse" to the main result. The following lemma is a very partial converse to theorem 2.2: only for sequences of eigenfunctions of A and C 0 = id. Lemma 3.5. Assume that (A, C) satisfies the observability cost estimate with exponent β > 0 and rate b > 0

e T A x 2 ≤ b 0 e 2b T β T 0 Ce tA x 2 dt, x ∈ D(A), T ∈ (0, T 0 ). ( 24
)
Any sequence (e n ) in D(A) such that -Ae n = λ n e n and lim λ n = +∞, must satisfy

e n 2 ≤ b 0 2λ n e 2aλ α n Ce n 2 , α = β β + 1 , a = β + 1 β α b 1 β+1 , λ n large enough. ( 25
)
In particular, if the sequence satisfies for some exponent α > 0 and rate a > 0:

e n 2 ≥ a 0 e 2aλ α n Ce n 2 , λ n large enough, ( 26
)
then the observability cost in (3) satisfies lim sup T →0 T β ln κ T > 0 with β = α 1-α .

Proof. Applying ( 24) to x = e n yields e -2T λn e n 2 ≤ b 0 e

2b T β T 0 Ce n 2 e -2tλn dt, hence e n 2 ≤ b 0 2λ n e 2h(T ) Ce n 2 , with h(T ) = b T β + T λ n . Minimizing h yields h(T n ) = β+1 β α b 1 β+1 λ α n at T n = βb λn 1 β+1 with T n < T 0 for λ n large enough.
We prove the last statement of lemma 3.5 by contradiction. If the observability cost in (3) satisfies lim sup T →0 T β ln κ T = 0 with β = α 1-α , then ( 24) holds for any b > 0 with T 0 small enough, hence [START_REF] Kannai | Off diagonal short time asymptotics for fundamental solutions of diffusion equations[END_REF] holds for any a > 0, which refutes (26).

+ • • • + ∂ 2 ∂x 2 d .
In this application, the state and input spaces are E = F = L 2 (M ), the growth spaces are the spectral spaces of § 3.6, the reference operator C 0 is the identity operator and the observation operator C is the multiplication by the characteristic function χ Ω of an open subset Ω = ∅ of M , i.e. it truncates the input function outside the control region Ω. If M is not compact, assume that Ω is the exterior of a compact set K such that K ∩ Ω ∩ ∂M = ∅.

For A = ∆, (6) holds with exponent α = 1 2 , cf. § 2.4 for compact M , and [34] otherwise. Hence for A = -(-∆) γ , (6) holds with exponent α = 1 2γ . Applying theorem 2.2 improves on [37, theorem 2]: Theorem 4.1. For all γ > 1/2, the anomalous diffusion:

∂ t φ + (-∆) γ φ = χ Ω u, φ(0) = φ 0 ∈ L 2 (M ), u ∈ L 2 ([0, T ] × M ), is null-controllable in any time T > 0. Moreover the cost κ T (cf. § 3.2) satisfies lim sup T →0 T β ln κ T < ∞ with β = 1 2γ-1 .
When the manifold M is the whole Euclidean space R d , the fractional Laplacian -(-∆) γ with γ ∈ (0, 1] generates the rotationally invariant 2γ-stable Lévy process. For γ = 1 this process is the Brownian motion B t on R d , and for γ < 1 it is subordinated to B t by a strictly γ-stable subordinator T t , so that it writes B Tt . The convolution kernels of the corresponding semigroups are the rotationally invariant Lévy stable probability distributions, in particular the Gaussian distribution for γ = 1 and the Cauchy distribution for γ = 1/2. For γ < 1 these distributions have "heavy tails", i.e. far away they decrease like a power as opposed to the exponential decrease found in the Gaussian, which accounts for the "superdiffusive" behavior of the semigroup. The more restrictive range γ ∈ (1/2, 1) is the most widely used to model anomalously fast diffusions, and it turns out that the controllability result theorem 4.1 applies to this range of fractional superdiffusions only.

When the manifold M is a domain of the Euclidean space R d , the Markov process generated by the fractional Dirichlet Laplacian -(-∆) γ with γ ∈ (0, 1] can be obtained by killing the Brownian motion on R d upon exiting the domain then subordinating the killed Brownian motion by the subordinator T t introduced above.

Structural damping.

Let A be a positive self-adjoint and boundedly invertible operator on another Hilbert space H (with norm still denoted • ). Let D(A) denote its domain with the norm ζ → Aζ . Since -A is normal, we may consider its spectral decomposition H, its spectral projections H µ = H({z ∈ σ(A) | Re z < µ}) and spectral spaces H µ = H µ (H). (cf. § 3.6). We consider an observation operator C in L(D(A), F) satisfying observability on H µ relative to the identity operator:

z 2 ≤ d 0 e 2dµ δ Cz 2 , z ∈ H µ , µ > 0, (27)
and the corresponding control operator

B = C * ∈ L(F, D(A) ) (D(A) denotes the dual space of D(A) in H).
To give a precise meaning to the solution of the structurally damped system

ζ(t) + ρA 2γ ζ(t) + A 2 ζ(t) = Bu(t), ζ(0) = ζ 0 ∈ D(A), ζ(0) = ζ 1 ∈ H, u ∈ L 2 ([0, T ], F), (28)
with structural dissipation power γ ∈ (0, 1), we write it as a first order system. The state space is E = D(A) × H. The semigroup generator A is

A = 0 I -A 2 -ρA 2γ , D(A) = (z 0 , z 1 ) ∈ E | Az 0 + ρA 2γ-1 z 1 ∈ D(A) .
It inherits from -A the necessary and sufficient properties of Lumer-Phillips for generating a contraction semigroup.

