Approximate controllability for a 2D Grushin equation with potential having an internal singularity
aa r X i v : . [ m a t h . O C ] O c t Approximate controllability for a Grushinequation with potential having an internalsingularity
Morgan
Morancey ∗† Abstract
This paper is dedicated to approximate controllability for Grushinequation on the rectangle ( x, y ) ∈ ( − , × (0 , with an inverse squarepotential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on R , studied byBoscain and Laurent. The operator is both degenerate and singular onthe line { x = 0 } .The approximate controllability is studied through unique continu-ation of the adjoint system. For the range of singularity under study,approximate controllability is proved to hold whatever the degeneracy is.Due to the internal inverse square singularity, a key point in this workis the study of well-posedness. An extension of the singular operator isdesigned imposing suitable transmission conditions through the singular-ity. Then, unique continuation relies on the Fourier decomposition of the solution in one variable and Carleman estimates for the heat equa-tion solved by the Fourier components. The Carleman estimate uses asuitable Hardy inequality. Keywords : unique continuation, degenerate parabolic equation, singular po-tential, Grushin operator, self-adjoint extensions, Carleman estimate. ∗ I2M UMR 7373, Université Aix-Marseille. email: [email protected] † The author was partially supported by the “Agence Nationale de la Recherche” (ANR),Projet Blanc EMAQS number ANR-2011-BS01-017-01 and by CMLA UMR 8536, ENSCachan, 61 avenue du Président Wilson, 94235 Cachan, FRANCE. Introduction
We consider for γ > the following degenerate singular parabolic equation ∂ t f − ∂ xx f − | x | γ ∂ yy f + cx f = u ( t, x, y ) χ ω ( x, y ) , ( t, x, y ) ∈ (0 , T ) × Ω ,f ( t, x, y ) = 0 , ( t, x, y ) ∈ (0 , T ) × ∂ Ω , (1.1)with initial condition f (0 , x, y ) = f ( x, y ) , ( x, y ) ∈ Ω . (1.2)The domain is Ω := ( − , × (0 , and ω , the control domain, is an opensubset of Ω . The function χ is the indicator function. The coefficient c of thesingular potential is real and will be restricted to (cid:0) − , (cid:1) . The degeneracy set { x = 0 } coincides with the singularity set ; it separates the domain Ω in twoconnected components. Due to the singular potential, the first difficulty is togive a meaning to solutions of (1.1). Through the study of an associated heat equation, we will design a suitable extension of the considered operatorgenerating a continuous semigroup. The solutions considered in this article willbe related to this semigroup. This is detailed in Section 2. Before stating thecontrollability result, we give some motivations and justify the range of constants c ∈ (cid:0) − , (cid:1) .In [4], Boscain and Laurent studied the Laplace-Beltrami operator for theGrushin-like metric given by the orthonormal basis X = (cid:18) (cid:19) and Y = (cid:18) | x | γ (cid:19) on R × T with γ > i.e. Lu := ∂ xx u + | x | γ ∂ yy u − γx ∂ x u. (1.3)They proved that this operator with domain C ∞ (cid:0) ( R \{ } ) × T (cid:1) is essentially self-adjoint on L ( R × T ) if and only if γ ≥ . Thus, for the heat equation associatedto this Laplace-Beltrami operator, no information passes through the singularset { x = 0 } when γ ≥ . This prevents controllability from one side of thesingularity.The change of variables u = | x | γ/ v , leads to study ∆ γ v = ∂ xx v + | x | γ ∂ yy v − γ (cid:16) γ (cid:17) vx . (1.4)The model (1.1) can then be seen as a heat equation for this operator. Bychoosing a coefficient c instead of γ (cid:0) γ + 1 (cid:1) we authorize a wider class of singularpotentials and decouple the effects of the degeneracy and the singularity for abetter understanding of each one of these phenomena. Adapting the argumentsof [4], one obtains that for any γ > , the operator − ∂ xx − | x | γ ∂ yy + cx with2omain C ∞ (Ω \{ x = 0 } ) is essentially self-adjoint on L (Ω) if and only if c ≥ .Thus, to look for controllability properties, our study focuses on the range ofconstants c < .The lower bound c > − for the range of constants considered comes fromwell posedness issues linked to the use of the following Hardy inequality (seee.g. [9] for a simple proof) Z z ( x ) x d x ≤ Z z x ( x ) d x, ∀ z ∈ H ((0 , , R ) with z (0) = 0 . (1.5)The critical case in the Hardy inequality c = − is not directly covered by thetechnics of this article.The notion of controllability under study in this article is given in the followingdefinition. Definition 1.1.
Let
T > and ω ⊂ Ω . The problem (1.1) is said to beapproximately controllable from ω in time T if for any ( f , f T ) ∈ L (Ω) , forany ε > , there exists u ∈ L ((0 , T ) × ω ) such that the solution of (1.1)-(1.2),in the sense of Proposition 2.5, satisfies || f ( T ) − f T || L (Ω) ≤ ε. The main result of this article is the following characterization of approximatecontrollability.
Theorem 1.1.
Let
T > , γ > and c ∈ ( − , ) . Let ω be an open subset of Ω . Then, (1.1) is approximately controllable from ω in time T in the sense ofDefinition 1.1. Except for the critical case of the Hardy inequality ( c = − ), this theoremfills the gap, for the approximate controllability property, between validity ofHardy inequality ( c ≥ − ) and the essential self-adjointness property of [4] for c ≥ . Remark . One key point for this approximate controllability result is to givea meaning to the solutions of (1.1). As it will be noticed (see e.g. Sect. 2.4)there are various possible definitions of solutions (depending mostly on whattransmission conditions are imposed at the singularity). The validity of theapproximate controllability property under study will strongly depend on thesetransmission conditions. This is why, in Definition 1.1, it is precised that thesolutions are understood in the sense of Proposition 2.5.
Remark . Going back to the Laplace-Beltrami operator studied by Boscainand Laurent (1.4), we would get approximate controllability for the heat equa-tion associated to the operator ∆ γ for any γ ∈ (0 , . To be closer to the settingthey studied one can notice that, essentially with the same proof, the approxi-mate controllability result of Theorem 1.1 also holds on ( − , × T . This willbe detailed in Remark 3.4. 3y a classical duality argument, approximate controllability will be studiedthrough unique continuation of the adjoint system. The unique continuation re-sult will be proved by a suitable Carleman inequality for an associated sequenceof problems. This Carleman estimate rely on a precise Hardy inequality.The model (1.1) can also be seen as an extension of [3] where Beauchard etal. studied the null controllability without the singular potential (i.e. in thecase c = 0 ). The authors proved that null controllability holds if γ ∈ (0 , anddoes not hold if γ > . In the case γ = 1 , for ω a strip in the y direction, nullcontrollability holds if and only if the time is large enough.The inverse square potential for the Grushin equation has already been takeninto account by Cannarsa and Guglielmi in [7] but in the case where both degen-eracy and singularity are at the boundary. With our notations, they proved nullcontrollability in sufficiently large time for Ω = (0 , × (0 , , ω = ( a, b ) × (0 , , γ = 1 and any c > − . They also proved that approximate controllability holdsfor any control domain ω ⊂ Ω , any γ > and any c > − . Thus, the factthat our model presents an internal singularity instead of a boundary singular-ity deeply affects the approximate controllability property as it does not holdwhen c > .As in [3], the results of this article will strongly use an associated sequenceof problems. As a by-product of the proof of Theorem 1.1, we obtain thefollowing approximate controllability result for the heat equation with asingular inverse square potential. Theorem 1.2.
