Precise estimates for biorthogonal families under asymptotic gap conditions

A classical and useful way to study controllability problems is the moment method developed by Fattorini-Russell, based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behaviour: the eigenvalues of the problem satisfy a uniform but rather 'bad' gap condition, and a rather 'good' but only asymptotic one. The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and so to measure the influence of the 'bad' gap condition and the good influence of the 'good' asymptotic one. To achieve our goals, we extend some of the general results of Fattorini-Russell concerning biorthogonal families, using complex analysis techniques developed by Seidman, G\"uichal, Tenenbaum-Tucsnak, and Lissy.


Presentation of the subject.
Biorthogonal families are a classical tool in analysis. In particular, they play a crucial role in the so-called moment method, which was developed by Fattorini-Russell [12,13] to study controllability for parabolic equations.
Given any sequence of nonnegative real numbers, (λ n ) n≥1 , we recall that a sequence (σ m ) m≥1 is biorthogonal to the sequence (e λnt ) n≥1 in L 2 (0, T ) if ∀m, n ≥ 1, The goal of this paper is to provide explicit and precise upper and lower bounds for the biorthogonal family (σ m ) m≥1 under the following gap conditions: • a 'global gap condition': (1. 1) ∀n ≥ 1, 0 < γ min ≤ λ n+1 − λ n ≤ γ max , • and an 'asymptotic gap condition': where γ * max − γ * min < γ max − γ min . Before explaining why we are interested in such a question, let us describe some of the main results of the literature on this subject.

The context.
Among the most important applications of biorthogonal families to control theory are those to the null controllability and sensitivity of control costs to parameters. Major contributions in such directions are the following: • Fattorini-Russell [12,13], Hansen [20], and Ammar Khodja-Benabdallah-González Burgos-de Teresa [1] studied the existence of biorthogonal sequences and their application to controllability for various equations; • for nondegenerate parabolic equations and dispersive equations, Seidman [34], Güichal [19], Seidman-Avdonin-Ivanov [35], Miller [30], Tenenbaum-Tucsnak [36], and Lissy [25,26] studied the dependence of the null controllability cost C T with respect to the time T (as T → 0, the so-called 'fast control problem') and with respect to the domain, obtaining extremely sharp estimates of the constants c(Ω) and C(Ω) that appear in e c(Ω)/T ≤ C T ≤ e C(Ω)/T ; • Coron-Guerrero [8], Glass [17], Lissy [26] investigated the vanishing viscosity problem: obtaining sharp estimates of the null controllability cost with respect to the time T , the transport coefficient M , the size of the domain L, and the diffusion coefficient ε; • in [5,6], we studied the dependence of the controllability cost with respect to the degeneracy parameter α for the degenerate parabolic equation There is a common feature in these works: they depend on some parameter p, and this parameter forces the eigenvalues to satisfy (1. 1) (sometimes after normalization) with gap bounds γ min (p) and γ max (p) such that γ min (p) → 0 and/or γ max (p) → ∞.
This fact makes it necessary to have general and precise estimates with respect to the main parameters that appear in the problem.
In [6], we proved the following general result: given T > 0 and a family (λ n ) n≥1 of nonnegative real numbers that satisfy the 'global gap condition' (1. 1), then: • every family (σ m ) m≥1 , biorthogonal to (e λnt ) n≥1 in L 2 (0, T ), satisfies the lower estimate  (1. 4) is in the spirit of [12,13] but the dependence with respect to T when T → 0 + is completely explicit, and assumption (1. 1) is a little more general than the asymptotic development of the eigenvalues used in Tenenbaum-Tucsnak [36] or Lissy [25,26]: Moreover, (1. 1) explains the role of γ min and γ max in the analysis of the biorthogonal family: γ min determines, essentially, the growth rate of the upper bound for (σ m ) m≥1 while γ max gives the lower bound.

