THE COST OF BOUNDARY CONTROLLABILITY FOR A PARABOLIC EQUATION WITH INVERSE SQUARE POTENTIAL

. The goal of this paper is to analyze the cost of boundary null controllability for the 1 − D linear heat equation with the so-called inverse square potential:

1. Introduction and main results.
1.1. Description of the problem. In this paper, we are interested in the linear 1 − D heat equation with an inverse square potential (that arises for example in the context of combustion theory and quantum mechanics): where u 0 ∈ L 2 (0, 1), T > 0 and µ is a real parameter. Here H represents some control term that aims to steer the solution to zero at time T . Our goal is not only to establish the existence of such control (which could be easily deduced from known results, see later) but also to provide sharp estimates of the cost of such control.

PATRICK MARTINEZ AND JUDITH VANCOSTENOBLE
Since the works by Baras and Goldstein [4,5], it is known that existence/nonexistence of positive solutions is determined by the value of µ with respect to the constant 1/4 appearing in the Hardy inequality [29,42]: ∀z ∈ H 1 0 (0, 1), When µ < 1/4, the operator z → −z xx − µx −2 z generates a coercive quadratic form in H 1 0 (0, 1). This allows showing the well-posedness in the classical variational setting of the linear heat equation with smooth coefficients. For the critical value µ = 1/4, the space H 1 0 (0, 1) has to be slightly enlarged as shown in [53] but a similar result of well-posedness occurs (see section 2.2 for details). Finally, when µ > 1/4, the problem is ill-posed (due to possible instantaneous blow-up) as proved in [4].
For these reasons, we concentrate on the two first cases and we assume throughout this paper that µ satisfies µ ≤ 1/4.
Here we are insterested in the study of null controllability using a boundary control acting at x = 1 (see problem (1)). More precisely, our aim is to provide sharp estimates of the cost of controllability in that case, analyzing its dependence with respect to the two parameters T > 0 and µ ∈ (−∞, 1 /4]. Observe that the existence of a boundary control for (1) could be deduced from [52], using for example the standard argument that consists in extending the domain (0, 1) into (0, 1+η), applying the result of null controllability for (3) in (0, 1+η) with a distributed control localized in (1, 1 + η) and concluding by taking the trace at x = 1 of such a control function. However this method based on Carleman estimates would not provide optimal estimates of the cost of the control, in particular when µ → −∞ (see Remark 3.5 in [52]).
For this reason, we turn here to methods based on decomposition in series and moment problems. These methods have been developped by Fattorini-Russell [18,19], and have been successfully applied/adapted to obtain sharp results in quite simple geometric situations, we refer for example the reader to [1,6,9,13,23,25,28,35,36,39,48,49].
Here we follow the classical strategy: • first of all, we transform the controllability question into a moment problem; THE COST OF BOUNDARY CONTROLLABILITY FOR A PARABOLIC ...

399
• to solve this moment problem and get some upper estimate of the cost of the control, we construct some suitable biorthogonal family (with best possible upper bounds of the norm of its elements); • to prove the sharpness of our estimate, we provide some lower bound of the cost of the control using lower bounds that are satisfied by any biorthogonal family.
Hence the tools that are required to solve our problem are : • the construction of some suitable biorthogonal family (with best possible upper bounds of the norm of its elements); • lower bounds of the norm of the elements of any biorthogonal family.
As seen in [9], the obtention of explicit and precise (upper and lower) estimates for such biorthogonal families is closely related to gap conditions on the eigenvalues, namely (Roughly speaking, the gap γ min gives the upper estimates whereas the gap γ max gives the lower estimates). Then here • we solve the eigenvalue problem, and we express the eigenvalues and eigenfunctions using Bessel functions and their zeros; • when µ ∈ [0, 1/4], the gap between the square root of successive eigenvalues satisfies some upper and lower bounds which are uniform with respect to the parameter µ; this enables us to use the results of [9]; • but when µ ≤ 0, the eigenvalues do not satisfy a good uniform gap condition from above; we use here some new tools developped in [10] in order to treat cases where the eigenvalues do not satify a good uniform gap condition but satisfy some better asymptotic gap condition; these new results have already been applied in the context of the degenerate heat equation with a strong degeneracy in [11], and in the present case they help us to provide a suitable lower bound of the cost.

