NULL CONTROLLABILITY OF STRONGLY DEGENERATE PARABOLIC EQUATIONS

. We consider linear one-dimensional strongly degenerate parabolic equations with measurable coeﬃcients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coeﬃcient a = a ( x ) in the principal part of the parabolic equation is such that the function x → x/a ( x ) is in L p (0 , 1) for some p > 1. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control


Introduction
We continue our investigation of the controllability of parabolic equations with measurable coefficients [24] (see also [2]) by studying the case of a strongly degenerate equation of the type (a(x)u x ) x + q(x)u = ρ(x)u t , x ∈ (0, 1), t ∈ (0, T ), where the nonnegative function a may vanish strongly at x = 0, and the potential q may be singular at x = 0.Only weakly degenerate (i.e.1/a ∈ L 1 (0, 1)) parabolic equations were covered by the theory developed in [24].
The null controllability of (weakly or strongly) degenerate parabolic equations was considered in e.g.[1, 4, 5, 9-11, 13-15, 26].Most of the papers were concerned with a parabolic equation with a(x) = x 2−ε , which is strongly (resp.weakly) degenerate for ε ∈ (0, 1] (resp.ε ∈ (1, 2)).More general choices for the coefficient a were considered in e.g.[14].However, several technical assumptions (e.g.x → a(x)/x γ nondecreasing for some exponent γ and a ∈ W 1,∞ (0, 1) in [14]) were required in order to derive some Carleman estimate to prove the null controllability of the parabolic equation.The purpose of this paper is to remove these technical assumptions in the derivation of the null controllability of the parabolic equation.
More precisely, we propose a general method based on the flatness approach to deal with quite general parabolic equations, displaying both a strong degeneracy of a and a singularity of the potential q at the same point, with measurable coefficients, and without any monotony assumption about a. Roughly, the main assumption is that the function x → x/a(x) is in L p (0, 1) for some p > 1.That assumption is slightly stronger (by Hölder inequality) than Trudinger assumption 1/ √ a ∈ L 1 (0, 1) (see e.g.[12,28]) which was made in order to investigate the degenerate elliptic system      −(au x ) x = f, x ∈ (0, 1), (au x )(0) = 0, u(1) = 0.
For the corresponding elliptic problem      −(x 2−ε u x ) x = f, x ∈ (0, 1), (x 2−ε u x )(0) = 0, u(1) = 0, the eigenfunctions and eigenvalues can be expressed in terms of Bessel functions, and the asymptotic behaviour of the eigenvalues is perfectly known [26].For a more general function a, however, Bessel functions cannot be used and, to the best knowledge of the authors, nothing is known about the sharp asymptotic behaviour of the eigenvalues.(See [3,17,18] for some results in that direction.)For the application of the flatness approach, what is needed is not a spectral gap, but merely that the eigenvalues tend to infinity faster than some power of the index of the eigenvalue.
1. Note that the main result in [26] corresponds to the case µ = 0. 2. Note that Proposition 1.4 is not a consequence of Proposition 1.3, since we cannot find any p in (1, +∞] with both (x → x ε−1 ) ∈ L p (0, 1) and (x → x −ε ) ∈ L p (0, 1). 3. Our computations suggest that for a(x) = x 2−ε and q(x) = µx −κ , κ should be at most ε.It would be interesting to see whether it is a necessary condition, or merely a technical assumption.
Let us say a few words about the proof of the main result.In a first step, we show that we can get rid of the term q(x)u in equation (1.1) by a change of variables, using assumption (1.9).Therefore we can restrict to the simplified parabolic equation We first prove that the boundary value problem possesses a unique solution in some weighted Sobolev space.Next, we pay some attention to the spectral properties of this boundary value problem.We show that the eigenvalue λ n grows at least as a power of n, and that the L ∞ -norm of the corresponding eigenfunction e n grows at most as a power of λ n .This is done by using a modified Prüfer method (see [6,29]) introducing a phase θ n associated with λ n .However, since 1/a ∈ L 1 (0, 1) in the interesting situation of a strong degeneracy, the classical argument relating λ n to the variation of the phase θ n has to be refined in using a splitting of the interval (0, 1) involving the frequency λ n .Roughly, we split r−p , C denoting some positive constant.We show that e n (x) remains close to e n (0) for x ∈ (0, A n ), so that the (bad) integral term An 0 dθn dx dx does not contribute too much in the variation of the phase θ n (1) − θ n (0).
With these spectral estimates at hand, we can prove that the eigenfunctions e n , n ≥ 0, can be expressed in terms of the generating functions g i , i ≥ 0, defined by g 0 (x) = 1 and the relation Finally, the trajectories of the control problems (1.1)-(1.4)can be expanded in the form for some function y ∈ G s ([0, T ]) (as in [24]), the series being convergent thanks to the spectral estimates.
The paper is organized as follows.Section 2 is devoted to the study of the corresponding elliptic problem.We introduce the appropriate weighted Sobolev space, derive some generalized Hardy inequality and obtain some estimates for both the eigenfunctions and the eigenvalues.In Section 3, we define and investigate the generating functions.The proof of the main results are given in Section 4. Finally, in some appendix we prove that the conditions (1.6) and (1.8) are independent, and we provide a class of functions for which equation (1.8) holds.