Motivations and main results of this paper.
Even though the aforementioned results give a fairly good picture of the properties of the family (σ m ) m , some delicate issues remain to be analysed and will be addressed in this paper. For instance, one would like to understand the dependence of the family (σ m ) m with respect to relevant parameters that come into play. Typical examples of such problems are the following ones.
• For the 1D degenerate parabolic equation the eigenvalues λ α,n of the associated elliptic operator (with suitable boundary conditions) can be expressed using the zeros of Bessel functions ( [18]) and depend on the degeneracy parameter α ∈ (0, 2). One can then prove (see [5,7]) that the global gap condition (1. 1) is satisfied only with hence it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 3) of the associated biorthogonal sequences, but the fact that is certainly to be taken into account. • In 2D problems such as the Grushin equation (see [2,3]), where the solution is decomposed into Fourier modes, one has to give uniform bounds for a certain sequence of elliptic problems, the eigenvalues of which satisfy (1. 1) and (1. 2) with some γ min (m), γ * min (m) and N * (m) such that once again, it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 4) of the associated biorthogonal sequence, but the fact that N * (m) → +∞ as m → ∞ is certainly to be taken into account. The above discussion motivates the general question whether estimates (1. 3) and (1. 4) can be improved when (1. 1) is combined with the asymptotic condition (1. 2). This is exactly what we prove in this paper: roughly speaking, (1. 3) and (1. 4) hold true replacing γ min by γ * min and γ max by γ * max . Moreover, the fact the 'good' gap condition (1. 2) holds true only after the N * first eigenvalues has a cost, and we obtain a precise estimate for that cost. Our main results (Theorem 2.1 and 2.2) are the following: under (1. 2), we prove that: • every family (σ m ) m≥1 , biorthogonal to (e λnt ) n≥1 in L 2 (0, T ), satisfies the lower estimate is a rational function of T that we determine explicitly, and • there exists a biorthogonal family that satisfies where C 0 > 0 is a universal constant and B * m (T, γ min , γ * min , N * , m) is a rational function of T that we determine explicitly.
Let us observe that the presence of the exponential factors e in (1. 5) and (1. 6) is quite natural and has already been pointed out by Seidman-Avdonin-Ivanov [35], Tenenbaum-Tucsnak [36], and Lissy [25,26] (see also Haraux [21] and Komornik [23] for a closely related context). On the other hand, the precise estimate of the behavior of b * m and B * m with respect to parameters, that we develop in this paper, is completely new and will be crucial for the sensitivity analysis of control costs to be performed in [7].
Our proofs are based on complex analysis techniques and Hilbert space methods developed by Seidman-Avdonin-Ivanov [35] and Güichal [19]. We have also used an idea from Tenenbaum-Tucsnak [36] and Lissy [25,26], based on the introduction of an extra parameter depending on T and the gap conditions. 1.4. Plan of the paper.
The paper is organized as follows: • in section 2, we state our results; • section 3 is devoted to the proof of Theorem 2.1 (construction of a biorthogonal family and derivation of upper bounds); • section 4 is devoted to the proof of Theorem 2.2 (lower bounds for biorthogonal families).

Existence of a suitable biorthogonal family and upper bounds.
We will establish the following results, that in some sense provide a more precise version of properties observed by Fattorini and Russell [12,13] (in short time), much in the spirit of Tenenbaum-Tucsnak [36] and Lissy [25,26] (with a slightly weakened assumption on the eigenvalues).
Theorem 2.1. Assume that ∀n ≥ 1, λ n ≥ 0, and that there is some 0 < γ min < γ * min such that Then there exists a family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ): Moreover, it satisfies: there is some universal constant C independent of T , γ min , γ * min , N * and m such that, for all m ≥ 1, we have Remark 2.1. Theorem 2.1 completes and improves several earlier results, in particular Theorem 1.5 of Fattorini-Russell [13] and [6], providing the explicit dependence of the L 2 bound with respect to γ min , γ * min in short time. It is useful in several problems, in which γ min → 0 with respect to some parameter, which occurs is several cases, see, e.g. [14], [2]. We will apply the construction used by Seidman, Avdonin and Ivanov in [35], which has the advantage to be completely explicit (which is not the case for the construction of [12,13,14,20,1], since there is a contradiction argument), combined with some ideas coming from the construction of Tenenbaum-Tucsnak [36] and Lissy [25], adding some parameter, in order to obtain precise results.