Let
T > and c ∈ (cid:0) − , (cid:1) . Let ω be an open subset of ( − , .Then approximate controllability holds for ∂ t f − ∂ xx f + cx f = u ( t, x ) χ ω ( x ) , ( t, x ) ∈ (0 , T ) × ( − , ,f ( t, −
1) = f ( t,
1) = 0 , t ∈ (0 , T ) ,f (0 , x ) = f ( x ) , x ∈ ( − , , (1.6) where the solutions of (1.6) are given by Proposition 2.2. The null controllability issue for the heat equation with such an internalinverse square singularity remains an open question. Like (1.1), it has to benoticed that the solutions of (1.6) are related to the semigroup generated by asuitable extension of the Laplace operator with a singular potential. Due to the internal singularity and the fact that the considered operators admitseveral self-adjoint extensions, the functional setting and the well posedness arecrucial issues in this article. Section 2 is dedicated to these questions.Section 3 is dedicated to the study of the unique continuation property forthe adjoint system. Using decomposition in Fourier series in the y variable andunique continuation for uniformly parabolic operator we reduce the problem to4he study of a singular problem with a boundary inverse square potential.Then we conclude proving a suitable Carleman inequality using an adaptedHardy’s inequality.We end this introduction by a brief review of previous results concerningdegenerate and/or singular parabolic equations. The first result for a heat equation with an inverse square potential c k x k dealswith well posedness issues. In [2], Baras and Goldstein proved complete in-stantaneous blow-up for positive initial conditions in space dimension N if c < c ∗ ( N ) := − ( N − . This critical value is the best constant in Hardy’sinequality. Cabré and Martel also studied in [6] the relation between blow-upof such equations and the existence of an Hardy inequality. Thus, most of thefollowing studies focus on the range of constants c ≥ c ∗ ( N ) . In this case, wellposedness in L (Ω) has been proved in [23] by Vazquez and Zuazua. Noticethat in those cases the singular set is the point { } (the singularity being at theboundary in the one dimensional case) whereas in this article the singular setis a line separating the domain in two connected components.The controllability issues were first studied for degenerate equations. In [8, 17,9, 10], Cannarsa, Martinez and Vancostenoble proved null controllability witha distributed control for a one dimensional parabolic equation degenerating atthe boundary. Then, they extended this result to more general degeneraciesand in dimension two. These results are based on suitable Hardy inequalitiesand Carleman estimates. More recently, Cannarsa, Tort and Yamamoto [11]proved approximate controllability for this one dimensional equation degenerat-ing at the boundary with a Dirichlet control on the degenerate boundary. Then,Gueye [15] proved null controllability for the same model. Its proof relies ontransmutation and appropriate nonharmonic Fourier series.Meanwhile, these Carleman estimates were adapted for heat equation with aninverse square potential c k x k in dimension N ≥ . In [22], Vancostenoble andZuazua proved null controllability in the case where the control domain ω con-tains an annulus centred on the singularity. Their proof relies on a decompo-sition in spherical harmonics reducing the problem to the study of a heatequation with an inverse square potential which is singular at the boundary.The geometric assumptions on the control domain were then removed by Erve-doza in [14] using a direct Carleman strategy in dimension N ≥ . Notice thatalthough these results deal with internal singularity they cannot be adapted toour setting. Indeed, in [22] it is crucial that the singularity of the prob-lem obtained by decomposition in spherical harmonics is at the boundary. TheCarleman strategy developed in [14] cannot be adapted in this article becauseour singularity is no longer a point but separates the domain in two connectedcomponents.For null controllability for a one dimensional parabolic equation both degener-ate and singular at the boundary we refer to [21] by Vancostenoble. The proof5xtends the previous Carleman strategy together with an improved Hardy in-equality.As the functional setting for this study is obtained through the design of a suit-able self-adjoint extension of our Grushin-like operator, we mention the work [5]conducted simultaneously to this study. In this paper, Boscain and Prandi stud-ied some extensions of the Laplace-Beltrami operator (1.3) for γ ∈ R . Amongother things, they designed for a suitable range of constants an extension calledbridging extension that allows full communication through the singular set.Even if the models under consideration are not exactly the same, the approxi-mate controllability from one side of the singularity given by Theorem 1.1 is inagreement with the existence of this bridging extension. The previous results of the literature dealing with an inverse square potentialwere obtained thanks to some Hardy-type inequality. For a boundary inversesquare singularity (as in [21]), the condition z (0) = 0 needed for (1.5) to hold iscontained in the homogeneous Dirichlet boundary conditions considered. Thus,in [21], the appropriate functional setting to study the operator − ∂ xx + cx with c > − is n f ∈ H loc ((0 , ∩ H (0 ,
1) ; − ∂ xx f + cx f ∈ L (0 , o . For an internal inverse square singularity one still has Z − z ( x ) x d x ≤ Z − z x ( x ) d x, ∀ z ∈ H ( − , such that z (0) = 0 . (2.1)This inequality ceases to be true if z (0) = 0 . Thus, the functional setting mustcontain some informations on the behaviour of the functions at the singularity.In this section, we design a suitable self-adjoint extension of the operator − ∂ xx − | x | γ ∂ yy + cx on C ∞ (Ω \{ x = 0 } ) . The next subsection deals withan associated one dimensional equation. Section 2.3 will then relate this onedimensional problem to the original problem in dimension two. In all whatfollows, the coefficient of the singular potential will be parametrized in the form c = c ν where c ν := ν − , for ν ∈ (0 , . (2.2) operator For n ∈ N ∗ , γ > and ν ∈ (0 , we consider the following homogeneous problem ∂ t f − ∂ xx f + c ν x f + ( nπ ) | x | γ f = 0 , ( t, x ) ∈ (0 , T ) × ( − , ,f ( t, −
1) = f ( t,
1) = 0 , t ∈ (0 , T ) . (2.3)6his equation is formally the homogeneous equation satisfied by the coefficientsof the Fourier expansion in the y variable done in [3] and will be linked to (1.1)in Sect. 2.3. From now on, we focus on the well posedness of (2.3). Remark . A naive functional setting for this equation is the adaptation of [21] n f ∈ L ( − ,
1) ; f | [0 , ∈ H loc ((0 , ∩ H (0 , ,f | [ − , ∈ H loc ([ − , ∩ H ( − , and − ∂ xx f + c ν x f ∈ L ( − , o . However, a functional setting where the two problems on ( − , and (0 , are well posed is not pertinent for the control problem from one side of thesingularity. It leads to decoupled dynamics on the connected components of ( − , ∪ (0 , .We study the differential operator A n f ( x ) := − ∂ xx f ( x ) + c ν x f ( x ) + ( nπ ) | x | γ f ( x ) . As ν ∈ (0 , , the results of [4] imply that A n defined on C ∞ (( − , ∪ (0 , admits several self-adjoint extensions. We here specify the self-adjoint extensionthat will be used. Let ˜ H ( − ,
1) := (cid:8) f ∈ H ( − ,
1) ; f (0) = f ′ (0) = 0 (cid:9) , and F s := n f ∈ L ( − ,
1) ; f = c +1 | x | ν + + c +2 | x | − ν + on (0 , and f = c − | x | ν + + c − | x | − ν + on ( − , o . Notice that for any f s ∈ F s , (cid:16) − ∂ xx + c ν x (cid:17) f s ( x ) = 0 , ∀ x ∈ ( − , ∪ (0 , . (2.4)The parametrization (2.2) of the coefficient of the singular potential by ν allowsto write easily the functions of F s .The domain of the operator is defined by D ( A n ) := n f = f r + f s ; f r ∈ ˜ H ( − , , f s ∈ F s such that f ( −
1) = f (1) = 0 ,c − + c − + c +1 + c +2 = 0 and ( ν + 12 ) c − + ( − ν + 12 ) c − = ( ν + 12 ) c +1 + ( − ν + 12 ) c +2 o . (2.5)Notice that for ν ∈ (0 , , D ( A n ) ⊂ L ( − , . In the following, this uniquedecomposition of functions of D ( A n ) will be referred to as the regular partfor f r and the singular part for f s . As this domain is independent of n , it7ill be denoted by D ( A ) in the rest of this article. The coefficients of thesingular part will be denoted by c +1 if there is no ambiguity and c +1 ( f ) otherwise.The conditions imposed on these coefficients in (2.5) will be referred to as the transmission conditions . These conditions are discussed in Remark 2.3, theirrole and origin are discussed in Sect. 2.2 and 2.4.This operator satisfies the following properties Proposition 2.1.