1.2.
Background and motivations. The problem of the cost of controllability (or of the minimal norm control) takes its origin from control theory in finite dimension.
In the case of ODE, it has been completely solved by Seidman in [46] and by Seidman and Yong in [47]. In [46], the author considers the linear time-invariant finite-dimensional system: A is a n × n matrix and B is a n × m matrix with m ≤ n. In this context, the author provides a formula that describes the blow-up rate of the minimal norm control when the control time T goes to 0. (This estimate involves the Kalman's rank of the system). In [47], an extended result is derived allowing to consider all the L p (0, T )-norms of the control instead of the only L 2 (0, T )-norm.
In the case of infinite-dimensional systems, an analysis of the lower bound of the controllability cost has been provided in [26] in the case of the heat equation. Let us also refer to [2] for a study that concerns a strongly damped wave equation.
Next, the case of PDEs systems has also been studied in several situations. In [32], Lasiecka and Seidman provide optimal blow-up rates for a non-scalar thermoelastic system (heat conduction coupled with Kirkhoff or Euler-Bernouilli plate model). They consider the cases of interior and boundary controls and they analyze the blow-up rate of the control cost both when the controllability time T goes to 0 and when the coupling parameter α goes to 0 (a question that can only arise for systems). Still in the case of thermoelastic plates, Avalos and Lasiecka [3] established optimal estimates of the blow-up of the cost function (both for a mechanic and a thermal control) as the controllability time T goes to 0. They give a very complete result since they consider all the various boundary conditions that may classicaly be associated to thermoelastic PDEs. (The most challenging cases concern the clamped and the free boundary conditions, that prevent them to use spectral analysis).
As for the motivation of studying the blow-up rates of controllability cost of PDEs (besides to be interesting in itself to evaluate the cost of controllability with respect to controllability time and to the parameters entering in the system), let us mention the link between this question and stochastic PDEs. As described in [14] and [15], the cost of controllability is linked to the regularity of some Markov semigroup, including Orstein-Uhlenbeck processes and related Kolmogorov equations. For some of these semi-groups (see e.g. Theorem 8.3.3 in [14]), null controllability is equivalent to the differentiability and regularizing effect of the Orstein-Uhlenbek process. Moreover the regularity of solutions of the Kolmogorov equation depends on the asymptotic behavior of the minimal control norm as T → 0.
Even in the deterministic case, it was shown in [24] that there is a connection between the asymptotic behavior of the cost and the regularity of the Bellman's function (that describes the minimal time control).
In the present paper, we consider the 1 − D heat equation with inverse square potential (1) that arises in combustion theory and quantum mechanics, with the parameter µ ≤ 1/4 ( the case where the problem is well-posed and null controllable). So it is natural here to investigate the null controllability cost as the control time T goes to 0 (as usual) but also as the parameter µ goes to the limit values 1/4 or −∞ (as in coupled system for the coupling parameter, see [32]). This is the purpose of this work.