The observation and control operators are the projection C 0 : E → H defined by C 0 (z 0 , z 1 ) = z 1 , C = CC 0 , and B defined in § 3.2. We assume that C is admissible for the semigroup generated by A, i.e. [START_REF] Arendt | Vector-valued Laplace transforms and Cauchy problems[END_REF]. Since the cost estimate for C 0 given in [START_REF] Avalos | Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation[END_REF] is polynomial in 1/T , (7) holds for any β > 0 and b > 0.

For µ > 0 and z = (z 0 , z 1 ) ∈ H × H, we denote H z0,z1 (µ) = H(µ)z 0 , z 1 where H is the spectral decomposition of A. We define the matrix valued function M and the positive Hermitian matrix valued measure E z,z by

M (µ) = 0 -1 µ 2 ρµ 2γ , E z,z = H z0,z0 H z0,z1 H z1,z0 H z1,z1
.

As proved in [START_REF]The cost of fast non-structural controls for a linear elastic system with structural damping[END_REF]Lemma 3], the roots λ ± = r ± s of P µ (λ) = det(M (µ) -λI) satisfy min {Re λ + , Re λ -} ≥ min ρ 2 , 1 ρ µ 2 min{γ,1-γ} for µ ≥ 1. Therefore we define the growth spaces as E λ = H µ × H µ with λ = min ρ 2 , 1 ρ µ 2 min{γ,1-γ} . Plugging in the spectral representation e tA z, z = σ(A) e -tM (µ) dE z,z (µ) the explicit formula e -tM (µ) = e -tr (t shc(st)M (µ) + (cosh(st) + rt shc(st)) I) , t > 0, (where the cardinal hyperbolic sine function is the continuous and even function defined by shc(0) = 1 and shc(t) = sinh(t)/t for t = 0) proves the growth condition [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] with any m > 0 (indeed the loss µ 2 is only polynomial in λ instead of exponential).

For this choice of growth spaces, [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF] [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF]. For all ρ > 0 and γ ∈ (δ/2, 1 -δ/2), for all ζ 0 and ζ 1 , there is an input u such that the solution ζ of (28) satisfies ζ(T ) = ζ(T ) = 0 and the cost estimate:

T 0 u(t) 2 dt ≤ b 0 exp 2b T β Aζ 0 2 + ζ 1 2 , ζ 0 ∈ D(A), ζ 1 ∈ H, T small, with β = 2 δ min {γ, 1 -γ} -1 -1
, and any b >

d β+1 min ρ 2 , 1 ρ β (β + 1) β(β+1) β β 2 .
We refer to [START_REF] Chen | A mathematical model for linear elastic systems with structural damping[END_REF][START_REF] Lasiecka | Exact null controllability of structurally damped and thermo-elastic parabolic models[END_REF] for the motivation of the abstract model [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. The main application is to the plate equation with square root damping and interior control in Ω with hinged boundary conditions on a manifold M , in the framework of § 4.1:

ζ -ρ∆ ζ + ∆ 2 ζ = χ Ω u on [0, T ] × M, ζ = ∆ζ = 0 on [0, T ] × ∂M, ζ(0) = ζ 0 ∈ H 2 (M ) ∩ H 1 0 (M ), ζ(0) = ζ 1 ∈ L 2 (M ), u ∈ L 2 ([0, T ] × M ). (29)
Applying theorem 4.2 instead of [36, theorem 1] to A = -∆ with δ = γ = 1 2 , hence β = 1, improves on the value of β in the first part of [36, theorem 2] (cf. also [START_REF] Avalos | Boundary and localized null controllability of structurally damped elastic systems, Control methods in PDE-dynamical systems[END_REF]). Under the geometrical optics condition in [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] that the length L Ω of the longest generalized geodesic in M which does not intersect Ω is not ∞, the second part of [36, theorem 2] estimates the cost rate: for all ρ ∈ (0, 2), the control cost of (29) satisfies the estimate in theorem 4.2 with β = 1 and any b > b 1 L 2 Ω for some b 0 and b 1 which do not depend on Ω and ρ (cf. [36, note added in proof]), hence e.g. (cf. [START_REF] López | Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations[END_REF], [START_REF] Edward | Internal null-controllability for a structurally damped beam equation[END_REF]Appendix]) the minimal null-control input u converges to the minimal null-control input for the undamped plate equation as ρ → 0. 4.3. Diffusion in a potential well. We consider a power k ∈ N * and the potential well In this application, the state and input spaces are E = F = L 2 (R d ), the growth spaces are the spectral spaces of § 3.6, the reference operator C 0 is the identity operator and the observation operator C is the multiplication by χ Γ as in § 4.1, i.e. it truncates the input function outside the control region Γ.