For any n ∈ N ∗ and ν ∈ (0 , , the operator ( A n , D ( A )) isself-adjoint on L ( − , . Moreover, for any f ∈ D ( A ) , h A n f, f i ≥ m ν Z − ∂ x f r ( x ) d x + ( nπ ) Z − | x | γ f ( x ) d x, (2.6) where m ν := min { , ν } . Before proving this proposition in Sect. 2.2, we give some comments on thisconstruction of the d operator. Remark . As noticed in (2.4), the functions of F s are chosen in the kernel ofthe singular differential operator − ∂ xx + c ν x . Thus, for any f ∈ D ( A ) , A n f = (cid:16) − ∂ xx f r + c ν x f r (cid:17) + ( nπ ) | x | γ f. As done in [1, Proposition 3.1], for any f r ∈ ˜ H ( − , , writing f r ( x ) = Z x ( x − s ) f ′′ r ( s )d s, and applying Minkowski’s integral inequality we get that the map x x f r ( x ) belongs to L ( − , . Thus, ( A n , D ( A )) is indeed an operator in L ( − , . Remark . The reason for imposing these particular transmission conditionsis threefold. First, it implies the self-adjointness of the operator under consid-eration. This will be pointed out in the proof of Proposition 2.1. This choice isguided by the general theory of self-adjoint extensions from the point of view ofboundary conditions as detailed by Zettl [24, Theorem 13.3.1, Case 5]. For thesake of clarity, the proof of self-adjointness is done independently of this generaltheory in Sect. 2.2. A discussion relating this general theory and the domain(2.5) together with other possible choices is done in Sect. 2.4.The second interest of these transmission conditions is to ensure the posi-tivity of the operator, as detailed in the proof of Proposition 2.2.Finally, these transmission conditions are really transmission conditions inthe sense that they allow some information to cross the singularity. In matrixform, the transmission conditions can be rewritten as (cid:18) c +1 ( f ) c +2 ( f ) (cid:19) = − ν (cid:18) − ν − ν + 1 1 (cid:19) (cid:18) c − ( f ) c − ( f ) (cid:19) , ∀ f ∈ D ( A ) . (2.7)Thus, the invertibility of the above matrix implies that if the singular part ofsome function f ∈ D ( A ) identically vanishes on one side of the singularity it alsovanishes on the other side. This is a crucial point for the proof of approximatecontrollability. 8sing Proposition 2.1, the well posedness of the one dimensional system (2.3)follows from Proposition 2.1 and the Hille-Yosida theorem (see e.g. [12, Theorem3.2.1]). Proposition 2.2.
For any n ∈ N ∗ and any f ∈ L ( − , , problem (2.3) withinitial condition f (0 , · ) = f has a unique solution f ∈ C ([0 , + ∞ ) , L ( − , ∩ C ((0 , + ∞ ) , D ( A )) ∩ C ((0 , + ∞ ) , L ( − , . This solution satisfies || f ( t ) || L ( − , ≤ || f || L ( − , . In all what follows, we denote by e − A n t the semigroup generated by − A n i.e.for any f ∈ L ( − , , the function t e − A n t f is the solution of (2.3) givenby Proposition 2.2. problem This subsection is dedicated to the proof of Proposition 2.1. The proof uses thefollowing two lemmas.The following lemma is proved in [24, Lemma 9.2.3].
Lemma 2.1.
For f, g ∈ ˜ H ( − , ⊕ F s , if we define [ f, g ]( x ) := ( f g ′ − f ′ g )( x ) , ∀ x = 0 , then Z − (cid:16) − ∂ xx f + c ν x f (cid:17) ( x ) g ( x )d x = Z − f ( x ) (cid:16) − ∂ xx g + c ν x g (cid:17) ( x )d x + [ f, g ](1) − [ f, g ](0 + ) + [ f, g ](0 − ) − [ f, g ]( − . The following lemma characterizes the behaviour of the regular part at thesingularity.
Lemma 2.2.
For any f ∈ H (0 , , satisfying f (0) = f ′ (0) = 0 , lim x → f ( x ) x = 0 and lim x → f ′ ( x ) x = 0 . The same holds for functions in ˜ H ( − , and both limits x → ± .Proof of Lemma 2.2. As f (0) = f ′ (0) = 0 , it comes that f ( x ) = Z x Z t f ′′ ( s )d s d t. Then, Cauchy-Schwarz inequality implies, | f ( x ) | ≤ Z x √ t (cid:18)Z t | f ′′ ( s ) | d s (cid:19) d t ≤ (cid:18)Z x | f ′′ ( s ) | d s (cid:19) x . The proof of the second limit is similar.9e now turn to the proof of Proposition 2.1.
Proof of Proposition 2.1.
We start by proving that ( A n , D ( A )) is a symmetricoperator. Thus, A ∗ n is an extension of A n and self-adjointness will follow fromthe equality D ( A ∗ n ) = D ( A n ) . First step : we prove that ( A n , D ( A )) is a symmetric operator. Let f, g ∈ D ( A ) . As f (1) = g (1) = f ( −
1) = g ( −
1) = 0 , it comes that [ f, g ](1) = [ f, g ]( −
1) = 0 . Lemma 2.2 implies that [ f, g ](0 + ) = [ f s , g s ](0 + ) = (cid:0) c +1 ( f ) c +2 ( g ) − c +2 ( f ) c +1 ( g ) (cid:1) [ | x | ν + , | x | − ν + ](0 + ) , and [ f, g ](0 − ) = [ f s , g s ](0 − )= (cid:0) c − ( f ) c − ( g ) − c − ( f ) c − ( g ) (cid:1) [ | x | ν + , | x | − ν + ](0 − )= − (cid:0) c − ( f ) c − ( g ) − c − ( f ) c − ( g ) (cid:1) [ | x | ν + , | x | − ν + ](0 + ) . Thus, using the matrix formulation (2.7) of the transmission conditions, we getthat for any f, g ∈ D ( A ) c +1 ( f ) c +2 ( g ) − c +2 ( f ) c +1 ( g ) = − (cid:0) c − ( f ) c − ( g ) − c − ( f ) c − ( g ) (cid:1) . This leads to [ f, g ](0 + ) = [ f, g ](0 − ) . Finally, Lemma 2.1 implies that for any f, g ∈ D ( A ) , h A n f, g i = h f, A n g i .Thus, to prove self-adjointness it remains to prove that D ( A ∗ n ) = D ( A ) .As D ( A ) is independent of n and x ( nπ ) | x | γ ∈ L ∞ ( − , it comes that D ( A ∗ n ) = D ( A ∗ ) . Second step : minimal and maximal domains.
