The cost of controlling degenerate parabolic equations by boundary controls

We consider the one-dimensional degenerate parabolic equation $$ u_t - (x^\alpha u_x)_x =0 \qquad x\in(0,1),\ t \in (0,T) ,$$ controlled by a boundary force acting at the degeneracy point $x=0$. First we study the reachable targets at some given time $T$ using $H^1$ controls, extending the moment method developed by Fattorini and Russell to this class of degenerate equations. Then we investigate the controllability cost to drive an initial condition to rest, deriving optimal bounds with respect to $\alpha$ and deducing that the cost blows up as $\alpha \to 1^-$.


Introduction
Null controllability of nondegenerate parabolic equations is by now well understood, either by locally distributed control or by a control acting on a part of the boundary, and we refer the reader to the seminal papers of Fattorini and Russell [15,16], and of Fursikov and Imanuvilov [18].
However, many problems that are relevant for applications are described by degenerate equations, with degeneracy occurring at the boundary of the space domain. We can mention the question of invariant sets for diffusion process in probability, the study of the velocity field of a laminar flow on a flat plate, the Budyko-Sellers climate models, the Fleming-Viot gene frequency model, referring, e.g., to [7] for details. The typical example of such degenerate problems is the equation where α > 0 is given. The diffusion coefficient in (1. 1) vanishes at x = 0. The question of null controllability by a locally distributed control has been solved in [6], and developped in several directions (more general classes of degenerate equations ( [1,3,4,28]), and in space dimension 2 ([7])). The question of null controllability by a boundary control acting where the degeneracy occurs, has been studied recently: • Cannarsa-Tort-Yamamoto [9] proved an approximate controllability result; • Gueye [19] proved the expected null controllability result, studying the degenerate wave equation and then applying the transmutation method ( [10,11]); • Martin, Rosier and Rouchon [27] proved a null controllability result using the flatness approach, which can be applied to general situations (degenerate or singular parabolic equations), and gives the control and the solution as series.
The main goal of this paper is to study the dependence of the controllability properties with respect to the degeneracy parameter α, in the typical example (1. 1) when the control acts on the boundary at the degeneracy point x = 0: • our first result concerns the reachable targets using H 1 controls: we prove that there is an explicit subset P α,T ⊂ L 2 (0, 1), which is dense in L 2 (0, 1), such that every u T ∈ P α,T is reachable with H 1 controls (see Theorem 2.1); • since the reachable set R α,T contains a subset P α,T which is dense in L 2 (0, 1), it is interesting to look for targets that could be reached for all parameter α ∈ [0, 1); however, we prove that α∈[0,1) P α,T = {0}, hence 0 is the only target that we are sure that can be reached for all parameter α ∈ [0, 1) (see Proposition 2.5); • since 0 is reachable for all parameter α ∈ [0, 1), it is interesting to measure the cost to drive an initial condition to 0 with respect to the parameter α; we prove that the controllability cost blows up with order 1 1−α as the degeneracy parameter goes to 1 − , providing (optimal) upper and lower bounds (see Theorem 2.2).
The proofs are based on the moment method developed by Fattorini and Russel [15,16]. We extend their method and some of their results to this degenerate case, and then we take advantage of the explicit expressions of the control that it provides (in terms of Bessel functions and their zeros) to obtain upper and lower bounds of the null controllability cost.
The paper is organized as follows: in section 2 we state precisely our results; in section 3, we summarize the definitions and properties of Bessel functions that are useful to solve the Sturm-Liouville problem; section 4 is devoted to the proof of Theorem 2.1 (concerning the subpart P α,T of reachable targets); section 5 is devoted to the proof of Proposition 2.5 (concerning α∈[0,1) P α,T ); section 6 is devoted to the proof of Theorem 2.2 (concerning the cost of null controllability).