V (x) = |x| 2k , x ∈ R d . The Schrödinger operator A = ∆ -V with domain D(A) = φ ∈ H 2 (R d ) | |V φ| 2 < ∞ is negative self-adjoint
In [START_REF]Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones[END_REF], [START_REF] Benabdallah | Null controllability of a thermoelastic plate[END_REF] with exponent α = 1 2 (1 + 1 k ) is proved and some radial eigenfunctions concentrating at some "equator" such that [START_REF] Lasiecka | Exact null controllability of structurally damped and thermo-elastic parabolic models[END_REF] holds are exhibited (cf. [40, § 4.2.2]) allowing to deduce from theorem 2.2 and lemma 3.5: Theorem 4.3. For all k > 1, the diffusion in the potential well V (x) = |x| 2k :

∂ t φ -∆φ + V φ = χ Γ u, φ(0) = φ 0 ∈ L 2 (R d ), u ∈ L 2 ([0, T ] × R d ),
is null-controllable in any time T > 0. Moreover the cost κ T (cf. § 3.2) satisfies: κ = lim sup T →0 T β ln κ T < ∞ with β = 1 + 2 k-1 . If there is a vector space of dimension 2 in R d which does not intersect the closure Ω 0 of the subset Ω 0 of the unit sphere defining the cone Γ then κ = 0.

When Γ is a bounded set instead of a cone, some radial eigenfunctions such that [START_REF] Kannai | Off diagonal short time asymptotics for fundamental solutions of diffusion equations[END_REF] As in § 4.1, the semigroup considered here is a well known model of diffusion. It can be interpreted as a Brownian diffusion on R d killed at the rate V .

5.

Lower bounds for the cost and spectral rates. The setting of this independent section is slightly more general than in § 2.4. As in § 4.1, M is a smooth complete Riemannian manifold and ∆ is the Laplace-Beltrami operator with Dirichlet boundary condition on H = L 2 (M ) which is both the state space E and the input space F. In this section we denote in the same way an open subset of M , its characteristic function and the multiplication by this function which is a bounded operator on H. With this notation, the observation operator is C = Ω where Ω = ∅ is an open subset of M . In this section A : D(A) ⊂ H → H still denotes the generator of a C 0 -semigroup (e tA ) t≥0 on H.

The main assumption in this section is the following L 2 Gaussian estimate: for all open subset ω ⊂ M and d < dist(Ω, ω) := inf (x,y)∈Ω×ω dist(x, y),

Ωe tA ω ≤ d 0 e -d 2 4t ω , t ∈ [0, T 0 ]. (30)
The lower bounds in this section are given in terms of the following distance:

d Ω = sup y∈M dist( Ω, y), (31)
i.e. the furthest from Ω a point of M can be. A simple example to keep in mind was considered at the end of § 3.6: M = R d , A = ∆, Ω is the exterior of a ball, hence d Ω in [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF] is the radius of this ball.

In the particular case of the heat semigroup on a compact manifold considered in § 2.4, Gaussian estimates were already the main tool in the geometric lower bound for the cost rate in [33, theorem 2.1] and the proof that the spectral rate is positive in [24, proposition 14.9] and [30, proposition 5.5]. But these proofs used pointwise Gaussian estimates and Weyl's asymptotics for eigenvalues.

The L 2 Gaussian estimate (30) provides simpler proofs where A need not even have eigenvalues. As shown in § 5.1, it does not only apply to A = ∆ but also e.g. to the linear Ginzburg-Landau equation on M complete or compact with Dirichlet boundary condition, real smooth potential V bounded from below and real ρ,

(1 + ρi)∂ t φ + (-∆ + V )φ = 0, t ≥ 0. (32)
N.b. [START_REF]The control transmutation method and the cost of fast controls[END_REF] gives an upper bound of the cost rate for this equation in terms of the length of the longest generalized geodesic in M which does not intersect Ω.

, Re z > 0, Ω ⊂ M, ω ⊂ M, (33)
where d = dist(Ω, ω), ω and Ω are open subsets. Following the theme of [START_REF] Cheeger | Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds[END_REF], this is an easy consequence of the propagation of the support with speed less than one for the (even) solution of the corresponding wave equation:

Ω cos(t √ -A)ω = 0, t ∈ (0, d), Ω ⊂ M, ω ⊂ M. (34)
The key idea is to represent the semigroup in terms of the wave group

e zA = 1 √ 4πz +∞ -∞ e -s 2 4z cos(s √ -A)ds, Re z > 0. (35)
Indeed, [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] Cf. [13, § 3] for a deeper study of these estimates and their relationship. The transmutation formula [START_REF]The control transmutation method and the cost of fast controls[END_REF] results directly from the integral representation of functions of A via spectral measures and the Fourier transform. In this context of short time asymptotics of diffusion semigroups, it was first used by Kannai in [START_REF] Kannai | Off diagonal short time asymptotics for fundamental solutions of diffusion equations[END_REF]. The control transmutation method in [START_REF]The control transmutation method and the cost of fast controls[END_REF] is based on analogous formulas for both the controlled solution and the input.

The L 2 Gaussian estimates [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] for real z are known as Davies-Gaffney estimates. Indeed, Gaffney's argument in [START_REF] Gaffney | The conservation property of the heat equation on Riemannian manifolds[END_REF] used to prove such estimates without [START_REF]Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds[END_REF] in [START_REF] Davies | Heat kernel bounds, conservation of probability and the Feller property[END_REF] needs very little smoothness, cf. [41, § 2] and [13, theorem 3.3].