First, we explicit the minimaland maximal domains in the case of a boundary singularity. Without loss ofgenerality, we study the operator on (0 , .Using [1, Proposition 3.1], the minimal and maximal domains associated to thedifferential expression A in L (0 , are respectively equal to H ([0 , (cid:8) y ∈ H ([0 , y (0) = y (1) = y ′ (0) = y ′ (1) = 0 (cid:9) and (cid:8) y ∈ H ([0 , y (0) = y ′ (0) = 0 (cid:9) ⊕ Span n x ν + , x − ν + o . Then, [24, Lemma 13.3.1] imply that the minimal and maximal domains asso-ciated to A on the interval ( − , are given by D min := n f ∈ ˜ H ( − ,
1) ; f ( −
1) = f (1) = f ′ ( −
1) = f ′ (1) = 0 o , (2.8)10nd D max := ˜ H ( − , ⊕ F s . (2.9)Besides, the minimal and maximal operators form an adjoint pair. Third step : self-adjointness.
The operator A being a symmetric extensionof the minimal operator it comes that D ( A ) ⊂ D ( A ∗ ) ⊂ D max . Let g ∈ D ( A ∗ ) be decomposed as g = g r + g s with g r ∈ ˜ H ( − , and g s ∈ F s . We prove that g satisfy the boundary and transmission conditions. By the definition of D ( A ∗ ) ,there exists c > such that for any f ∈ D ( A ) , |h A f, g i| ≤ c || f || L . Let f ∈ D ( A ) ∩ ˜ H ( − , be such that f ≡ in ( − , . Then, Lemma 2.1implies that h A f, g i = h f, A g i + [ f, g ](1) = h f, A g i + f ′ (1) g (1) . Thus, g (1) = 0 . Symmetric arguments imply that g ( −
1) = 0 .We now turn to the transmission conditions. Let f ∈ D ( A ) be such that itssingular part is given by c +1 ( f ) := 12 ν , c +2 ( f ) := − ν . Then, the transmission conditions imply c − ( f ) = 12 ν , c − ( f ) = − ν . By Lemma 2.1 h A f, g i = h f, A g i + [ f, g ](0 − ) − [ f, g ](0 + ) . Using Lemma 2.2 it comes that the regular parts have no contribution at i.e. [ f, g ](0 − ) = [ f s , g s ](0 − ) and [ f, g ](0 + ) = [ f s , g s ](0 + ) . Straightforward computa-tions lead to [ f, g ](0 + ) = − c +1 ( g ) − c +2 ( g ) , [ f, g ](0 − ) = c − ( g ) + c − ( g ) . We thus recover the first transmission condition. The second transmission condi-tion follows from similar computations with the choice of a particular f ∈ D ( A ) satisfying c +1 ( f ) := − ν − ν , c +2 ( f ) := − ν + ν . Thus, D ( A ∗ ) ⊂ D ( A ) . This proves that ( A n , D ( A )) is a self-adjoint operator. Fourth step : positivity.
We end the proof of Proposition 2.1 by proving(2.6). Let f ∈ D ( A ) . 11sing Lemma 2.1 and integration by parts it comes that h A n f, f i = Z − (cid:16) − ∂ xx f r + c ν x f r (cid:17) ( x ) f ( x )d x + Z − ( nπ ) | x | γ f ( x )d x, = Z − ( ∂ x f r ) ( x ) + c ν x f r ( x )d x + Z − ( nπ ) | x | γ f ( x )d x + ( − ∂ x f r )(1) f r (1)+ ∂ x f r ( − f r ( −
1) + [ f r , f s ](1) − [ f r , f s ](0 + ) + [ f r , f s ](0 − ) − [ f r , f s ]( − . Using Lemma 2.2, it comes that [ f r , f s ](0 + ) = [ f r , f s ](0 − ) = 0 . Gathering theboundary terms and using f (1) = f ( −
1) = 0 it comes that h A n f, f i = Z − ( ∂ x f r ) ( x ) + c ν x f r ( x )d x + Z − ( nπ ) | x | γ f ( x )d x + f r (1) ∂ x f s (1) − f r ( − ∂ x f s ( − . (2.10)As f (1) = f ( −
1) = 0 , it comes that f r (1) ∂ x f s (1) = − (cid:0) c +1 ( f ) + c +2 ( f ) (cid:1) (cid:18)(cid:18) ν + 12 (cid:19) c +1 ( f ) + (cid:18) − ν + 12 (cid:19) c +2 ( f ) (cid:19) ,f r ( − ∂ x f s ( −
1) = (cid:0) c − ( f ) + c − ( f ) (cid:1) (cid:18)(cid:18) ν + 12 (cid:19) c − ( f ) + (cid:18) − ν + 12 (cid:19) c − ( f ) (cid:19) . Thus, a sufficient condition to ensure that A n is non-negative is (cid:0) c − ( f ) + c − ( f ) (cid:1) (cid:18)(cid:18) ν + 12 (cid:19) c − ( f ) + (cid:18) − ν + 12 (cid:19) c − ( f ) (cid:19) = − (cid:0) c +1 ( f ) + c +2 ( f ) (cid:1) (cid:18)(cid:18) ν + 12 (cid:19) c +1 ( f ) + (cid:18) − ν + 12 (cid:19) c +2 ( f ) (cid:19) . (2.11)This follows directly from the transmission conditions. Thus, (2.10) implies h Af, f i ≥ Z − ( ∂ x f r ) ( x ) + c ν x f r ( x )d x + Z − ( nπ ) | x | γ f ( x )d x. (2.12)If c ν ≥ , we get (2.6) with m ν = 1 . If c ν < , using Hardy’s inequality (2.1),it comes that Z − ( ∂ x f r ) ( x ) + c ν x f r ( x )d x = (1 + 4 c ν ) Z − ( ∂ x f r ) ( x )d x − c ν Z − (cid:18) ( ∂ x f r ) ( x ) − f r ( x ) x (cid:19) d x ≥ (1 + 4 c ν ) Z − ( ∂ x f r ) ( x )d x. This gives (2.6) with m ν = 4 ν . This ends the proof of Proposition 2.1.12 .3 Semigroup associated to the problem Let f ∈ L (Ω) . For almost every x ∈ ( − , , f ( x, · ) ∈ L (0 , and thus canbe expanded in Fourier series as follows f ( x, y ) = X n ∈ N ∗ f n ( x ) ϕ n ( y ) , (2.13)where ( ϕ n ) n ∈ N ∗ is the Hilbert basis of L (0 , of eigenvectors of the Laplaceoperator on H (0 , with homogeneous boundary conditions i.e. ϕ n ( y ) := √ nπy ) , ∀ n ∈ N ∗ , and f n ( x ) := Z − f ( x, y ) ϕ n ( y )d y. For any t ∈ (0 , T ) , we define the following operator ( S ( t ) f )( x, y ) := X n ∈ N ∗ f n ( t, x ) ϕ n ( y ) , (2.14)where for any n ∈ N ∗ , f n ( t ) := e − A n t f n . Then, the following proposition holds. Proposition 2.3. S ( t ) defined by (2.14) is a continuous semigroup of contrac-tion in L (Ω) .Proof of Proposition 2.3. By Proposition 2.2, S ( t ) is well defined, with valuein L (Ω) , it is a semigroup and satisfies the contraction property. For any f ∈ L (Ω) , we have || S ( t ) f − f || L (Ω) = X n ∈ N ∗ || f n ( t, · ) − f n || L ( − , . By Proposition 2.2 it comes that || f n ( t, · ) − f n || L ( − , −→ t → , || f n ( t, · ) − f n || L ( − , ≤ || f n || L ( − , . Thus, by the dominated convergence theorem, S ( t ) f −→ t → f in L (Ω) .Recall that the infinitesimal generator A of S ( t ) is defined on D ( A ) := (cid:26) f ∈ L (Ω) ; lim t → S ( t ) f − ft exists (cid:27) , by A f := lim t → S ( t ) f − ft . L norm. Then, from [18, Theorems 1.3.1and 1.4.3] it comes that ( A , D ( A )) is a closed dissipative densely defined oper-ator and satisfies for any λ > , R ( λI − A ) = L (Ω) . The following propositionlinks the system (1.1) and the semigroup S ( t ) . Proposition 2.4.