General lower bounds.
We generalise a result by Güichal [19] to prove the following Theorem 2.2. Assume that ∀n ≥ 1, λ n ≥ 0, and that there are 0 < γ min ≤ γ * max ≤ γ max such that Then any family (σ + m ) m≥1 which is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ) (hence that satisfies (2. 4)) satisfies: where b * is rational in T (and explictly given in the key Lemma 4.4).
Remark 2.2. Theorem 2.2 completes a result of Güichal [19] and is useful in several problems, in which γ max → ∞ with respect to some parameter, which occurs is several cases, see, e.g. [14], and [7]. It is to be noted that the behaviour with respect to m can perhaps be improved, comparing with Theorem 1.1 of Hansen [20]. It would be interesting to investigate this.

Proof of Theorem 2.1
3.1. The general strategy. It begins with the following remarks: if the family (σ + m ) m≥1 is biorthogonal to the family (e λnt ) n≥1 , then ∀m, n ≥ 1, and Now we recall the Paley-Wiener theorem ( [38]): if f : C → C is an entire function of exponential type, such that there exist nonnegative constants C, A such that One of the objects of [35] is to prove the existence of a sequence (f m ) m of entire functions satisfying (see Theorem 2 and Lemma 3 in [35]) under some general assumptions on the sequence (λ n ) n . If we can apply such a result in our context (hence with our sequence (λ n ) n ), then the two last properties together with the Paley-Wiener theorem will imply that there exists some will be biorthogonal to the family (e λnt ) n in L 2 (0, T ), as desired. Moreover using the Parseval theorem. Now, it remains to construct such entire functions f m . The idea is to consider the natural infinite product that satisfies the first condition of (3. 1), f m (−iλ n ) = δ mn , and to multiply it by a so-called 'mollifier', in such a way that the other two conditions of (3. 1) will be also satisfied. Hence one has to estimate the growth of the natural infinite product, and then to choose a choose a suitable mollifier. This is what is performed in [35]. For our problem, our task will be to add the dependency into the parameters γ min , γ * min and T , and to understand specifically the behaviour of the natural infinite product, the mollifier and at the end of σ + m L 2 (0,T ) with respect to γ min and T . We will modify a little the construction of [35], in order to obtain optimal results in our context, see Lemma 3.4, and specifically the definition (3. 19) of the mollifier, where the additional parameter N ′ will be chosen of the size

The counting function.
Consider We prove the following: b) Assume that the gap assumptions (2. 1)-(2. 2) are satisfied; then • when n = N * : • when n > N * : , as ρ → +∞, and to compute all the needed additional constants.
• When n = N * : from the previous study, we see that this gives that which gives (3. 3). • When n > N * : now we have which gives (3. 4). • When n < N * : now we have which gives (3. 5), and similar estimates when λ n ≥ λ N * − λ n , which give (3. 6). Before going further, let us give another estimate of the counting function, which reveals to be more practical and more natural, since it gives a better understanding of the role of the different parameters: • when n > N * : and also Remark 3.2. Lemma 3.2 enlightens the role of the quantity (1 − γmin (2. 3)); when γ min = γ * min or if N * = 1, this quantity is equal to zero, and we logically find estimates similar to the ones of Lemma 3.1 (i.e. the "1 gap condition"); in the more interesting case where γ min < γ * min and N * > 1, this quantity measures the increase of the counting function with respect to the "1 gap condition".
Let us note also that we expect that (3. 9) holds true with 2 instead of 1 + √ 2, however we could not prove it in full generality.
Proof of Lemma 3.2.
• When n = N * , it is sufficient to note that this estimate and (3. 4) imply (3. 8). • When n < N * , we obtain (3. 10) proceeding in the same way: when ρ ≤ λ N * , then clearly N n (ρ) is less than the number of terms that would be at both sides, for which the gap of their square root would be γ min , hence when ρ ≥ λ N * , then clearly one has all the N * − 1 first terms, and the others, for which the gap of their square root is γ * min , hence which gives (3. 10); • finally we prove (3. 9): in the same way, if ρ ≤ max{λ n , λ N * − λ n } one has immediately when ρ ≥ max{λ n , λ N * − λ n }, then we already know from (3. 5) and (3. 6) that we deduce that which is (3. 9). This concludes the proof of Lemma 3.2.