If A satisfies (33) then (1 + iρ) -1 (A + λ 0 I), with ρ ∈ R and λ 0 ≥ 0 satisfies (30) with d 0 = e T0λ0 (with d 0 = 1 if λ 0 ≤ 0). In particular, for a potential V ∈ C ∞ (M ) such that V (x) ≥ -λ 0 , for all x ∈ M , A = ∆ -V -λ 0 (defined by Friedrichs extension from C ∞ c (M )) satisfies [START_REF]Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds[END_REF], hence [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], therefore the generator (1 + iρ) -1 (∆ -V ) satisfies (30) (n.b. [13, theorem 3.3] proves that A still satisfies [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF] for V ∈ L 1 loc (M ) on a complete M ). Hence theorems 5.1, 5.2 and 5.3 apply to the linear Ginzburg-Landau equation [START_REF] López | Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations[END_REF].

When it is not assumed that A is self-adjoint, but only that it is the generator of a cosine operator function Cos, then the transmutation formula [START_REF]The control transmutation method and the cost of fast controls[END_REF] holds with s → Cos(s) replacing s → cos(s √ -A), cf. e.g. [2, Weierstrass formula (3.102)], [START_REF] Sova | Cosine operator functions[END_REF][START_REF] Bragg | Related problems in partial differential equations[END_REF][START_REF] Fattorini | Ordinary differential equations in linear topological spaces I[END_REF]. Since a cosine operator function satisfies a growth bound Cos(s) L(H) ≤ M 0 e M s , s ≥ 0, the finite propagation speed [START_REF]Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds[END_REF] for Cos implies the weaker L 2 Gaussian estimate [START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF] where t is bounded and the limit value d = dist(Ω, ω) is excluded. E.g. theorem 5.1 still applies to the diffusion semigroup with generator,

Aφ = d j,k=1 ∂ xj (g jk ∂ x k φ) + d j=1 b j ∂ xj φ + V φ, D(A) = φ ∈ H 1 0 (M ) | Aφ ∈ L 2 (M ) ,
where M is a C 2 connected bounded domain in R d , b j and V are complex valued and bounded on M , g jk ∈ C 1 (M ), the matrix G = (g ij ) is real symmetric and 0 < G ≤ I uniformly on M . Indeed these assumptions ensure that A is a generator of a cosine operator function, cf. [2, theorem 7.2.3], and that the support propagates with speed less than one, cf. [START_REF] Alinhac | Hyperbolic partial differential equations[END_REF][START_REF] Taylor | Partial differential equations[END_REF]. N.b. if b j = 0 and V is real then A is self-adjoint, theorems 5.2 and 5.3 also hold and [START_REF] Boyer | Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations[END_REF] proves that (39) does hold. In the remaining part of § 5, we need spectral subspaces to state our results. Therefore we assume that A is the generator of a normal semigroup (cf. § 3.6) and H λ is the spectral subspace of H relative to z ∈ σ(-A) | Re z > λ 2 .

(

N.b. λ was an "eigenvalue" in E λ , now it is a "square-root of an eigenvalue" in H λ . The next theorem makes a weaker assumption than the previous one but draws the same conclusion when taking the limit ε → 0. Both theorems 5.1 and 5.2 were proved in [33, theorem 2.1] in the setting of § 2.4.

5.3.

Lower bound for the spectral rate.

Theorem 5.3. If A is the generator of a normal semigroup, satisfies the Gaussian estimate [START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF] and the spectral observability estimate on H λ defined in [START_REF]On the controllability of anomalous diffusions generated by the fractional Laplacian[END_REF] v ≤ a 0 e aλ Ωv , λ > 0, v ∈ H λ ,

then a ≥ d Ω /2 where d Ω is the distance defined in [START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF].

Proof. Let d and ω be as in the proof of theorem 5.1. For any λ > 0 and t ≤ T 0 , we consider φ = e tA ω and its projection v on H λ , i.e. v = 1 -A<λ 2 φ and φ -v = 1 A≤-λ 2 e tA ω. The spectral representation of functions of A and (30) yield Ω(φ -v) ≤ φ -v ≤ e -tλ 2 ω and Ωφ ≤ e -d 2 4t ω .

We choose t = d/(2λ) to make the right-hand sides of these inequalities equal. Plugging them in (39) and taking the limit λ → ∞ yield a contradiction for a < d/2: 0 = ω ← v ≤ a 0 e aλ Ωv ≤ a 0 e aλ ( Ωφ + Ω(φ -v) ) ≤ 2a 0 ω e (a-d/2)λ → 0.

Hence a ≥ d/2. Taking the limit d → d Ω completes the proof.

2. 4 .

 4 Original example. For A = ∆ the Laplace-Beltrami operator with Dirichlet boundary condition on a compact smooth connected Riemannian manifold M , E = F = L 2 (M ), E λ = the spectral spaces of A (cf. § 3.6), C 0 the identity operator, and C the multiplication by the characteristic function of an open subset Ω = ∅ of M , [29, theorem 3

Lemma 3 . 1 .