The infinitesimal generator A of S ( t ) is characterized by D ( A ) = (cid:26) f ∈ L (Ω) ; f = X n ∈ N ∗ f n ( x ) ϕ n ( y ) with f n ∈ D ( A ) and X n ∈ N ∗ || A n f n || L ( − , < + ∞ (cid:27) , (2.15) and A f = − X n ∈ N ∗ ( A n f n )( x ) ϕ n ( y ) . (2.16) This operator extends the Grushin differential operator in the sense that A f = ∂ xx f + | x | γ ∂ yy f − c ν x f, ∀ f ∈ C ∞ (Ω \{ x = 0 } ) . (2.17) Proof of Proposition 2.4.
Let f ∈ D ( A ) . Then, A f ∈ L (Ω) and S ( t ) f − f t −→ t → A f , in L (Ω) . As A f ∈ L (Ω) , it can be decomposed in Fourier series in the y variable i.e. A f ( x, y ) = X n ∈ N ∗ ( A f ) n ( x ) ϕ n ( y ) . Thus, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ( t ) f − f t − A f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (Ω) = X n ∈ N ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f n ( t ) − f n t − ( A f ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ( − , −→ t → . This implies that for any n ∈ N ∗ , f n ∈ D ( A ) and ( A f ) n = − A n f n . We thus get −A f = X n ∈ N ∗ ( A n f n )( x ) ϕ n ( y ) . Conversely, let g ∈ L (Ω) be such that for any n ∈ N ∗ , g n ∈ D ( A ) and P n ∈ N ∗ || A n g n || L ( − , < + ∞ . Let f ∈ D ( A ) . Then, |hA f, g i| ≤ X n ∈ N ∗ |h A n f n , g n i| ≤ X n ∈ N ∗ || f n || L ! X n ∈ N ∗ || A n g n || L ! . g ∈ D ( A ∗ ) . Finally, self-adjointness of S ( t ) and thus of A ends the proof of (2.15). Straightforward computations lead to (2.17) and thusends the proof of Proposition 2.4.Using Proposition 2.4, we rewrite (1.1)-(1.2) in the form ( f ′ ( t ) = A f ( t ) + v ( t ) , t ∈ [0 , T ] ,f (0) = f , (2.18)where v ( t ) : ( x, y ) ∈ Ω u ( t, x, y ) χ ω ( x, y ) . The following proposition is classi-cal (see e.g. [18]) and ends this well posedness section Proposition 2.5.
For any f ∈ L (Ω) , T > and v ∈ L ((0 , T ); L (Ω)) ,system (2.18) has a unique mild solution given by f ( t ) = S ( t ) f + Z t S ( t − τ ) v ( τ )d τ, t ∈ [0 , T ] . In the following a solution of (1.1) will mean a solution of (2.18).
This subsection is dedicated to enlighten the choices made in the constructionof the functional setting leading to the definition (2.5) of D ( A ) .The question of finding the self-adjoint extensions of a given closed symmet-ric operator is classical. In [19, Theorem X.2] such extensions are characterizedby means of isometries between the deficiency subspaces. The particular caseof Sturm-Liouville operators has been widely studied : most of these resultare contained in [24]. The self-adjoint extensions are characterized by means ofgeneralized boundary conditions. In our case, we are concerned with the Sturm-Liouville operator − d d x + c ν x on the interval ( − , . This fits in the setting of[24, Chapter 13]. The number of boundary conditions to impose is given by thedeficiency index. Following [1, Proposition 3.1], it comes that our operator onthe interval (0 , has deficiency index . This is closely related to the fact that ν ∈ (0 , . Then, [24, Lemma 13.3.1] implies that the deficiency index for theinterval ( − , is . We thus get the following proposition which is simply arewriting of [24, Theorem 13.3.1 Case 5]. Proposition 2.6.
Let u and v in D max be such that their restriction on (0 , (resp. ( − , ) are linearly independent modulo H (0 , (resp. H ( − , ) and [ u, v ]( −
1) = [ u, v ](0 − ) = [ u, v ](0 + ) = [ u, v ](1) = 1 . Let M , . . . , M be × complex matrices. Then every self-adjoint extensionof the minimal operator is given by the restriction of D max to the functions f satisfying the boundary conditions M (cid:18) [ f, u ]( − f, v ]( − (cid:19) + M (cid:18) [ f, u ](0 − )[ f, v ](0 − ) (cid:19) + M (cid:18) [ f, u ](0 + )[ f, v ](0 + ) (cid:19) + M (cid:18) [ f, u ](1)[ f, v ](1) (cid:19) = 0 , here the matrices satisfy ( M M M M ) has full rank and M EM ∗ − M EM ∗ + M EM ∗ − M EM ∗ = 0 , with E := (cid:18) −
11 0 (cid:19) . Conversely, every choice of such matrices defines a self-adjoint extension.
We end this section by giving the choice of such matrices that we made andgive another functional setting that would lead to well posedness but that is notadapted to controllability issues. We define on (0 , u and v to be solutions of − f ′′ ( x ) + c ν x f ( x ) = 0 with ( u (1) = 0 , u ′ (1) = 1) and ( v (1) = − , v ′ (1) = 0) i.e. u ( x ) = 12 ν x ν + − ν x − ν + ,v ( x ) = − ν − ν x ν + − ν + ν x − ν + . Thus for any f ∈ D max , [ f, u ](1) = f (1) and [ f, v ](1) = f ′ (1) , and for any x ∈ [0 , , [ u, v ]( x ) ≡ . We design u and v similarly on ( − , i.e. u ( x ) = − ν | x | ν + + 12 ν | x | − ν + ,v ( x ) = − ν − ν | x | ν + − ν + ν | x | − ν + . Due to the choice of functions u and v , the homogeneous Dirichlet conditionsat ± are implied by the choice M = , M = M , M = M , M = . Then, the conditions of Proposition 2.6 are satisfied if and only if the matrix ( ˜ M ˜ M ) has rank and det( ˜ M ) = det( ˜ M ) . Straightforward computationslead to, for any f ∈ D max [ f, u ](0 + ) = c +1 + c +2 , [ f, v ](0 + ) = (cid:18) ν + 12 (cid:19) c +1 + (cid:18) − ν + 12 (cid:19) c +2 , [ f, u ](0 − ) = c − + c − , [ f, v ](0 − ) = − (cid:18) ν + 12 (cid:19) c − − (cid:18) − ν + 12 (cid:19) c − . Construction of D ( A ) . The choice ˜ M = ˜ M = (cid:18) (cid:19) D ( A ) in (2.5). The computations done in the fourthstep of the proof of Proposition 2.1 (see (2.12)) prove the positivity and thus,Proposition 2.1 could also be seen as an application of Proposition 2.6. Other construction.
At this stage, there is another choice that would leadto a self-adjoint positive extension. If, we set ˜ M = (cid:18) (cid:19) and ˜ M = (cid:18) (cid:19) , then the domain with conditions c +1 = − c +2 , c − = − − ν + ν + c − , (2.19)give rise to a self-adjoint positive operator. However, from a point of view of con-trollability, this domain does not seem interesting as this conditions couple thecoefficients on each side on the singularity. As it can be noticed from the proofof Proposition 3.1, once the domain of A n is defined to ensure well-posedness,the only requirement on the transmission condition to obtain the approximatecontrollability result of Theorem 1.1 is c − = c − = 0 = ⇒ c +1 = c +2 = 0 . This is not satisfied for the transmission conditions (2.19) and we cannot applythe results developed in this article to this functional setting.As a matter of fact, one gets from [24, Proposition 10.4.2] (characterizingself-adjoint extensions in the case of a boundary singularity, similarly to Propo-sition 2.6 for internal singularity) that the two problems on ( − , and (0 , with conditions (2.19) are well-posed. Thus the dynamics really are decoupledand approximate controllability from one side of the singularity does not holdfor the transmission conditions (2.19). Remark . Notice that the goal here is not to give an exhaustive characteri-zation. This will be pointless with regards to the main goal of giving a meaningto (1.1). Indeed, in the construction of the semigroup S we here imposed thesame transmission conditions for each Fourier component. As soon as we havedifferent transmission conditions ensuring to have a self-adjoint extension A n ,there is infinitely many extensions A generating a semigroup. Symmetry of transmission.