A Weierstrass product.
Motivated by [35], we consider Then the growth in k of λ k ensures that this infinite product converges uniformly over all the compact sets, hence F m is well-defined and entire over C. Moreover We are going to estimate the growth of F m . We prove the following b) Assume that the gap assumptions (2. 1)-(2. 2) are satisfied. Then the function F m satisfies the following growth estimate: there is some uniform constant C u (independent of m, γ min , γ * min and z), such that  . 15) and (3. 16)), that will help us in the following. Comparing with (3. 13), this gives a better idea of the improvement brought by 'large' gap γ * min and of the price to pay due to the 'small' gap γ min for the N * first eigenvalues. In fact we will first prove the following better estimates: (3. 14) holds true with and this easily implies (3. 15) and (3. 16).

A suitable mollifier.
Motivated by [35], we made in [6] the following construction: consider T ′ > 0, N ′ ≥ 1, a k := and finally (3. 19) Then we have the following (2) The behaviour of P N ′ ,T ′ over R: there exist θ 0 > 0, θ 1 > 0, both independent of N ′ and T ′ such that P N ′ ,T ′ satisfies (3) The behaviour of P N ′ ,T ′ over iR + : there is some constant θ 2 > 0, independent of N ′ and T ′ , such that P N ′ ,T ′ satisfies The Proof of Lemma 3.4 follows by elementary analysis techniques. In the following we are going to use the mollifier P N ′ ,T ′ to construct the biorthogonal family.
3.6. The resulting biorthogonal sequence. With our choices, the function x → f m,N ′ ,T ′ (−x)e −ixT /2 is in L 2 (R), and we can consider its Fourier transform φ m,N ′ ,T ′ : It is well-defined since f m,N ′ ,T ′ ∈ L 2 (R), and the Paley-Wiener theorem ( [38] p.
To obtain good results, we will choose N ′ satisfying the stronger property: Then we have the following Lemma 3.6. Take T ′ and N ′ satisfying (3. 24) and (3. 30), and consider Then the family (σ + m,N ′ ,T ′ ) m≥1 is biorthogonal to the family (e λnt ) n≥1 in L 2 (0, T ): Moreover, it satisfies: there is some universal constant C u independent of T , γ min , γ * min , N * and m such that, for all m ≥ 1, we have where B(T, γ min , γ * min , N * , m) is given by (2. 6).
Proof of Lemma 3.6. The Fourier inversion theorem gives that This gives (3. 32). Concerning (3. 33), we note that the Parseval equality gives We need to estimate precisely the last integral. Denote
Since (from (3. 24)) 1 T ′ ≤ 1 (γ * min ) 2 (T ′ ) 2 , we obtain that: One can easily check that we obtain Finally, we see that there exists some C u independent of m, γ min , γ * min , N * and T such that
It follows from (2. 7) that and then it is well-known ( [32,31]) that E(Λ, T ) is a proper subspace of L 2 (0, T ). Moreover, given m ≥ 1, denote Λ m := (λ k ) k =m , and E(Λ m , T ) the smallest closed subspace of L 2 (0, T ) containing the functions ε λ k , with k ≥ 1 and k = m (it does not include ε λm ). Then consider p m the orthogonal projection of ε λm on E(Λ m , T ), and d T,m the distance between ε λm and E(Λ m , T ): we have Then ε λm − p m is orthogonal to E(Λ m , T ) , which implies that Moreover it is optimal in the following sense: if (σ − m ) m≥1 is another biorthogonal family for the set (ε λn ) n≥1 in L 2 (0, T ), then for all m ≥ 1,σ − m − σ − m is orthogonal to all ε λn , hence to E(Λ, T ), hence to σ − m since σ − m ∈ E(Λ, T ). Hence Hence 1 dT,m is a lower bound of every biorthogonal sequence (σ − m ) m≥1 ; and a bound from above for d T,m gives a bound from below for every biorthogonal sequence.
At last, we note that if the sequence of functions (σ + m ) m≥1 is a biorthogonal family for the set (e λnt ) n≥1 in L 2 (0, T ), then Hence e −λmT dT,m is a lower bound of every biorthogonal sequence (σ + m ) m≥1 . In the following (Lemma 4.4), we provide a bound from above for d T,m , that will give a bound from below for every biorthogonal sequence (σ + m ) m≥1 .

