New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence

New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence Farid Ammar Khodja∗ Assia Benabdallah † Manuel Gonz´ alez-Burgos ‡ and Luz de Teresa§ June 12, 2015 Abstract We consider the null controllability problem for two coupled parabolic equations with a space-depending coupling term. We analyze both boundary and distributed null controllability. In each case, we exhibit a minimal time of control, that is to say, a time T0∈[0,∞] such that the corresponding system is null controllable at any time T > T0and is not if T < T0. In the distributed case, this minimal time depends on the relative position of the control interval and the support of the coupling term. We also prove that, for a fixed control interval and a time τ0∈[0,∞], there exist coupling terms such that the associated minimal time is τ0. Contents 1 Introduction and main results 2 2 Some preliminary results 8 3 Boundary controllability problem 13 3.1 Boundary approximate controllability ......................... 14 3.2 Boundary null controllability .............................. 15 3.2.1 Positive boundary controllability result .................... 16 3.2.2 Negative boundary controllability result .................... 17 4 Distributed approximate controllability 18 5 Proof of Theorem 1.3: The positive null controllability result 20 5.1 The moment problem .................................. 20 5.2 Construction of the functions f1and f2........................ 22 5.3 Solving the moment problem .............................. 24 5.3.1 The case k∈Λ1................................. 25 5.3.2 The case k∈Λ2................................. 26 5.3.3 The case k∈Λ3................................. 27 5.4 Conclusion ........................................ 27 ∗Laboratoire de Math´ematiques de Besan¸con, UMR 6623, Universit´e de Franche-Comt´e, 16 route de Gray, 25030 Besan¸con cedex, France. E-mail: [email protected] †Aix Marseille Universit´e, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. E-mail: [email protected] ‡Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. Supported by grant MTM2013- 41286-P, Ministry of Economy and Competitiveness (Spain). E-mail: [email protected] §Instituto de Matem´aticas, UNAM, Circuito Exterior, C.U. 04510 D.F., M´exico. Supported by project IN102799 of D.G.A.P.A. (Mexico). E-mail: [email protected] 1 6 Proof of Theorem 1.3: The negative null controllability result 28 7 Complementary results. Some examples 29 A Proof of Lemma 7.1 35 1 Introduction and main results This paper deals with the controllability of non-scalar parabolic equations with a reduced number of controls. The control of parabolic systems is a challenging issue, which has attracted the interest of the control community in the last decade. These parabolic systems arise, for example, in the study of chemical reactions and in a wide variety of mathematical biology and physical situations (see e.g. [23], [33], [16], ...). More precisely, the aim of this paper is to investigate the relationship between the location of the controls and the action of the coupling terms. We will see that in this framework new phenomena arise. To this end, let us fix T > 0 and ω= (a, b)⊂(0, π) and consider the following control problems:      yt−yxx +q(x)A0y= 0 in QT:= (0, π)×(0, T), y(0,·) = Bu, y(π, ·) = 0 on (0, T), y(·,0) = y0in (0, π), (1.1) and      yt−yxx +q(x)A0y=Bv1ωin QT, y(0,·)=0, y(π, ·) = 0 on (0, T), y(·,0) = y0in (0, π), (1.2) where A0∈L(R2) and B∈R2are respectively given by: A0=0 1 0 0 and B=0 1.