Study of the elliptic problem
Through the paper, we denote u L p for u L p (0,1) (1 ≤ p ≤ ∞), and u L p (x1,x2) for the L p norm of u on an interval (x 1 , x 2 ) = (0, 1).
Then the following result holds.
Lemma 2.5.The embedding Proof of Lemma 2.5: Let (u n ) be a sequence in H a and let u ∈ H a be such that u n → u weakly in H a .We have to show that u n → u strongly in L 2 (0, 1).Since for δ ∈ (0, 1) the embedding ) is compact for any δ ∈ (0, 1), and hence u n → u in L 2 (δ, 1).Let ε > 0 be given.By equation (2.14), there exists some δ ∈ (0, 1) such that Using the fact that u Ha ≤ sup n≥0 u n Ha and equation (2.15), we obtain Proof of Lemma 2.6: By equation (1.7) one may pick some numbers δ ∈ (0, 1  2 ) and C > 0 such that The fact that H a,ρ ⊂ L 2 ρ continuously comes from the definition of the spaces H a,ρ and L 2 ρ and of their norms.Using equation (4.2) and the lines above, we infer that the map u ∈ H a,ρ → θu ∈ L 2 ρ is compact.On the other hand, the embedding W 1,1 (0, 1) ⊂ L 2r (0, 1) is compact, and by Hölder inequality ) is dense in H a (0, 2), so that we can pick a sequence (ϕ n ) in C ∞ ([0, 2]) with ϕ n → u in H a (0, 2), and also in L 2 (0, 2).This gives By equation (4.2), we have ) and in L 2r (2δ, 1), we also have that The proof of Proposition 2.1 is complete.
Next, we investigate the elliptic problem equations (2.1)-(2.3).Introduce the symmetric bilinear form where be endowed with the norm • Ha,ρ .By equation (2.9), H is a closed subspace of H a,ρ , and the bilinear form a is continuous on H × H.To prove that the bilinear form is coercive, we need the following lemma.
Lemma 2.7.There exist a constant C > 0 such that )  Thus u = 0, but this contradicts the condition 1 0 |u| 2 ρ dx = 1.The proof of Lemma 2.7 is complete.We have to prove that for some C > 0, which gives again equation (2.22).Thus the bilinear form a is coercive.Let f ∈ L 2 ρ be given (with also  Finally, using we infer from equation (2.1) that au ∈ W 1,1 (0, 1) for any value of (α, β).
We are in a position to study the spectral problem associated with equations (2.1)-(2.3).
We are now interested in the asymptotic behavior of the eigenvalues λ n , n ≥ 0. Indeed, to apply the flatness approach, we need to prove that λ n ≥ Cn κ for some C, κ > 0 and all n ≥ 0. The estimate we shall derive is likely not sharp, but it is sufficient for the sequel.Theorem 2.9.Let a, ρ, (α, β) and the sequences (e n ) n≥0 , (λ n ) n≥0 be as in Theorem 2.8.Then (i) e n ∈ W 1,1 (0, 1) and ae n ∈ W 1,r (0, 1) for all n ≥ 0; (ii) there exists some constant C 1 > 0 such that (2.27) Then there exists some constant C 2 > 0 such that (2.28) Remark 2.10.The estimates equations (2.27) and (2.28) are not sharp, but sufficient for our aim which is to apply the flatness approach.In the classical case when a(x) = x 2−ε , q(x) = 0 and ρ(x) = 1 for all x ∈ (0, 1) with (α, β) = (1, 0), so that e n (1) = 0, it follows from the proof of Proposition 1.4 (see below (2.30) On the other hand, it is well known (see e.g.[26]) that, letting ν = ε −1 − 1 > 0, we have where is the Bessel function of order ν of the first kind, and (j ν,n ) n∈N * is the increasing sequence of zeros of J ν , which are real and satisfy is not far) from the exponent 2 in equation (2.36) as ε → 0 + (resp.as ε → 1 − ).On the other hand, we know from [26] that ) (see again [26]).It follows that and that ) of λ n is much smaller than the lower bound Proof.(i) We need several lemmas.
The proof is similar to those of Lemma 2.11 by applying the contraction principle to the map Γ from (ρu)(s)ds when δ > 0 is small enough, and by propagating the uniqueness up to [x 0 − δ, x 0 + δ] when δ is as in the statement of the lemma.Lemma 2.14.Let n ≥ 0, and let e n and λ n be as in Theorem 2.8.If u is the function associated with λ = λ n in Corollary 2.12, then e n (x) = e n (0)u(x) ∀x ∈ (0, 1), (2.48) so that e n ∈ W 1,1 (0, 1) and ae n ∈ W 1,r (0, 1).
The proof of Lemma 2.14 is complete.This ends the proof of (i) in Theorem 2.9.
(ii) In what follows, we fix some n ≥ 0 for which λ n > 0 and denote e = e n and λ = λ n to simplify the writing.The letter C will denote a constant that may change from line to line.
• Assume first that α = 0. We first check that Note also that But for 0 < x < 1 we have that and where s a(s) L 1 denotes (2.53) On the other hand, for 0 < x < 1, by Hölder inequality p .The function u(x) still denoting the solution to equations (2.45)-(2.47),we have that and hence for such x It follows that Then, by equations (2.54) and (2.55), 2(p −r) .Since by Cauchy-Schwarz inequality we obtain with equation (2.52) that  • Assume now that α = 0. Then The estimate equation (2.57) for I 1 is still valid without any change.For I 2 , we first notice that for x > (4Cλ) ≤ s a(s) , so that equation (2.58) holds again.Therefore equations (2.59) and (2.60) hold.
• Since λ n > 0 for n ≥ 1 and since the number of λ n 's in (0, 1) is finite, we infer that equation (2.60) is still valid for all the eigenvalues λ n > 0 by replacing the constant C by a larger one denoted C 1 .The proof of (ii) is complete.