1.3.
Main results and comments. Let us define the notion of cost of controllability. For any T > 0, µ ≤ 1/4 and u 0 ∈ L 2 (0, 1), we introduce the set of admissible controls: where u (H) denotes the solution of (1). Then we consider the controllability cost for any u 0 ∈ L 2 (0, 1) which is the minimal value to drive u 0 to 0. Finally, we define the global notion of controllability cost: Then we prove the following results Theorem 1.1. Given µ ∈ [0, 1 4 ], T > 0, and u 0 ∈ L 2 (0, 1), there exists H ∈ H 1 (0, T ) such that the solution of (1) satisfies u(·, T ) ≡ 0. Moreover, we have THE COST OF BOUNDARY CONTROLLABILITY FOR A PARABOLIC ... 401 the following estimates of the controllability cost: there exists 0 < c < C both independent of µ ∈ [0, 1/4] and of T > 0 such that and Theorem 1.2. Given µ ≤ 0, T > 0, and u 0 ∈ L 2 (0, 1), there exists H ∈ H 1 (0, T ) such that the solution of (1) satisfies u(·, T ) ≡ 0. Moreover, we have the following estimates of the controllability cost: there exists 0 < c < C both independent of µ ≤ 0 and of T > 0 such that and Remark 1. Several observations can be made from Theorems 1.1 and 1.2: • we deduce from (4) that the null controllability cost is uniformly bounded when µ ∈ [0, 1 4 ], in particular it does not blow-up as µ → 1/4; this is natural and expected since it has been proved in [52] (in the case of a distributed control) that null controllability holds true for any µ ≤ 1/4; however, the proof of (4) holds on the fact that 1 4 − µ ≥ 0, and the final estimate (4) puts this in evidence; it also implies the following simpler estimate C ctr−bd (µ, T ) ≤ 2Ce C/T e −T /C , but where the condition 1 4 − µ ≥ 0 is hidden; • (5) allows to measure how good is the qualitative behavior given by the upper bound (4), once again putting in evidence that the condition 1 4 − µ ≥ 0 is not to be forgotten; the proofs of (4) and of (5) give explicit values of all the coefficients, hence in particular of the coefficients of 1/T in the exponential factors, but in this paper we are mainly interested in the qualitative behavior with respect to µ and T ; once again, (5) implies the following simpler estimate C ctr−bd (µ, T ) ≥ ce c/T e −2T /c , but where the condition 1 4 − µ ≥ 0 is hidden; • (6) gives an upper bound interesting when T → 0, T → ∞, and µ → −∞; it implies the more compact form but we stated (6) to put it in parallel with (4); note that (6) implies that C ctr−bd (µ, T ) → 0 as µ → −∞: this comes from the fact the parameter here has the good sign (it makes the energy decreasing); this could not have been deduced from the result obtained by Carleman estimates since those estimates does not allow us to take into account the "good" or "bad" sign of potential terms (the constant appearing in Carleman estimates would grow up as µ → −∞, see Remark 3.5 in [52]); • (7) allows to measure how good is the qualitative behavior given by the upper bound (6), the main difference coming from the last exponential factor; this allows us to precise the behavior of C ctr−bd (µ, T ) → 0 as µ → −∞.

2.
Well-posedness of the problem.

Preliminary transformation.
Let µ ≤ 1/4 be given. To define the solution of the boundary value problem (1), we transform it (as done for instance in [9] in the context of a degenerate parabolic equation) into a problem with homogeneous boundary conditions and a source term. Let us define Observe that p(0) = 0, p(1) = 1 and Formally, if u is a solution of (1), then the function defined by is solution of Reciprocally, given h ∈ L 2 (0, T ), consider the solution of x ∈ (0, 1).
This motivates the fact that we will first establish results of well-posedness for an auxiliary problem with Dirichlet homogeneous boundary conditions and source term (see section 2.2). Then we will define the solutions of the boudary value problem (1) via this auxiliary problem (see section 2.3).

2.2.
Homogeneous boundary conditions and a source term. Let us first consider the system with homogeneous boundary conditions and a source term x ∈ (0, 1).
We define H 1 0 (µ) the Hilbert space obtained as the closure of H 1 0 (0, 1) with respect to the norm (Thanks to the Hardy inequality (2), this defines a norm for any value of the paramater µ ≤ 1/4).
3. Proof of the main results (Theorems 1.1 and 1.2). In this section, we prove our main result. In order to straight the main lines of the proof, some computations are postponed in the technical part in section 4.

3.1.
Eigenvalues and eigenbasis of the Sturm-Liouville Problem. In order to transform the question of null controllability into a moment problem, let us first determine the eigenfunctions and eigenvalues associated to the operator L µ . This means that we aim at solving the following boundary value problem for any suitable As this problem is closely related to Bessel's equations, we will need to use throughout this paper some standard definitions, notations and properties coming from Bessel's theory. For reader convenience, we recall and summarize the elements that are useful in the appendix in section 5.