At this stage, one can wonder if there arenon-symmetric transmission conditions i.e. a choice of matrices ˜ M and ˜ M such that c − = c − = 0 = ⇒ c +1 = c +2 = 0 ,c +1 = c +2 = 0 = ⇒ c − = c − = 0 . Rewriting the condition ˜ M (cid:18) [ f, u ](0 − )[ f, v ](0 − ) (cid:19) + ˜ M (cid:18) [ f, u ](0 + )[ f, v ](0 + ) (cid:19) = (cid:18) (cid:19) ˆ M (cid:18) c − c − (cid:19) + ˆ M (cid:18) c +1 c +2 (cid:19) = (cid:18) (cid:19) we get that the condition det( ˜ M ) = det( ˜ M ) implies det( ˆ M ) = det( ˆ M ) . It isthen not possible to have only one of the matrices ˆ M and ˆ M invertible. Thus,in this setting, there is no choice of non-symmetric transmission conditions. We consider the adjoint system of (1.1) ∂ t g − ∂ xx g − | x | γ ∂ yy g + c ν x g = 0 , ( t, x, y ) ∈ (0 , T ) × Ω ,g ( t, x, y ) = 0 , ( t, x, y ) ∈ (0 , T ) × ∂ Ω ,g (0 , x, y ) = g ( x, y ) , ( x, y ) ∈ Ω . (3.1)Here again this system is understood in the sense of the self-adjoint extension A designed i.e. for any g ∈ L (Ω) , the solution of (3.1) is given by S ( t ) g .This section is dedicated to the study of the following unique continuationproperty Definition 3.1.
Let
T > and ω ⊂ Ω . We say that the unique continuationproperty from ω holds for system (3.1) if the only solution of (3.1) vanishing on (0 , T ) × ω is identically zero on (0 , T ) × Ω i.e. (cid:16) g ∈ L (Ω) , χ ω S ( t ) g = 0 for a.e. t ∈ (0 , T ) (cid:17) = ⇒ g = 0 . By a classical duality argument, Theorem 1.1 is equivalent to the followingunique continuation theorem.
Theorem 3.1.
Let
T > , γ > and ν ∈ (0 , . Let ω be an open subset of Ω .The unique continuation property from ω holds for the adjoint system (3.1) inthe sense of Definition 3.1. Without loss of generality, we may assume that ω is an open subset of oneconnected component of Ω \{ x = 0 } . As it will be noticed in Remark 3.1, if ω intersects both connected components of Ω \{ x = 0 } then the proof is simpler.In the following we assume that ω ⊂ ( − , × (0 , .The rest of this section is dedicated to the proof of Theorem 3.1. In Sect. 3.1,we prove that if g ( t ) := S ( t ) g is vanishing on (0 , T ) × ω then it is vanishing on (0 , T ) × ( − , × (0 , . This will imply that any Fourier component g n has nosingular part and is identically zero on [ − , .Then, we are left to study a one dimensional equation on the regular partwith a boundary inverse square singularity. Dealing with the regular part, weknow furthermore that the function under study has the H regularity andsatisfies ∂ x g n ( t,
0) = 0 . This will be used in Sect. 3.2 to prove a suitable Car-leman estimate, relying on an adapted Hardy inequality, to end the proof ofTheorem 3.1. 18 .1 Reduction to the case of a boundary singularity
The goal of this section is the proof of the following proposition
Proposition 3.1.
Let
T > , γ > , ν ∈ (0 , and ω be an open subset of ( − , × (0 , . Assume that g ∈ L (Ω) is such that g ( t ) := S ( t ) g is vanishingon (0 , T ) × ω . Then g is vanishing on (0 , T ) × ( − , × (0 , . Moreover, forany n ∈ N ∗ , the n th Fourier component satisfies c − ( g n ) = c − ( g n ) = c +1 ( g n ) = c +2 ( g n ) = 0 ,g n ( t, x ) = χ (0 , ( x ) g n,r ( t, x ) for every ( t, x ) ∈ (0 , T ) × ( − , , where g n,r is the regular part of g n .Proof of Proposition 3.1. Let ε > be such that ω ⊂ Ω − ε := ( − , − ε ) × (0 , . For every t ∈ [0 , T ] , ( S ( t ) g )( x, y ) = X n ∈ N ∗ g n ( t, x ) ϕ n ( y ) , where g n is the solution of (2.3) with initial condition g n .We check that on Ω − ε , the operator A is uniformly elliptic. Let h ∈ D ( A ) and φ ∈ C ∞ (Ω − ε ) . Then, hA h, φ i L (Ω − ε ) = Z − ε − Z A h ( x, y ) φ ( x, y )d y d x = − X n ∈ N ∗ h A n h n , φ n i L ( − , − ε ) = − X n ∈ N ∗ h h n , A n φ n i L ( − , − ε ) = h h, (cid:16) ∂ xx + | x | γ ∂ yy − c ν x (cid:17) φ i L (Ω − ε ) . Thus, h ∈ D ( A ) implies that A h D ′ (Ω − ε ) = (cid:16) ∂ xx + | x | γ ∂ yy − c ν x (cid:17) h. As h ∈ D ( A ) , this equality also holds in L (Ω − ε ) . In particular, this impliesthat ∂ xx h + | x | γ ∂ yy h ∈ L (Ω − ε ) , and also that A is uniformly elliptic on Ω − ε . Thus, using classical unique contin-uation results for uniformly parabolic operators with variable coefficients (seee.g. [20, Theorem 1.1]), it comes that S ( t ) g = 0 for every t ∈ (0 , T ] in L (Ω − ε ) .19hen, it comes that S ( t ) g = 0 for every t ∈ (0 , T ] in L (Ω − ) . If, for any n ∈ N ∗ , we decompose g n in regular and singular part (as defined in (2.5)) weget c − ( g n ( t )) = c − ( g n ( t )) = 0 , ∀ t ∈ (0 , T ) , (3.2) g n,r ( t, x ) = 0 , ∀ ( t, x ) ∈ (0 , T ) × ( − , . (3.3)Using the transmission conditions in (2.5), it also comes that c +1 ( g n ( t )) = c +2 ( g n ( t )) = 0 and thus the singular part is identically zero on (0 , T ) × ( − , .This ends the proof of Proposition 3.1. Remark . Notice that Proposition 3.1 proves that if ω intersects both con-nected components of Ω \{ x = 0 } , then unique continuation from ω hold for any ν ∈ (0 , .Proposition 3.1 implies that if χ ω S ( t ) g is identically zero then for any n ∈ N ∗ , g n ∈ C ((0 , T ] , H ∩ H (0 , ∩ C ((0 , T ] , L (0 , is solution of ∂ t g n − ∂ xx g n + (cid:16) c ν x + ( nπ ) x γ (cid:17) g n = 0 , ( t, x ) ∈ (0 , T ) × (0 , ,g n ( t,
0) = g n ( t,
1) = 0 , t ∈ (0 , T ) ,∂ x g n ( t,
0) = 0 , t ∈ (0 , T ) . (3.4)We prove in Sect. 3.2 that this leads to g n ≡ using a suitable Carlemanestimate. This subsection is dedicated to the proof of the following Carleman type in-equality.