 31 The conclusion 2c = lim sup T →0 T β ln κ T ≤ 2c * of theorem 2.2 is still valid if we replace the assumption that C ∈ L(D(A), F) satisfies the admissibility condition (2) by the following time smoothing effect assumption: ∀x ∈ E, ∀t > 0, e tA x ∈ D(A), and lim sup t→0 t β ln Ae tA = 0. (15)

with input u ∈ L 2 (

 2 [0, T ], F) and control operator B = C * ∈ L(F, D(A * ) ) (A * denotes the adjoint of A and D(A * ) denotes the dual space of D(A * ) in E). Since C satisfies the admissibility condition (2), B satisfies T 0 e tA * Bu(t)dt 2 ≤ K T T 0 u(t) 2 dt, and the solution of (18) is f (T ) = e T A * f 0 + T 0 e (T -t)A * Bu(t)dt. More precisely, if (

0 e

 0 [START_REF] Gaffney | The conservation property of the heat equation on Riemannian manifolds[END_REF] holds for c = 0 and a > (bβ)1 β+1 α by choosing δ such that a > a δ . On the other hand, if λ ≤ bβ T β+1 then T -2j λ (t) dt ≥ T δT e -2j λ (t) dt ≥ (1 -δ)T e -2j λ (δT ) , δ ∈ (0, 1), with j λ (δT ) ≤ b (δT ) β + (δT ) bβ T β+1 = c δ T β , c δ = b 1 δ β + δβ δ→1 ---→ b(β + 1),hence[START_REF] Gaffney | The conservation property of the heat equation on Riemannian manifolds[END_REF] holds for a = 0 and c > b(β + 1) by choosing δ such that c > c δ .

  φ is the restriction to the real axis of an entire function φ such that | φ(z)| ≤ ce √ λ|Im z| by the Paley-Wiener theorem. When C 0 is the identity operator, C is the multiplication by the characteristic function of the exterior of a ball and F = E, [34] proves (6) with exponent α = 1 2 by Carleman estimates as in § 2.4. It is an open problem to obtain an explicit bound on the rate a in (

4. Applications. 4 . 1 .

 41 Anomalous diffusions. Let M be a smooth connected complete d-dimensional Riemannian manifold with metric g and boundary ∂M . When ∂M = ∅, M denotes the interior and M = M ∪ ∂M . Let ∆ denote the Laplace-Beltrami operator on L 2 (M ) with domain D(∆) = H 1 0 (M ) ∩ H 2 (M ) defined by g. N.b. the results are already interesting when (M, g) is a smooth connected domain of the Euclidean space R d , so that ∆ = ∂ 2 ∂x 2 1

  and has compact resolvent. Let χ Γ denote the multiplication by the characteristic function of any non empty open cone Γ = x ∈ R d | |x| > r 0 , x/|x| ∈ Ω 0 , where r 0 ≥ 0 and Ω 0 is an open subset of the unit sphere.

  fails are exhibited in [40, § 4.2.3] allowing to deduce from lemma 3.5 that κ = lim sup T →0 T β ln κ T = +∞ with β = 1 + 2 k-1 . Whether null-controllability from bounded sets Γ holds for k > 1 remains open.

5. 2 .Theorem 5 . 1 .00

 251 Lower bound for the cost rate. If A satisfies the Gaussian estimate[START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF] and the cost bounde T A v 2 ≤ c 0 e 2c T T Ωe tA v 2 dt, v ∈ D(A), T ∈ (0, T 0 ), (36)then c ≥ d 2 Ω /4where d Ω is the distance defined in[START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF]. Proof. Given d < d Ω , by the definition[START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF], there is an open ball ω ⊂ M such that dist(Ω, ω) > d. Taking v = ω in[START_REF]The cost of fast non-structural controls for a linear elastic system with structural damping[END_REF], applying[START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF] and taking the limit T → 0 yields a contradiction for c ≤ d 2 /4:0 = ω 2 ← e T A v 2 ≤ c 0 e 2c T T Ωe tA v 2 dt ≤ T c 0 d 2 0 ω 2 e 2(c-d 2 /4) T → 0.Hence c > d 2 /4. Taking the limit d → d Ω completes the proof.

Theorem 5 . 2 .0

 52 If A is the generator of a normal semigroup, satisfies the Gaussian estimate[START_REF] Rousseau | Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques. Applications au prolongement unique et au contrôle des équations paraboliques[END_REF] and the cost bound for some ε ∈ (0, 4/d 2 Ω ]:e T A v 2 ≤ c 0 e 2c T T Ωe tA v 2 dt, v ∈ H 1 εT , T ∈ (0, T 0 ),(38)then(1 + ε)c ≥ d 2 Ω /4where d Ω is the distance defined in[START_REF] Rousseau | Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations[END_REF]. Proof. Let d and ω be as in the proof of theorem 5.1. We consider φ = e εT A ω and its projection v on H 1 εT , i.e. v = 1 -A<(εT) -2 φ and φ -v = 1 A≤-(εT ) -2 e εT A ω. The spectral representation of functions of A and ε ≤ 4/d 2 Ω yield e tA (φ -v) ≤ e -t+εT (εT ) 2 ω ≤ e -1 (εT ) ω ≤ e -d 2 Ω /(4(1+ε)T ) ω .Plugging this and (30) for φ in (38) yields (1 + ε)c ≥ d 2 /4 as in the proof of theorem 5.1. Taking the limit d → d Ω completes the proof.