Setting of the problem and main results
We are interested in the controllability properties of the problem (2. 1) that is when the control acts at the degeneracy point 0 through a nonhomogeneous Dirichlet boundary condition. First, we recall that, in a general way, the well-posedness of degenerate parabolic equations is stated in weighted Sobolev spaces. We will consider the problem when α ∈ [0, 1); in this case, the Dirichlet boundary control makes sense, as we explain in the following.  Then, formally, if u is a solution of (2. 1), then the function v defined by satisfies the auxiliary problem Reciprocally, given g ∈ L 2 (0, 1), consider the solution v of (2. 4) Then the function u defined by This motivates the following definition of what is the solution of the boundary value problem (2. 1), as we explain in the following.

2.2.
Main results: the reachable set and the cost of null controllability.
2.2.a. The controllability problem. The first problem we address concerns the boundary controllability of equation (2. 1) using a control acting at the degeneracy point. Given α ∈ [0, 1), T > 0, u 0 , u T ∈ L 2 (0, 1), we wish to find G ∈ H 1 (0, T ) that drives the solution u of (2. 1) from u 0 to u T in time T .
(Of course, due to the regularizing effect, it is clear that we will not be able to reach targets u T with low regularity.) We will also be interested in the targets that can be reached for all the parameters α. If such a target exists, it makes sense to evaluate the cost to reach it with respect to the degeracy parameter α.
We will denote by P α,T the set of u T that satisfy (2. 11).

Remark 2.3.
• Of course condition (2. 11) is satisfied if µ T α,n = 0 for all n large enough. Since finite linear combinations of Φ α,n are dense in L 2 (0, 1), the reachable targets form a dense subset of L 2 (0, 1), which was already known from [9]. Our result allows us to be more precise on the reachable targets.
• We underline the fact that (2. 11) is independent of T . Indeed, it is well-known in a general setting that the reachable set R T of the targets that can be attained at time T does not depend on T , see Seidman [31]. • A condition like (2. 11) already appears in the pioneering works of Fattorini and Russell [15,16]. Ervedoza and Zuazua [10] proved a similar (in fact a slightly better) condition in a general context. One could provide an explicit estimate of the constant K that appears in (2. 11), but we emphasize the fact that it does not depend on α. This is worth to be noted since we are interested in the behavior of the reachable set with respect to the degeneracy parameter α.

2.2.c.
The regularity of the targets and the question of the targets that are reachable for all α ∈ [0, 1). Fattorini and Russell [15] noted that in the case of the heat equation, i.e. when α = 0, a reachable target is the restriction to [0, 1] of an analytic function. Let us study what can be said in our case. We prove the following regularity result: Then u T has the following property: there exists an even function F α , holo- This regularity result extends in a natural way the result of Fattorini and Russell [15], and it has the following consequences: if the Fourier coefficients of u T satisfy (2. 11), then u T is reachable, and there exists an even function F α , holomorphic in the strip {z ∈ C, |ℑz| < K π } such that (2. 12) holds.
Remark 2.4. The problem of establishing whether zero is the only target that can be reached for all α ∈ [0, 1) is widely open.
2.2.d. The cost of null controllability. Finally, since 0 can be reached for all α ∈ [0, 1), it is interesting to measure the cost to drive any u 0 to 0 in time T , with respect to α. We define the controllability costs in the following way: given u 0 ∈ L 2 (0, 1), we consider the set of admissible controls that drive the solution u of (2. 1) to 0 in time T : where u (G) denotes the solution of (2. 1); we consider the controllability cost which is the minimal value to drive u 0 to 0. We also consider a global notion of controllability cost: (2. 14) Similar notions were already being considered, see in particular Fernandez-Cara and Zuazua [17]. Then we prove the following 1), and M 2 independent of u 0 and α such that Remark 2.5. This shows that the controllability cost blows up as α → 1 − , and that our upper estimate is optimal.
2.3. Additional comments and related questions.

2.3.a.
The question of uniformly reachable targets. As in [10], and of course as in [15], we obtain a subpart P α,T of the reachable set R α,T , and we prove a somewhat negative result concerning α∈[0,1) P α,T in Proposition 2.5. It would be interesting to obtain a result concerning α∈[0,1) R α,T .