(1.3) In systems (1.1) and (1.2), q∈L∞(0, π) is a given function, y0is the initial datum and u∈L2(0, T) and v∈L2(QT) are the control functions. Let us remark that for every u∈L2(0, T) (resp., v∈L2(QT)) and y0∈H−1(0, π;R2) (resp., y0∈L2(0, π;R2)), system (1.1) (resp., system (1.2)) possesses a unique solution defined by transposition (resp., a unique weak solution) which satisfies y∈L2(QT;R2)∩C0([0, T]; H−1(0, π;R2)) (resp., y∈L2(0, T;H1 0(0, π;R2)) ∩C0([0, T]; L2(0, π;R2))) and depends continuously on the data uand y0, i.e., there exists a constant C=C(T)>0 such that kykL2(QT;R2)+kykC0([0,T ];H−1(0,π;R2)) ≤Cky0kH−1(0,π;R2)+kukL2(0,T ) (resp., kykL2(0,T ;H1 0(0,π;R2)) +kykC0([0,T ];L2(0,π;R2)) ≤Cky0kL2(0,π;R2)+kvkL2(QT)). Let us recall that the function y∗∈L2(QT;R2)∩C0([0, T]; H−1(0, π;R2)) (resp., the function y∗∈L2(0, T;H1 0(0, π;R2)) ∩C0([0, T]; L2(0, π;R2))) is a trajectory of system (1.1) (resp., of system (1.2)) if y∗is the solution of (1.1) (resp., of (1.2)) corresponding to the data u∗∈L2(0, T) and y∗ 0∈H−1(0, π;R2) (resp., v∗∈L2(QT) and y∗ 0∈L2(0, π;R2)). With the previous notations, we define: Definition 1.1. 1. It will be said that system (1.1) (resp., system (1.2)) is approximately controllable in H−1(0, π;R2) (resp., in L2(0, π;R2)) at time Tif for every y0, yd∈H−1(0, π;R2) (resp., y0, yd∈L2(0, π;R2)) and for every ε > 0, there exists a control u∈L2(0, T) (resp., v∈L2(QT)) such that the solution yto (1.1) (resp., to (1.2)) satisfies ky(·, T)−ydkH−1(0,π;R2)≤ε(resp., ky(·, T )−ydkL2(0,π;R2)≤ε). 2 2. It will be said that system (1.1) (resp., system (1.2)) is null controllable at time Tif for every y0∈H−1(0, π;R2) (resp., y0∈L2(0, π;R2)), there exists a control u∈L2(0, T) (resp., v∈L2(QT)) such that the solution yto (1.1) (resp., to (1.2)) satisfies y(·, T) = 0 in H−1(0, π;R2) (resp., in L2(0, π;R2)). 3. Finally, it will be said that system (1.1) (resp., system (1.2)) is exactly controllable to trajectories at time T > 0 if for every y0∈H−1(0, π;R2) and every trajectory y∗of system (1.1) (resp., for every y0∈L2(0, π;R2) and every trajectory y∗of system (1.2)), there exists a control u∈L2(0, T) (resp., v∈L2(QT)) such that the solution yto (1.1) (resp., to (1.2)) satisfies y(·, T) = y∗(·, T ) in H−1(0, π;R2) (resp., in L2(0, π;R2)). In this work we are interested in studying the controllability properties of systems (1.1) and (1.2). Let us observe that we are exerting only one control force on the systems (a boundary or distributed control) but we want to control the corresponding state ywhich has two components. In fact, the first equation in (1.1) and (1.2) is indirectly controlled by means of the term q(x)y2. Of course, this coupling term must be different from zero, i.e., q6≡ 0.On the other hand, using the linearity of systems (1.1) and (1.2), it is easy to see that the null controllability property at time Tof the previous systems is equivalent to the exact controllability to trajectories at time T for these systems. Systems (1.1) and (1.2) are particular classes of more general n×nparabolic control systems of the form:      yt−D∆y+A(x, t)y=Bv1ωin QT:= Ω ×(0, T), y=Cu1Γ0,on ΣT:= ∂Ω×(0, T), y(·,0) = y0in Ω, (1.4) where ωand Γ0are, respectively, open subsets of the smooth bounded domain Ω ⊂RNand of its boundary ∂Ω, D= diag (d1,· · · , dn)∈L(Rn), with n≥1, is a positive matrix, B, C ∈L(Rm,Rn), with m≤n, are given matrices, and A= (aij )1≤i,j≤2∈L∞(QT;L(Rn)) is a matrix-valued function. When m<n, the issue for this system is to control the whole components of the system with a control function acting, locally in space or on a part of the boundary, only on some of them. We refer to [5] for a review of results for the controllability problem of system (1.4). The first results on controllability of the scalar case, n= 1, concerns the one-dimensional case N= 1. They have been established by H.O. Fattorini and D.L. Russell (see [19,20]) through the moment method. The controllability of the N-dimensional case, still for the scalar equation (n= 1), has been established later by G. Lebeau and L. Robbiano in [31] and by A. Fursikov and O. Yu. Imanuvilov in [22] using Carleman estimates. It is interesting to point out that the boundary and distributed null controllability of scalar parabolic problems is valid for any positive time T, for any Γ0⊂∂Ω and for any ω⊂Ω. Let us also underline the reference [18], where the author proves the existence of a minimal control time for the one-dimensional heat equation with controls on the form f(x)u(t), with f∈H−1(0, π), a given fixed function, and u∈L2(0, T). The first results on controllability of coupled parabolic equations (n > 1) have been established in [35,13,3,25]. They concern mainly system (1.4) with n= 2, C= 0 (distributed control) and B=0 1. In all the previous works the authors use Carleman inequalities for the corresponding adjoint system to (1.4). The main assumption on the matrix-valued function Ais that there exist an open subset ω0⊂ωand a positive constant σsuch that a12 ≥σ > 0 or a12 ≤ −σ < 0 in ω0×(0, T).