Introduction of the generating functions
We shall see later that the zero-order term q(x)u in equation (1.1) can be removed thanks to a change of variables.Consider the simplified system and search for a solution of it in the form where y is the flat output and the g i 's are the generating functions.
It follows from Corollary 2.12 that ẽ = u for λ = λ n , and from Lemma 2.14 that We are in a position to prove the main results in the paper.

Step 2: Flatness approach
We follow closely [24].We assume that q = 0 in equation (1.1).Let u 0 ∈ L 2 ρ .As (e n ) n≥0 is an orthonormal basis in L 2 ρ , we can expand u 0 as a series of the e n 's: where the sequence (c n ) n≥0 of real numbers satisfies ∞ n=0 c 2 n < ∞.Using equations (2.27) and (2.28), we notice that the map z → n≥0 c n e n (0)e −λnz is analytic in the set {z = t + it ; t > 0, t ∈ R}.It follows that the map t → n≥0 c n e n (0)e −λnt is real analytic in (0, ∞), and its restriction to [ , T ] belongs to G 1 ([ , T ]) ⊂ G s ([ , T ]) for all ∈ (0, T ) and all s ∈ (1, 2).Pick s ∈ Pick any ∈ (0, T /3).Then y ∈ G s ([ , T ]), and there exist some positive numbers M , R such that Combined with equation (3.9), this yields Furthermore, and au x (0, t) = 0. We notice that for 0 < t ≤ T /3, we have that y for all x ∈ [0, 1].The above computations are valid, since for 0 < δ ≤ t ≤ T and 0 where we used the estimate x i /i! ≤ e x for x = δλ n /2 ≥ 0 and i ∈ N. It follows from equation (4.1) that u is the free evolution (i.e. with a null control) of the parabolic equation for 0 < t ≤ T /3.Therefore lim t→0 + u(., t) = u 0 in L 2 ρ .
It follows from equation (4.1) that h(t) = 0 for 0 < t ≤ T /3 and from equation (3.9) combined with the choice of s that h ∈ C ∞ ([0, T ]) with h (j) (t) = for some constant C which does not depend on j and t, where we used (i + j)! ≤ 2 i+j i!j!.As h(t) = 0 for 0 ≤ t ≤ T /3, we conclude that h ∈ G s ([0, T ]).Finally The proof of Theorem 1.1 is complete.
Remark 4.1.We stress that assumption (1.9) was used only in Step 1 the get rid of the term q(x)u in equation (1.1).If q ≡ 0 in equation (1.1), then Theorem 1.1 is still valid with the assumptions (1.5)-(1.8).