  The crucial lemma to bound this cost here is (cf. a continuous version in § 3.3)

Lemma 2.1. If the approximate observability estimate ( § 3.5 justifies this name)

  Theorem 4.2. Recall that δ and d are the exponent and rate in the main assumption

			implies the relative observability (6) with
	exponent α =	δ 2 min{γ,1-γ} and rate a =	d min{ ρ 2 , 1 ρ }	α .
	Applying theorem 2.2 and corollary 1(ii) improves on [36, theorem 1]:

  5.1. Semigroups satisfying L 2 Gaussian estimates. When A is a nonpositive self-adjoint operator, the semigroup satisfies this stronger L 2 Gaussian estimate Ωe zA ω L(H) ≤ e -d 2

	4 Re 1 z

N.b. ν, α or β may be increased so that the match β = α 1-α = ν 1-ν is achieved as in the statement of theorem

2.2. 2 Indeed [46, theorem 2.4] states c ≤ 2a 2 , but correcting the factor 2 into 1 2 in the definition of d(s) in [46, theorem 2.1] only proves c ≤ 8a 2 .

Acknowledgement. After this manuscript was submitted, Marius Tucsnak and Gerald Tenenbaum submitted their manuscript "On the null-controllability of diffusion equations". I am grateful to Marius Tucsnak for sending it to me in exchange for mine. Their first theorem is related to our main result without the admissibility assumption (in a setting otherwise less general) and the addition of lemma 3.1 was prompted by this statement. In the case where the manifold M and the observation region Ω are intervals, the lower bound in theorem 5.3 is nicely complemented by the upper bound given in their second theorem. I am grateful to Xu Zhang for sending me the manuscript "Observability Estimate on Sums of Eigenfunctions for Elliptic Operators: A Global Carleman Estimate Based Approach" by Hongheng Li and Qi Lü, which concerns C 2 principal part coefficients and general conditions on a C 2 boundary (this pointwise approach should be compared with the other global approach in [7]).

 [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF]was expanded and some references added. This work was partially supported by ANR grant JCJC06-137283.


EA Luc Miller Modal'x
email: luc.miller@math.polytechnique.fr

On the cost of fast control for heat-like semigroups: spectral inequalities and transmutation

Keywords: Observation of operator semigroups, estimates on sums of eigenfunctions, L 2 Gaussian estimates. Field: Control of PDEs Presentation: Plenary lecture

The Lebeau-Robbiano strategy deduces the null-controllability of parabolic systems of linear PDEs from some spectral inequalities: we implement it so simply that it yields valuable estimates on the cost of the distributed control as the time available to perform it tends to zero. The solutions and controls of some parabolic equations can be expressed in terms of those of the hyperbolic equation obtained by changing the time derivative into a second oder derivative: such transmutation formulas yield geometric bounds on both the spectral and cost rates.

Heat-like control problems

The basic null-controllability problem we shall consider is: for any given initial temperature distribution in a domain M , find a heat source located in a given region Ω such that the temperature is null everywhere after heat has diffused during a given time T . The cost is the ratio κ T of the size of the input over the size of the initial state: it blows up as T tend to zero.

In § 2, we present the quite general method introduced in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] to simultaneously prove null-controllability and estimate this blow-up. For this basic problem, it proves that the cost κ T blows up like exp(c/T ) and it is based on an observability estimate for sums of eigenfunctions of the Dirichlet Laplacian proved by some Carleman estimates in joint papers of Lebeau with Jerison, Robbiano and Zuazua. Moreover, it bounds the rate c from above in terms of the rate a in this estimate.

In § 3, we present some bounds on the spectral rate a and the cost rate c in terms of distances related to the configuration of the control region Ω within the domain M . The lower bounds are based on Gaussian estimates in L 2 rather than pointwise. This approach introduced in [Mil09] also simplifies and generalizes earlier results. The upper bound on c is based on the control transmutation method, cf. [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF].

Here are some control problems to which these results apply, cf. [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF].

Notations: We denote in the same way an open subset Ω = ∅ of M , its characteristic function and the multiplication by this function which is a bounded operator on the Hilbert space H = L 2 (M ) with norm • . For simplicity, we always consider Dirichlet condition on the boundary ∂M of M , we solve equations in Hilbert spaces of square integrable functions and achieve control by input functions

u in L 2 ([0, T ] × M ) = L 2 ([0, T ]; H).
Problem 1: Consider the heat equation as an ODE in H,

∂ t f -∆f = Ωu, where ∆ = ∂ 2 x 1 +• • •+∂ 2
x d is the usual Laplacian and M is a smooth connected bounded domain in R d . (∆ could also be the Laplace-Beltrami operator on a compact smooth connected Riemannian manifold M .)

Problem 2: Consider the same heat equation where = R d and Ω is the exterior of a ball with center 0 and radius R. (∆ could also be the Laplace-Beltrami operator on a complete smooth connected Riemannian manifold M with Ω the exterior of a compact K such that K ∩ Ω∩∂M = ∅.)

