Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

In this paper we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm-Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier-Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

The goal of this work is to prove the null controllability of the following system, with a control f (t) ∈ L 2 (0, T ) acting at the left endpoint, where our Lagrange form [·, ·] is given by [u, v] (x) = (upv ′ − vpu ′ )(x), with p(x) = x α+β , and ′ = d dx .
and its corresponding norm is denoted by · β .
Here, we use some results from the singular Sturm-Liouville theory to see the well-posedness of the system (4) with initial data in L 2 β (0, 1), although the solution u(t) lives in an interpolation space H −s . We say the system (4) is null controllable in L 2 β (0, 1) at time T > 0 with controls in L 2 (0, T ), if for any u 0 ∈ L 2 β (0, 1) there exists f ∈ L 2 (0, T ) such that the corresponding solution satisfies u(·, T ) ≡ 0.
We are also interested in the behavior of the cost of the controllability. Consider the set of admissible controls given by U (T, α, β, µ, u 0 ) := {f ∈ L 2 (0, T ) : u is solution of the system (4) that satisfies u(·, T ) ≡ 0}.
The main result of this work is the following.
(3) Lower bound of the cost There exists a constant c > 0 such that where j ν+1,2 is the second positive zero of the Bessel function J ν+1 .
We also analyze the null controllability of a similar system but the control acting at the right endpoint, Consider the corresponding set of admissible controls U (T, α, β, µ, u 0 ) = {f ∈ L 2 (0, T ) : u is solution of the system (6) that satisfies u(·, T ) ≡ 0}, and the cost of the controllability given by where X is a subspace in L 2 β (0, 1). Theorem 2. Let T > 0, β ∈ R, 0 ≤ α < 2, and µ satisfying (2). The next statements hold.
(2) Upper bound of the cost There exists a constant c > 0 such that for every δ ∈ (0, 1) we have (3) Lower bound of the cost There exists a constant c > 0 such that In [7] the authors prove the null controllability of the equation (1) with a weighted Dirichlet boundary condition at the left endpoint, provided that α + β < 1. In the case α + β > 1, in [8] they get the null controllability of the equation (1) with a weighted Neumann boundary condition at the left endpoint. They consider initial data in L 2 β (0, 1) in both cases. In these works the authors prove suitable versions of a Hardy inequality to assure the well-posedness of their systems, but in the case α + β = 1 is necessary to consider some results from the singular Sturm-Liouville theory, see [8]. Here, we use that approach to show the well-posedness of our system. This paper is organized as follows. Section 2 uses some results from the singular Sturm-Liouville to show that the operator A given in (7) is self-adjoint. There, we also use Fourier-Dini expansions to show that A is diagonalizable, this allows us to consider initial data in some interpolation spaces. Next, we introduce a notion of a weak solution for both systems and then show the well-posedness of these systems.
In Section 3 we prove Theorem 1 by using the moment method introduced by Fattorini & Russell. Here, the idea is to construct a biorthogonal sequence to a family of exponentials involving the eigenvalues of A. To do this we use some results from complex analysis to construct a suitable complex multiplier. As a consequence, we get an upper estimate of the cost of the controllability. Finally, we use a representation theorem, Theorem 13, to obtain a lower estimate of the cost of the controllability.
In Section 4 we proceed as before to solve the case when the control acts at the right endpoint.

Now we introduce the operator A given by
From the theory developed in [11] we can build a self-adjoint domain D(A) for the operator A.
For µ satisfying (2), 0 ≤ α < 2, and β ∈ R, we set D max := u ∈ AC loc (0, 1) | pu x ∈ AC loc (0, 1), u, Au ∈ L 2 β (0, 1) , and Recall that the Lagrange form associated with M is defined as follows, The next result shows that A is a diagonalizable operator whose Hilbert basis of eigenfunctions can be written in terms of the function x 1/2+ν , the Bessel function of the first kind J ν and the corresponding positive zeros j ν+1,k , k ≥ 1, of the Bessel function J ν+1 , see the proof of Proposition 14. In the appendix, we give some properties of Bessel functions of the first kind and their zeros.
The assumption implies that This concludes the first part of the proof.
where y 0 is given in (9).
Then (A, D(A)) is the infinitesimal generator of a diagonalizable self-adjoint semigroup in L 2 β (0, 1). Thus, we can consider interpolation spaces for the initial data. For any s ≥ 0, we define and we also consider the corresponding dual spaces It is well known that H −s is the dual space of H s with respect to the pivot space L 2 Equivalently, H −s is the completion of L 2 β (0, 1) with respect to the norm It is well known that the linear mapping given by For δ ∈ R and a function h : (0, 1) → R we introduce the notion of δ-generalized limit of h at x = 0 as follows 2.1. Notion of weak solutions for both systems. Now we consider a convenient definition of a weak solution for the system (4). Let τ > 0 be fixed. We multiply the equation in (4) by integrate by parts (formally) and by using the boundary conditions for u, ϕ, see Remark 4, we get Definition 6. Let T > 0, 0 ≤ α < 2, β ∈ R, µ < µ(α + β), and a given by (3). Let f ∈ L 2 (0, T ) and u 0 ∈ H −s for some s > 0. A weak solution of (4) is a function u ∈ C 0 ([0, T ]; H −s ) such that for every τ ∈ (0, T ] and for every z τ ∈ H s we have The next result shows the existence of weak solutions for the system (4) under suitable conditions on the parameters α, β, µ, and s, its proof is similar to the proof of Proposition 10 in [7]. (3). Let f ∈ L 2 (0, T ) and u 0 ∈ H −s such that s > ν, with ν given in (5). Then, formula (10) defines for each τ ∈ [0, T ] a unique element u(τ ) ∈ H −s that can be written as Furthermore, the unique weak solution u on [0, T ] to (4) (in the sense of (10)) belongs to C 0 ([0, T ]; H −s ) and fulfills Proof. Fix τ > 0. Let u(τ ) ∈ H −s be determined by the condition (10), hence We claim that ζ(τ ) is a bounded operator from L 2 (0, T ) into H −s : consider z τ ∈ H s given by therefore By using Lemma A.3 and (67) we obtain that there exists a constant C = C(α, β, µ) > 0 such that Finally, we fix f ∈ L 2 (0, T ) and show that the mapping τ → ζ(τ )f is right-continuous on [0, T ). Let h > 0 small enough and z ∈ H s given as in (11). Thus, proceeding as in the last inequalities, we have Since 0 ≤ I(τ, k, h) ≤ 1/2 uniformly for τ, h > 0, k ≥ 1, the result follows by the dominated convergence theorem.