2.3.b.
The cost of null controllability. It would be interesting to improve (if possible) (2. 15) of Theorem 2.2 to obtain a lower bound that depends on u 0 L 2 (0,1) .

2.3.c.
The question of locally distributed controls. In [8], we will study the same questions when the control is locally distributed in (0, 1).

Useful tools from Bessel's theory for the Sturm-Liouville problem
In this section, we recall existing tools, that we will need to prove our results, stated in section 2.2. Note that one can observe that if λ is an eigenvalue, then λ > 0: indeed, multiplying (2. 8) by y and integrating by parts, then which implies first λ ≥ 0, and next that y = 0 if λ = 0.

The link with the Bessel's equation.
There is a change a variables that allows one to transform the eigenvalue problem (2. 8) into a differential Bessel's equation (see in particular Kamke [21, section 2.162, equation (Ia), p. 440], and Gueye [19]): assume that Φ is a solution of (2. 8) associated to the eigenvalue λ; then one easily checks that the function Ψ defined by is solution of the following boundary problem:

Eigenvalues and eigenfunctions.
Consider α ∈ [0, 1), and let Φ be the solution of (2. 8) associated to the eigenvalue λ. Define ν α := 1−α 2−α ∈ (0, 1 2 ]. Hence ν α / ∈ N, and Bessel's functions J να and J −να are particular solutions of (3. 6) Since they are also linearly independent, all the solutions of equation (3. 6) are linear combination of J να and J −να . Hence there exists constants C + and C − such that In particular, Then, using the series expansion of J να and J −να , one obtains Next one easily verifies that Φ + , . But the boundary conditions allow us to obtain information on C + and C − : να,m . Therefore, if Φ is an eigenfunction associated to the eigenvalue λ, then for some C + and m ∈ N, m ≥ 1, we have . Conversely, one easily verifies that, for all m ≥ 1 and all C Now consider Φ α,n given by (2. 10). The family (Φ α,n ) n≥1 forms an orthonormal basis of L 2 (0, 1): indeed, they are the eigenfunctions of the operator T α : is the solution of the problem −Au f = f , and T α is selfadjoint and compact (see Appendix in [1]). The fact that their L 2 norm is equal to 1 comes from a classical identity on Bessel functions, see [26], formula (5.14.5), p. 129: 3.4. Some bounds on J ν and on its zeros.

3.4.b.
Some bounds on the zeros of J ν . Using McMahon's formula (see [32, section 15.53, p. 506] applied in the case θ = 0 i.e. for C ν = J ν ), we can give the following asymptotic expansion of the zeros of J ν for any fixed ν ≥ 0: We will also use the following bounds on the zeros, proved in Lorch and Muldoon [25]: ).

Proof of Theorem 2.1
Let α ∈ [0, 1) be given and consider T > 0 and u 0 ∈ L 2 (0, 1). Following the ideas of [22] (in the context of the wave equation), we may reduce the control problem (2. 1) to a moment problem. Then, we will solve this moment problem, using ideas and results of [15,16], and of course properties of the eigenvalues and eigenfunctions given in section 3.
To prove the existence of such a function G, it will be necessary to know if r α,n = 0 for all n. We prove this property in the following section. Moreover, since we want a solution of the moment problem that belongs to H 1 (0, T ), it will be more interesting to see what its derivative has to satisfy. Integrating by parts, we have Hence the derivative G ′ has to satisfy (4. 3) − r α,n λ α,n T 0 G ′ (t)e λα,nt dt = −µ 0 α,n + µ T α,n e λα,nT − r α,n λ α,n G(T )e λα,nT − G(0) .
We will provide a solution of this problem that satisfies G(0) = 0 = G(T ).