Problem 3: Consider the diffusion equation,

∂ t f -P f = Ωu, P f = d j,k=1 ∂ x j (g jk ∂ x k f ) + d j=1 ∂ x j (b j f ) + V f,
where M is a C 2 connected bounded domain in R d , b j and V are complex valued and bounded on M , g jk ∈ C 1 (M ), the matrix G = (g ij ) is real symmetric and 0 < G I uniformly on M . Problem 4: Consider the linear Ginzburg-Landau equation,

(1 + ρi)∂ t f + (-∆ + V )f = Ωu,
where V ∈ L 1 loc (M ) is a real potential bounded from below, ρ ∈ R, and M as in problems 1 or 2.

Problem 5: Consider thermoelastic plates without rotatory inertia,

     ∂ 2 t g + ∆ 2 g + α∆f = Ωu 1 on (0, T ) × M, ∂ t f -∆f -α∆∂ t g = Ωu 2 on (0, T ) × M, g = ∆g = f = 0 on (0, T ) × ∂M,
for § 2, where either u 1 = 0 or u 2 = 0, α > 0, M and Ω are as in problem 1.

The direct Lebeau-Robbiano strategy

Each of these problems can be formulated in terms of a continuous semigroup (e tA ) t 0 on a Hilbert space with suitable generator A. By a duality argument, null-controllability is equivalent to the final-observability inequality (stated here for problems 1 to 4, but similar in problem 5),

e T A φ 2 κ T T 0 Ωe tA φ 2 dt, φ ∈ H, T > 0, (FinalObs)
where κ T is the cost, i.e. the ratio of the size T 0 u 2 dt of the input over the size f (0) 2 of the initial state which it annihilates.

In problem 1, A = ∆ with domain

D(A) = H 2 (M ) ∩ H 1 0 (M ) ⊂ H and the Lebeau-Robbiano strategy is based on the spectral inequality, v a 0 e aλ Ωv , λ > 0, v ∈ H λ , (SpecObs)
where H λ denotes the set of finite sums of eigenfunctions of -∆ with eigenvalues less than λ 2 . More generally H λ is the spectral subspace of H relative to the part of the spectrum of A with real part greater than -λ 2 . E.g. (SpecObs) holds for problem 2: v ∈ H λ means that the Fourier transform v(ξ) vanishes for |ξ| > λ, i.e. v is the restriction to the real axis of an entire function ṽ such that |ṽ(z)| ce λ|Im z| by the Paley-Wiener theorem.

Theorem 1 (SpecObs) implies (FinalObs) with, for any c > 4a 2 , the cost bound: κ T c 0 e 2c/T .

With c > 8a 2 , this theorem is due to Seidman. Indeed [START_REF] Seidman | How violent are fast controls. III[END_REF] states an abstract theorem which also applies to problem 5. The statement in [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF] has a simpler cost bound and is more general (e.g. it applies to [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems[END_REF] and to linear elastic systems with structural damping for which λ in (SpecObs) and T in the cost bound have fractional powers). In his survey [START_REF] Zuazua | Controllability and Observability of Partial Differential Equations: Some results and open problems[END_REF], which covers the Lebeau-Robbiano strategy and the cost of fast controls, Zuazua remarked: "Actually, as far as we know, there is no direct proof of the fact that the spectral observability inequality implies the observability inequality for the heat equation. The existing proof is that due to Lebeau and Robbiano and passes through the property of null controllability and duality". Whereas Seidman's proof of this theorem still uses approximate null-controllability, our direct proof does not have recourse to controllability. (cf. [START_REF] Tenenbaum | On the null-controllability of diffusion equations[END_REF] for a less general direct proof based on the moment problem)

For problem 3, with V real and b j = 0 to ensure self-adjointness, (SpecObs) is proved in [START_REF] Boyer | Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations[END_REF] and theorem 1 still holds, including with ∂ t replaced by (1 + ρi)∂ t as in problem 4. The cost bound in theorem 1 is still valid for problem 5, keeping (SpecObs) as in problem 1 and replacing Ω and H in (FinalObs) by the suitable bounded observation operator.

We explain the direct Lebeau-Robbiano strategy on a simple example.

Proof that (SpecObs) with a = 1 implies the cost bound κ T e 18/T , for T small enough, for problems 1 and 2. First note that κ T 1/T e 1/T when Ω = M . Hence (SpecObs) with a = 1 implies

e T ∆ v 2 e λ+1/T T 0 Ωe t∆ v 2 dt, T > 0, λ > 0, v ∈ H λ .
On the other hand, for w ⊥ H λ the time-decay rate increases with λ:

e t∆ w e -tλ 2 w . To prove (FinalObs), we decompose φ = v + w with λ = 3/T , and use e T ∆ φ 2 = e T ∆ v 2 + e T ∆ w 2 e T ∆ v 2 + e -9/T w 2 . Since there is no decay as t → 0, we observe on [T /2, T ] but not on [0, T /2]:

         e T ∆ v 2 e λ+2/T T T 2 Ωe t∆ v 2 dt = e 5/T T T 2 Ωe t∆ v 2 dt, v ∈ H λ , T T 2 Ωe t∆ w 2 dt T 2 e T 2 ∆ w 2 T e -T λ 2 w 2 = T e -9/T w 2 , w ⊥ H λ . Hence e T ∆ φ 2 2e 5/T T T 2
Ωe t∆ φ 2 dt + T e -9/T w 2 + e -9/T w 2 .