Proposition 3.2.
Let
T > and Q T := (0 , T ) × (0 , . There exist R , C > such that for any R ≥ R , for every γ > , ν ∈ (0 , and n ∈ N ∗ , any g ∈ C ((0 , T ] , L (0 , ∩ C ((0 , T ] , H ∩ H (0 , with ∂ x g ( t, ≡ on (0 , T ) satisfies C R Z Z Q T t ( T − t )) exp (cid:18) − Rx b t ( T − t ) (cid:19) g ( t, x )d x d t ≤ Z Z Q T |P n,ν,γ g ( t, x ) | exp (cid:18) − Rx b t ( T − t ) (cid:19) d x d t, (3.5) where P n,ν,γ := ∂ t − ∂ xx + (cid:16) c ν x + ( nπ ) x γ (cid:17) , and b satisfies ( if ν ∈ (cid:0) , (cid:3) , b ∈ (0 , , if ν ∈ (cid:0) , (cid:1) , b := 2 − ν ∈ (0 , . (3.6)20efore proving Proposition 3.2 we show that it ends the proof of Theorem 3.1.Let g ∈ L (Ω) be such that χ ω S ( t ) g ≡ . Using Proposition 3.1 and the finalcomment of Sect. 3.1, it comes that for any n ∈ N ∗ , g n ∈ C ((0 , T ] , L (0 , ∩ C ((0 , T ] , H ∩ H (0 , with ∂ x g n ( t, ≡ on (0 , T ) . As, g n is solution of (3.4),it comes that P n,ν,γ g n ≡ on (0 , T ) × (0 , . Then, Proposition 3.2 implies that g n ≡ and thus, as g n ∈ C ([0 , T ] , L (0 , , we recover g = 0 . Remark . Contrarily to Carleman estimates proved by Vancostenoble [21],there are no boundary terms in the right-hand side of the inequality. Actually,the homogeneous Neumann boundary condition at x = 0 is crucial for inequality(3.5) to hold.The proof will rely on the following Hardy type inequality. Proposition 3.3.
For any z ∈ H ∩ H (0 , with z ′ (0) = 0 , (1 − α ) Z x α − z ( x ) d x ≤ Z x α z ′ ( x ) d x < ∞ , ∀ α ∈ [ − , . The statement and the proof are classical (see for example [21, Theorem2.1]). The main novelty here is that due to the extra information z ′ (0) = 0 , wecan prove the Hardy inequality with singular potential up to α ≥ − . Proof of Proposition 3.3.
Applying the generalized Hardy inequality [9] we getProposition 3.3 for α ∈ [0 , . Let α ∈ [ − , . Applying this generalized Hardyinequality to z ′ , we have ( α + 1) Z x α z ′ ( x ) d x ≤ Z x α +2 z ′′ ( x ) d x < ∞ . For any c ∈ R , ≤ Z (cid:16) x α z ′ ( x ) + cx α − z ( x ) (cid:17) d x = Z x α z ′ ( x ) + c x α − z ( x ) + cx α − ( z ) ′ ( x )d x. From Lemma 2.2, integration by parts lead to Z x α − ( z ) ′ ( x )d x = − Z ( α − x α − z ( x ) d x. Thus, for any c ∈ R , Z x α z ′ ( x ) d x ≥ (cid:0) c ( α − − c (cid:1) Z x α − z d x. Choosing c = ( α − ends the proof of Proposition 3.3.21e set some notations that will used throughout the proof.Let θ : t ∈ (0 , T ) t ( T − t ) . Let σ ( t, x ) := θ ( t ) x b where b satisfies (3.6). Noticethat as b ∈ (0 , every space derivative of the weight function is singular at x = 0 . This will be useful to handle the singular potential. The idea of usingsuch a weight is inspired by [11]. In Remark 3.3, we give further comments onthis choice of weight function. To simplify the notations, we denote the partialderivatives by subscripts: g x stands for ∂ x g . Proof of Proposition 3.2.
We set for
R > , z ( t, x ) := e − Rσ ( t,x ) g ( t, x ) . (3.7)From the definition of σ we get that, for any x ∈ (0 , , z (0 , x ) = z ( T, x ) = z t (0 , x ) = z t ( T, x ) = 0 . The boundary conditions on g also imply that for any t ∈ (0 , T ) , z ( t,
0) = z ( t,
1) = z x ( t,
0) = 0 .By Proposition 3.3, these boundary conditions imply that x z ( t,x ) x ∈ L (0 , , x z x ( t,x ) x ∈ L (0 , and using Lemma 2.2 we get z ( t, x ) x −→ x → , z x ( t, x ) x −→ x → . (3.8)Straightforward computations lead to e − Rσ P n,ν,γ g = P + R z + P − R z where P + R z : = ( Rσ t − R σ x ) z − z xx + (cid:16) c ν x + ( nπ ) x γ (cid:17) z,P − R z : = z t − Rσ x z x − Rσ xx z. Then,
Z Z Q T P + R zP − R z d x d t ≤ Z Z Q T e − Rσ |P n,ν,γ g | d x d t. (3.9)The rest of the proof follows the classical Carleman strategy [16] (see [13] for apedagogical presentation). We just pay attention to the singular terms. First step : integrations by part lead to Z Z Q T e − Rσ |P n,ν,γ g | d x d t ≥ Z Z Q T P + R zP − R z d x d t = R Z T σ x ( t, z x ( t, t − R Z Z Q T σ xx z x d x d t + Z Z Q T (cid:18) − R σ tt + 2 R σ x σ xt − R σ x σ xx + R σ xxxx (cid:19) z d x d t + R Z Z Q T (cid:16) − c ν x + 2 γ ( nπ ) x γ − (cid:17) σ x z d x d t. (3.10)22erforming integrations by parts, it is easily seen that h P + R z, P − R z i = I + · · · + I ,where I := h ( Rσ t − R σ x ) z − z xx , z t i = Z Z Q T (cid:16) − R σ tt + R σ x σ xt (cid:17) z d x d t,I : = − R h σ t z, σ x z x + σ xx z i = − R Z T (cid:2) σ t σ x z (cid:3) d t + R Z Z Q T σ xt σ x z d x d t,I : = R h σ x z, σ x z x + σ xx z i = R Z T (cid:2) σ x z (cid:3) d t − R Z Z Q T σ x σ xx z d x d t,I : = R h z xx , σ x z x + σ xx z i = R Z T (cid:2) σ x z x (cid:3) d t − R Z Z Q T σ xx z x d x d t + R Z T [ σ xx zz x ] d t − R Z Z Q T (cid:0) σ xx z x + σ xxx zz x (cid:1) d x d t = R Z T (cid:16)(cid:2) σ x z x (cid:3) + [ σ xx zz x ] − (cid:2) σ xxx z (cid:3) (cid:17) d t + R Z Z Q T (cid:18) σ xxxx z − σ xx z x (cid:19) d x d t. and I : = h (cid:16) c ν x + ( nπ ) x γ (cid:17) z, z t − Rσ x z x − Rσ xx z i = − R Z T h(cid:16) c ν x + ( nπ ) x γ (cid:17) σ x z i d t + R Z Z Q T (cid:16) c ν x + ( nπ ) x γ (cid:17) x σ x z d x d t. Z Z Q T e − Rσ |P n,ν,γ g | d x d t ≥ Z Z Q T P + R zP − R z d x d t = − R Z T (cid:2) σ t σ x z (cid:3) d t + R Z T (cid:2) σ x z (cid:3) d t + R Z T (cid:2) σ x z x (cid:3) d t + R Z T [ σ xx zz x ] d t − R Z T (cid:2) σ xxx z (cid:3) d t − R Z T h(cid:16) c ν x + (2 nπ ) | x | γ (cid:17) σ x z i d t − R Z Z Q T σ xx z x d x d t + Z Z Q T (cid:18) − R σ tt + 2 R σ x σ xt − R σ x σ xx + R σ xxxx (cid:19) z d x d t + R Z Z Q T (cid:16) − c ν x + 2 γ ( nπ ) x γ − (cid:17) σ x z d x d t. The weight being regular at x = 1 and z ( t,
1) = 0 every boundary term at x = 1 vanishes except R Z T σ x ( t, z x ( t, t. Using (3.8) and b > we get that every boundary term at x = 0 vanish. Forexample σ xxx z = b ( b − b − x b z ( x ) x −→ x → . Thus, we get (3.10).