3.2.
Useful estimates on eigenvalues and eigenfunctions. Next we give some estimates on the eigenvalues (proved later in section 4.2) that will be useful in the analysis of the problem: (ii) When µ ≤ 0, then and Observe that (15) and (16) ensure a lower estimate of the gap that is uniform for any µ ≤ 1/4: This will enable us to use the standard methods developped by Fattorini-Russell [18,19] to give an upper estimate of the cost. More precisely, we use a result proved in [9] that makes explicit the one obtained in [18,19] (in short time). For this part, we will able to treat the two cases µ ∈ [0, 1/4] and µ ≤ 0 in the same way.
As for the obtention of a lower estimate of the controllability cost, we will have to treat separately the two cases µ ∈ [0, 1/4] and µ ≤ 0, using extensions of Guichal [26], that we proved in [9] and [10].
3.3. Transformation into a moment problem. Following the strategy initiated by Fattorini and Russell [18,19], we reduce here the controllability question to some moment problem. In this part, we analyze the problem with formal computations.
On the other hand, we observe that w µ,n (x, t) := Φ µ,n (x)e λµ,n(t−T ) is solution of the adjoint problem: Multiplying (1) by w µ,n and (18) by u, we obtain Taking into account that u(T ) ≡ 0 and the boundary conditions satisfied by u and w, we obtain Therefore the question reduces into the following moment problem : find H such that where we set r µ,n = Φ µ,n (x = 1).
As we look for a control H belonging to H 1 (0, T ) such that H(0) = 0 = H(T ), we rather write the problem satisfied by H (t) which is simply obtained by some integration by part in time: 3.4. Formal solution of the moment problem. Set artificially λ µ,0 := 0, and assume for a moment that we are able to construct a family (σ + µ,m ) m≥0 of functions σ + µ,m ∈ L 2 (0, T ), which is biorthogonal to the family (e λµ,nt ) n≥0 , which means that: Then let us define and Then it is easy to show that, at least formally, K solves the moment problem for the derivative (20). Moreover, if K ∈ L 2 (0, T ), then clearly H ∈ H 1 (0, T ), H = K, and H(0) = 0. Moreover H(T ) = 0 thanks to the additional property that the family (σ + µ,m ) m≥1 is orthogonal to e λµ,0t = 1. So H will be in H 1 (0, T ) such that H(0) = 0 = H(T ) and will satisfy the moment problem (19).
It remains to check that all this makes sense. For this purpose, we will have to prove the existence of a biortogonal family (σ + µ,m ) m≥0 together with suitable L 2 bounds (so that we can prove that K ∈ L 2 (0, T )).

3.5.
Existence of a suitable biorthogonal family. We will use the following result: Theorem 3.2. (see Theorem 2.4 in [9]) Assume that ∀n ≥ 0, λ n ≥ 0, and that there is some γ min > 0 such that Then there exists a family (σ + m ) m≥0 which is biorthogonal to the family (e λnt ) n≥0 in L 2 (0, T ): Moreover, it satisfies: there is some universal constant C u independent of T , γ min and m such that, for all m ≥ 0, we have σ + with

PATRICK MARTINEZ AND JUDITH VANCOSTENOBLE
Remark 3. Theorem 2.4 in [9] is formulated in the following way: with and this is clearly equivalent to (25)- (26). Its proof is based on complex analysis tools, combining the approach of Seidman-Avdonin-Ivanov [48] with the addition of some parameter, as in Tucsnak-Tenenbaum [49].
One deduces that Next we use the following Lemma: Lemma 3.4. There is some constant (independant of ν and of T ) such that : Proof. The proof of Lemma 3.4 is based on classical analysis estimates, see [11].