A general result for sums of exponentials.
Clearly, d T,m ≤ e −λms − p(s) L 2 (0,T ) for all p ∈ E(Λ m , T ). The idea used in Güichal [19] is to chose a particular element p ∈ E(Λ m , T ) in order to provide an upper bound of d T,m . The first thing to note is the following: consider M ≥ m and A i e −λis with coefficients A 1 , · · · , A M+1 . Then q ∈ E(Λ m , T ) if and only if A m = 0, and when A m = 0, then We will choose the coefficients A 1 , · · · , A M+1 so that The following lemma is essentially extracted from Güichal [19]: Consider M ≥ 0, and 0 < λ 1 < · · · < λ M+1 . a) There exist coefficients A 1 , · · · , A M+1 so that the function q defined by The coefficients are given by the following formulas: . b) With this choice of coefficients, we have The only difference with Güichal [19] is the estimate (4. 7) which is more precise than the one obtained in [19], Lemma 4: In the following, we prove (4. 7), and in a sake of completeness, we give the main arguments for part a) of Lemma 4.1.
Proof of Lemma 4.1.
a) We write the linear system This can be written The (M + 1) × (M + 1) matrix A that appears in the left hand side of (4. 8) is invertible: indeed, its determinant is of Vandermonde type, and Hence the system (4. 8) is invertible, and the Cramer's formula gives where B is the (M + 1) × (M + 1) matrix obtained from A putting the right-hand side member of (4. 8) at the place of the k th -column of A. But then, we can develop det B with respect to the k th -column and we find again a Vandermonde determinant. Then using the formula of Vandermonde determinant, one gets (4. 6). b) We prove (4. 7) by induction. When M = 0, (4. 7) is true. Assume that it is true for some M , and let us prove that it is true for M + 1: take where the coefficients A 1 , · · · , A M+2 , are chosen so that q(0) = q ′ (0) = q ′′ (0) = · · · = q (M) (0) = 0, q (M+1) (0) = 1.
Then the Taylor developments of q and q ′ say that Theñ But the last term in the series is clearly equal to 0, henceq is a sum of M + 1 exponentials. Moreover, and we can apply the induction assumption toq: then We deduce first that s → q(s)e λM+2s is increasing. Since its value in 0 is 0, then q is positive on (0, +∞). Next, we obtain that which completes the induction argument and the proof of Lemma 4.1.

4.3.
A precise estimate of the remaining part of the exponential function.
It turns out that we will need an estimate for the remaining part of the exponential function ∞ n=N x n n! in function of x and N . We prove the following general and precise result: We have the following estimates: x .
Proof of Lemma 4.2. Denote Let us prove by induction that First, of course f 0 (x) = e x , and then Next, assume that We note that The study of the variations of the function x → N +1 (1+x) 2 + x 1+x gives Then and since the values at 0 are 0, we obtain that This proves the first part of (4. 9). For the second part (which is not necessary for us here), we note that Assume that Then To conclude, note that Hence, we obtain that which concludes the induction, and the proof of (4. 9).

4.4.b.
Estimate under the uniform gap condition (2. 7) and the asymptotic gap condition (2. 8). Now, taking into account the "asymptotic gap" given by (2. 8), we will be able to improve the previous estimate, roughly speaking replacing γ 2 max by (γ * max ) 2 in the exponential factor.   The starting point is of course (4. 10) and (4. 12). Concerning the estimate of the product, we proceed in the same way as previously, distinguishing several cases. We investigate what can be said when m ≤ N * < M + 1: in this case, • first we see that ∀i ≥ N * + 1, • next, similarly we have