Second step : lower bounds on the right-hand side of (3.10).
Recall that σ ( t, x ) = θ ( t ) x b with b satisfying (3.6). Boundary term. As b > , we have σ x ( t, > and thus R Z T σ x ( t, z x ( t, t ≥ . (3.11) Potential coming from the degeneracy. As σ x ( t, x ) = bθ ( t ) x b − ≥ on Q T and R z x d x < ∞ , it comes that ≤ R Z Z Q T γ ( nπ ) x γ − σ x z d x d t < ∞ . (3.12) Regular term.
Let I r := Z Z Q T (cid:0) − R σ tt + 2 R σ x σ xt − R σ x σ xx (cid:1) z d x d t.
24e prove that for R large enough the leading term in I r is the one in R . Fromstraightforward computations we have − R Z Z Q T σ x σ xx z d x d t = 2 b (1 − b ) R Z Z Q T θ x b − z d x d t. (3.13)Notice that this term is non-negative as b ∈ (0 , . Classical computations implythat | θ tt ( t ) | + | θ ( t ) θ t ( t ) | ≤ Cθ ( t ) , ∀ t ∈ (0 , T ) . Here and in the following, C denotes a positive constant that may vary eachtime it appears. Thus, (cid:12)(cid:12)(cid:12)(cid:12)Z Z Q T (cid:0) − R σ tt + 2 R σ x σ xt (cid:1) z d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Z Q T θ (cid:0) R x b − + Rx b (cid:1) z d x d t. As b ∈ (0 , , for every x ∈ (0 , , x b − ≤ x b − and x b ≤ x b − . Hence, assoon as R ≥ , (cid:12)(cid:12)(cid:12)(cid:12)Z Z Q T (cid:0) − R σ tt + 2 R σ x σ xt (cid:1) z d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ CR Z Z Q T θ x b − z d x d t. Together, with (3.13), we get I r ≥ C ( R − R ) Z Z Q T θ x b − z d x d t. Using, x b − ≥ on (0 , , we get the existence of C and R positive constantssuch that for R ≥ R , I r ≥ C R Z Z Q T θ z d x d t. (3.14) Singular potential.
Let I s := − R Z Z Q T σ xx z x d x d t + R Z Z Q T (cid:18) σ xxxx − c ν σ x x (cid:19) z d x d t. Notice that the two singular potentials are of the same order. Indeed, σ xxxx − c ν σ x x = b b − b − b − − c ν ) x b − . We prove that I s ≥ . (3.15)25rom the Hardy’s inequality given in Proposition 3.3, with α := b − , we get I s = 2 Rb (1 − b ) Z T θ Z x b − z x d x d t + b R Z T θ Z (( b − b − b − − c ν ) x b − z d x d t ≥ b R Z T θ Z (cid:0) (1 − b )( b − − (1 − b )( b − b − − c ν (cid:1) x b − z d x d t = b R Z T θ Z ((1 − b )(3 − b ) − c ν ) x b − z d x d t Recall that c ν = ν − . Thus, if ν ∈ (cid:0) , (cid:3) we have c ν ≤ and then (1 − b )(3 − b ) − c ν ≥ (1 − b )(3 − b ) ≥ , for any b ∈ (0 , . This gives (3.15).If ν ∈ (cid:0) , (cid:1) , setting b = 2 − ν we still have b ∈ (0 , and (1 − b )(3 − b ) − c ν = 0 . This gives (3.15).
Conclusion.
Gathering (3.11), (3.12), (3.14) and (3.15) in (3.10) we getthat for R ≥ R , Z Z Q T e − Rσ |P n,ν,γ g | d x d t ≥ C R Z Z Q T θ z d x d t. This ends the proof of Proposition 3.2.
Remark . We here point out some of the differences between Proposition 3.2and the Carleman estimates established in the case of a boundary inverse squaresingularity in [22, 21]. In both estimates the singular potential appears as
Z Z Q T σ x x z d x d t. In [22], the weight is defined by p ( x ) = 1 − x . Thus, the singular potential canbe treated with some classical Hardy type inequalities.In our situation, using the extra information z x ( t,
0) = 0 , we are able to dealwith a weight function with singular derivatives. The weight is chosen concaveso that the term in R is the leading one. The weight is chosen increasing todeal with the boundary term σ x ( t, z x ( t, . At the same time, this allows todeal with the potential coming from the degeneracy (3.12). The price to pay isthat we have to handle very singular terms of the form Z Z Q T θx b z x d x d t. z x ( t,
0) = 0 . Remark . To be closer to the setting studied by Boscain and Laurent [4], wecould study for ( t, x, y ) ∈ (0 , T ) × ( − , × T , ∂ t f − ∂ xx f − | x | γ ∂ yy f + γ (cid:16) γ (cid:17) x f = u ( t, x, y ) χ ω ( x, y ) ,f ( t, − , y ) = f ( t, , y ) = 0 ,f (0 , x, y ) = f ( x, y ) . (3.16)Defining the semigroup as T ( t ) f ( x, y ) := X n ∈ Z f n ( t, x ) e iny , with f n := e − tA nπ f n we get that its infinitesimal generator is an extension of thesingular Grushin operator on C ∞ ([( − , ∪ (0 , × T ) . As in Proposition 2.5,this semigroup leads to a unique mild solution of (3.16). As < γ (cid:0) γ + 1 (cid:1) < for γ ∈ (0 , , essentially with the same proof as Sect. 3, we would obtainapproximate controllability for any γ ∈ (0 , . In this paper we have investigated the approximate controllability propertiesfor a Grushin equation which presents both a degeneracy and an inversesquare singularity on the internal set { x = 0 } . As the associated operatorpossesses several self-adjoint extensions, the functional setting in which we studythe well posedness and unique continuation for the adjoint system is crucial.This functional setting relies on a precise study of the associated operatorsand the design of a self-adjoint extension of the singular operator with suitabletransmission conditions across the singularity.Using classical unique continuation results for uniformly parabolic operators,the study of unique continuation is reduced to the study of a problem with aboundary inverse square singularity. The proof of unique continuation is endedwith a suitable Carleman type estimate that relies on a Hardy inequality.An interesting open problem coming from this work is the question of nullcontrollability in the case ν ∈ (0 , . The classical strategy would be to proveuniform observability for the adjoint systems. This has been done in thecase where there is no singular potential in [3] and with a boundary singularpotential in [7]. The Carleman type estimate we proved in this paper might notbe directly used as it holds true only for the regular part of the coefficient g n .Dealing with the singular part in Carleman type estimates is quite tricky as wecannot perform integrations by part on the singular part. The other difficultyrelies on the fact that we want these estimates to be uniform with respect to n .27 cknowledgements : The author thanks K. Beauchard for having drawnhis attention to this problem and for fruitful discussions. The author thanksM. Gueye and D. Prandi for interesting discussions.
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