Control at the left endpoint
3.1. Upper estimate of the cost of the null controllability. Here we use the moment method, introduced by Fattorini & Russell in [6], to prove the null controllability of the system (4). The first step is to construct a biorthogonal family {ψ k } k≥0 ⊂ L 2 (0, T ) to the family of exponential functions e −λ k (T −t) k≥0 on [0, T ], i.e that satisfies This construction will help us to get an upper bound for the cost of the null controllability of the system (4).
Assume that for each k ≥ 0 there exists an entire function F k of exponential type T /2 such that F k (x) ∈ L 2 (R), and F k (iλ l ) = δ kl , for all k, l ≥ 0.
The L 2 -version of the Paley-Wiener theorem implies that there exists η k ∈ L 2 (R) with support in [−T /2, T /2] such that F k (z) is the analytic extension of the Fourier transform of η k . Then we have that is the family we are looking for. Now, we proceed to construct the family F k , k ≥ 0. Consider the Weierstrass infinite product From (65) we have that j ν+1,k = O(k) for k large, thus the infinite product converges absolutely in C. Hence Λ(z) is an entire function with simple zeros at iλ k , k ≥ 0.
From [10, Chap. XV, p. 498, eq. (3)], we have for ν > −1 that In [7] was proved that Therefore, In particular, It follows that is a family of entire functions that satisfy (16). Since Ψ k (x) is not in L 2 (R), we need to fix this by using a suitable "complex multiplier", thus we follow the approach introduced in [9].
Lemma 11. The function H ω,θ fulfills the following inequalities where c > 0 does not depend on ω and θ.
When k = 0 we have then we integrate on R and the result follows.
We also have that Consider the entire function F (z) given by for some δ > 0 that will be chosen later on. Clearly, From (37), (40) and (41) we obtain We recall the following representation theorem, see [4, p. 56].
Theorem 13. Let g(z) be an entire function of exponential type and assume that Let {d ℓ } ℓ≥1 be the set of zeros of g(z) in the upper half plane ℑ(z) > 0 (each zero being repeated as many times as its multiplicity). Then, We apply the last result to the function F (z) given in (41). In this case, (40) implies that A ≤ T /2. Also notice that ℑ (b k ) > 0, k ≥ 0, to get By using the definition of the constants b k 's we have where we have used Lemma A.2 and made the change of variables x.
From (43) we get the estimate From (15) and (49) it follows that Consider the entire function v : C → C given by v(s) := Therefore, Moreover, (50) implies that v(iλ k ) = 0 for all k ≥ 0, k = 1, and v(iλ Consider the entire function F (z) given by Clearly, From (49), (51) and (52) we obtain We apply Theorem 13 to the function F (z) given in (52). Then, (51) implies that A ≤ T /2, hence From (54) we get the estimate the result follows by letting ε → 0 + .

Appendix A. Bessel functions
We introduce the Bessel function of the first kind J ν as follows where Γ(·) is the Gamma function. In particular, for ν > −1 and 0 < x ≤ √ ν + 1, from (57) we have (see [1, 9.1.7, p. 360]) A Bessel function J ν of the first kind solves the differential equation Bessel functions of the first kind satisfy the recurrence formulas (see [1, 9.1.27]): Recall the asymptotic behavior of the Bessel function J ν for large x, see [5,Lem. 7.2,p. 129].