Taking T small enough, setting τ = T , q = 2/3 and f (T ) = e -6/T , this yields the approximate observability inequality,

f (τ ) e τ ∆ φ 2 -f (qτ ) φ 2 τ 0 Ωe t∆ φ 2 dt, φ ∈ H, τ ∈ (0, ε 0 ). (1)
Consider the partition (0,

T ] = ∪ k∈N (T k+1 , T k ] with T k+1 -T k = τ k = qτ k-1 .
Applying (1) on each [T k+1 , T k ] with τ = τ k and adding these telescoping inequalities yield, since f (T ) → 0 as T → 0:

f ((1 -q)T ) e T ∆ φ 2 = f (τ 0 ) e T 0 ∆ φ 2 -0 × φ 2 T 0 Ωe t∆ φ 2 dt.
This completes the proof of (FinalObs) with κ T 1/f ((1q)T ) = e 18/T .

Transmutations

The lower bounds in this section are given in terms of the following distance:

d Ω = sup y∈M dist( Ω, y),
i.e. the furthest from Ω a point of M can be, e.g. in problem 2:

d Ω = R.
Theorem 2 (FinalObs) with κ T = c 0 e 2c/T implies c d 2 Ω /4.

Theorem 3 (SpecObs) implies a d Ω /2.

Theorem 2 holds for problems 1 to 4. Theorem 3 holds for problems 1 to 4 with b j = 0 and V real to ensure selfadjointness and natural spaces H λ .

The upper bound in theorem 4 is given in terms of the length L Ω of the longest generalized geodesic in M which does not intersect Ω (these geodesics behave like the rays of geometrical optics), e.g. in problem 2:

L Ω = 2R.
When M is a segment of length L it is known that null-controllability with input at one end holds with the cost bound κ T c 0 e 2c/T for any c > c * L 2 . Tenenbaum and Tucsnak proved in 2007 that c * 3/4. Theorem 4 (FinalObs) holds with κ T c 0 e 2c/T for any c > c * L 2 Ω .

N.b. theorem 4 is only relevant when L Ω is finite. The proof of theorem 4 is based on this result of Bardos, Lebeau and Rauch: L > L Ω is sufficient (and almost necessary) for the exact controllability of the wave equation ∂ 2 t g -∆g = Ωu in the context of problem 1. This result, hence theorem 4, extend to problem 2. Burq extended this result to ∂ 2 t g -P g = Ωu, when M is C 3 , P is as in problem 3 with g jk ∈ C 2 (M ), b j = 0 and V = 0, replacing the geodesics by bicharacteristics of G, hence theorem 4 holds under these assumptions.

The key idea in this section is that the wave equation ∂ 2 t w = Aw describes "the geometry" of the diffusion equation ∂ t φ = Aφ in short times. For A = ∆, Kannai implemented this idea with the transmutation formula (in order to prove an asymptotic expansion of the heat kernel as t → 0),

e t∆ φ = +∞ -∞ k(t, s) cos(s √ -∆)φds, k(t, s) = 1 √ 4πt e -s 2 /(4t) , (2)
which gives the solution of the heat equation as a weighted integral over all times of the (even) solution of the wave equation with the same initial condition. This formula results directly from the integral representation of functions of ∆ via spectral measures and the Fourier transform of the Gaussian. Cheeger, Gromov and Taylor fully exploited such transmutations in combination with the finite speed of propagation. Indeed the support of these waves propagate with speed less than one, i.e. Ω cos(t √ -∆)ω = 0, t ∈ (0, δ), where ω and Ω are open subsets of M and δ = dist(Ω, ω). Plugging this in (2) implies the L 2 Gaussian estimate on which theorems 1 and 2 are based, 4t) ds ω e -δ 2 /(4t) .

Ωe t∆ ω ω 1 √ 4πt |s| δ e -s 2 /(
(3)

Proof of theorem 2 for problems 1 and 2: Given δ < d Ω , there is an open ball ω ⊂ M such that dist(Ω, ω) = δ. Taking φ = ω in (SpecObs), applying (3) and taking the limit T → 0 yields a contradiction for c δ 2 /4: Hence c > δ 2 /4. Taking the limit δ → d Ω completes the proof of theorem 2.

The proof of theorem 3 is similar (Jerison and Lebeau proved a > 0 for problem 1 by a pointwise Gaussian estimate and Weyl's law for eigenvalues).

The proof of theorem 4 uses the control transmutation method, cf. [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF]. It replaces the one dimensional fundamental heat solution k in (2) by the solution of the heat equation obtained from an initial Dirac mass with an input on the boundary of [-L, L] which performs null-controllability in time T . Thus both the solution and the input of the wave equation are transmuted into those of the heat equation (this idea was first used by Phung for Schrödinger equation). Cf. [START_REF] Munch | Numerical approximation of null controls for the heat equation through transmutation[END_REF] for a recent numerical application.

An upper bound for a in problem 1 with d = 1 is given in [START_REF] Tenenbaum | On the null-controllability of diffusion equations[END_REF].

0

  0 = ω 2 ← e T A φ 2 c 0 e 2c T T Ωe tA φ 2 dt T c 0 d 2 0 ω 2 e 2(c-δ 2 /4) T → 0.