4.2.
The generalized derivative of the eigenfunctions at the degeneracy point.

4.3.
Existence and L 2 -bound for the biorthogonal sequence. In order to solve the moment problem (4. 2), we will use a sequence (σ α,n ) n≥1 in L 2 (0, T ) which is biorthogonal to (e λα,nt ) n≥1 , that is The existence of such a sequence follows from general results of Fattorini and Russell [15,16]. More precisely, we are going prove the following Theorem 4.1. Let (λ α,n ) n≥1 be defined by (2. 9). Then there exist positive constants denoted B T (depending on T ) and K (independent of T ), both independent of α ∈ [0, 1), and a sequence (σ α,n ) n≥1 of functions of L 2 (0, T ) satisfying the following properties: Remark 4.1. The sequence (σ α,n ) n≥1 will be the basis that allows us to write a solution G of the moment problem (4. 2). The L 2 -bounds (4. 10) will be useful to ensure the convergence of the associated series giving the control G. The orthogonality condition (4. 9) is interesting to construct a control G in H 1 (0, T ). Finally, the fact that the constants B T and K are independent of α ∈ [0, 1) will allow us to estimate the controllability cost (Theorem 2.2).
Proof of Theorem 4.1.
To have the existence of the biorthogonal family and an estimate of the L 2 -norm explicit with respect to the degeneracy parameter α, we will use results from [16]: Let (λ n ) n be a sequence such that, for some ℓ > 0, (4. 11) λ 0 ≥ ℓ, and, for all n ≥ 0, λ n+1 − λ n ≥ ℓ.

4.5.
Rigorous study of the moment problem and of the controllability problem.

4.5.b.
The associated solution is driven from the initial state to the prescribed target. Now, from the definition of the solution u of the boundary control problem (2. 1), it is natural to consider the problem (see (2. 4)) where we recall that Fix ε ∈ (0, T ). Then the regularity noted in Remark 2.1 allows us to see that Letting ε → 0 + , we obtain λ α,n r α,n −µ 0 α,n + µ T α,n e λα,nT Lemma 4.2. The following identity holds: Proof of Lemma 4.2.
We recall that Let us study the behavior of u T near 0. Using the series expression of J να , we derive from (5. 1) that Formally, exchanging the sums, we obtain This is precisely of the form We will now provide a rigorous proof of the above reasoning. We will need the following Lemma 5.1. If (µ T α,n e Kn ) n remains bounded for some K > 0, then the function F α is holomorphic in the disc {z ∈ C, |z| < K π } and is even. Proof of Lemma 5.1. We recall that, by (4. 7), we have hence there exists C * ≥ 0 independent of m such that where we used the fact that (3. 10) implies that (5. 4) ∀ν ∈ (0, 1 2 ], ∀n ≥ 1, j ν,n ≤ πn.
Note that, if (µ T α,n ) n satisfies the assumption of Proposition 2.4, and x κα < K π , then the series να,n is convergent. Therefore our previous argument is justified and (5. 2) is valid, with F α holomorphic in a neighborhood of 0. We are going to be a little more precise, proving that F α is in fact holomorphic in the horizontal strip {z ∈ C, |ℑz| < K π }. We note that which is holomorphic in C. Hence, coming back to the expression of u T we have µ T α,n C α,n x (1−α)/2 (j να,n x κα ) να L να (j να,n x κα ).
Let us prove the following Lemma 5.3. If the sequence (µ T α,n e Kn ) n≥1 is bounded for some K > 0, then the functionF α is holomorphic in the horizontal strip {z ∈ C, |ℑz| < K π }, and even.
If u T is nonzero, then consider p α the first integer such that F Hence the quantity κ α p α + 1 − α has to remain constant on [0, 1). This obliges α → p α to be locally constant, but this is not sufficient to ensure that κ α p α +1−α remains constant. Therefore u T has to be identically 0.
Proof of Lemma 6.1. First we check that the family (Φ 1,n ) n≥1 forms an orthonormal family of L 2 (0, 1). First we consider n = m. By change of variables, we have 2yJ 0 (j 0,n y)J 0 (j 0,m y) dy.