PATRICK MARTINEZ AND JUDITH VANCOSTENOBLE
Applying Lemma 3.4, it follows that: (4) and (6). 3.7. Lower bound of the cost when µ ∈ [0, 1 4 ]: proof of (5). Given m ≥ 1, consider u 0 = Φ µ,m , and let H m be any control that drives the solution of (1) to 0 in time T . Then (19) gives that Hence the sequence (r µ,m H m ) m≥1 is biorthogonal to the set (e λµ,nt ) n≥1 . There exist several lower bounds in the literature for such biorthogonal sequences, in particular Guichal [26] and Hansen [28]. In this case we are going to use the following generalization of Guichal [26], proved in [9]: Theorem 3.5. (Theorem 2.5 in [9]) Assume that ∀n ≥ 1, λ n ≥ 0, and that there is some 0 < γ min ≤ γ max such that Then there exists c u > 0 independent of T , and m such that: any family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ) satisfies: and (The proof of Theorem 3.5 is a natural generalization of the Hilbertian techniques used in [26].) When µ ∈ [0, 1 4 ], we are in position to apply Theorem 3.5. Indeed, using (15), we see that the assumption (31) is satisfied with γ min := 7π 8 , and γ max := π.
THE COST OF BOUNDARY CONTROLLABILITY FOR A PARABOLIC ...
(The proof of Theorem 3.6 is a natural generalization of the Hilbertian techniques used in [26]; Theorem 3.6 was motivated by several problems where it appears that the global gap is quite bad, while there is a much better asymptotic gap, see [11] for an application to degenerate parabolic equations.) We are in position to apply Theorem 3.6: indeed, (38) is satisfied with γ min := π, and γ max := j ν(µ),2 − j ν(µ),1 , and (39) is satisfied choosing N * and γ * max as in (37). Then, applying Theorem 3.6, we obtain that any family (σ + µ,m ) m≥1 which is biorthogonal to the family (e λµ,nt ) n≥1 in L 2 (0, T ) satisfies where , and These expressions seem be a little frightening, but we are looking for the behavior as µ → −∞ i.e. ν(µ) → +∞, and this is not difficult to study (see [11]), and to obtain that, when m = 1: This gives the following lower bound of the cost when µ ≤ 0: So, using (64), we get This proves (7).

4.1.
Proof of proposition 2. Let us first observe that any admissible eigenvalue λ satisfies λ > 0 (use for instance Theorem 2.1). Therefore in the following we assume λ = 0. Using the following changes of variables, one can easily see that φ satisfies (14) if and only if ψ is solution of Hence ψ is a solution of the Bessel equation (59) (see section 5.1) of order Let us now solve (43).
In the following, we study these three functions that appear in the formula of Φ +,− . First, let us study It satisfies
(ii) Next consider ν ≥ 1/2. Here we use the fact (see section 5.4) that, for ν ≥ 1 2 , the sequence (j ν,n+1 − j ν,n ) n is nonincreasing and converges to π. This ensures that j ν,n+1 − j ν,n ≥ π. (iii) Finally, (iii) is proved in Lemma 5.1 of [11] (in the spirit of Komornik-Loreti [31] p. 135, using classical Sturm theory for second order differential equations). Hence The above equation is called Bessel's equation for functions of order ν. Of course the fundamental theory of ordinary differential equations says that the solutions of (59) generate a vector space S ν of dimension 2. In the following we recall what can be chosen as a basis of S ν .

Fundamental solutions of Bessel's equation when ν /
∈ N. Assume that ν / ∈ N. When looking for solutions of (59) of the form of series of ascending powers of x, one can construct two series that are solutions: where Γ is the Gamma function (see [54,  When ν ∈ N, the two functions J ν and J −ν are linearly independent and therefore the pair (J ν , J −ν ) forms a fundamental system of solutions of (59), (see [54, section 3.12, eq. (2), p. 43]).

5.3.
Fundamental solutions of Bessel's equation when ν = n ∈ N. Assume that ν = n ∈ N. When looking for solutions of (59) of the form of series of ascending powers of x, one sees that J n and J −n are still solutions of (59), where J n is still defined by (60)  For any ν ∈ R + , the two functions J ν and Y ν always are linearly independent, see [54, section 3.63, eq. (1), p. 76]. In particular, in the case ν = n ∈ N, the pair (J n , Y n ) forms a fundamental system of solutions of the Bessel's equation for functions of order n.
In the case ν = n ∈ N, it will be useful to expand Y n under the form of a series of ascending powers. This can be done using Hankel's formula, see [54,  In the case n = 0, the first sum in (62) should be set equal to zero.