(1.5) 3 It is interesting to point out that in [25], under the weaker assumption |a12| ≥ σ > 0 in ω0×(0, T ),(1.6) the authors prove a null controllability result at time T > 0 for some generalizations of system (1.4). The previous controllability results have been extended in [26] to n≥2 when system (1.4) has a particular structure: cascade systems. To this end, the authors assume a generalization of assumption (1.5) on the coupling matrix A(·,·) and, again, use Carleman inequalities for the adjoint problem for proving the null controllability result. In [4], a necessary and sufficient condition for the approximate and null controllability at time T > 0 is established when Ais a constant matrix. This condition does not depend on Tand generalizes the algebraic Kalman condition (see [28]), well-known for the controllability of finite dimensional systems. In the case n= 2, this necessary and sufficient condition reduces to a12 6= 0. Let us now describe the existing results on boundary controllability of system (1.4) (B= 0). There are few results on this framework and most of them concern the one-dimensional case (N= 1), D=Id and Aa constant matrix. When D=Id and Ais a constant matrix, a necessary and sufficient condition is exhibited in [21] and [6]. This condition is different from the one that characterizes the distributed null controllability of system (1.4) in the constant case (see [4]). As a consequence and unlike the scalar case, we deduce that the distributed and boundary null controllability properties of non-scalar parabolic systems are in general not equivalent. The boundary null controllability problem for system (1.4) in the constant case is more intricate if D6=Id. When n= 2, the boundary null controllability property holds if Tis greater than a minimal time T0∈[0,∞] which depends on the coefficients of the constant matrices Dand A (see [10]). For instance, if the diffusion matrix D, the coupling matrix Aand the control vector Care given by D= diag (1, d2), d 6= 0,1, A =0 1 0 0 , C =0 1,(1.7) then system (1.4) is approximately controllable at time T > 0 if and only if dis an irrational number and the minimal time T0of null controllability depends on the diophantine approximation of d. Let us also underline that this phenomenon (minimal time of controllability for parabolic equations) has been observed for the first time in the scalar case in [18], but concerning pointwise controls. We would like to comment that, for system (1.4) with the previous data (1.7), it is possible to select positive numbers d > 0 for which system (1.4) is approximately controllable at any positive time Tand never null controllable (see [10]). Unlike the scalar case, from the results in [10] we infer that the approximate and null boundary controllability properties for non-scalar parabolic systems are, in general, not equivalent. In [21], [6] and [10], the authors use the moment method (see [19,20]) to prove the positive null controllability result at time T. They carry out a study on bounds of biorthogonal families to exponentials associated to complex sequences. In fact, the previous minimal time T0is related to the index of condensation of the sequence of eigenvalues of the operator associated to the system (see [10]). Finally, in [12] the authors extend the one-dimensional boundary null controllability results from [21] and [6] to the N-dimensional case when the domain Ω is a cylindrical domain. Unlike the distributed controllability problem for system (1.4), Carleman estimates for the corresponding adjoint system seem not to be suitable when dealing with the boundary null controllability problem of system (1.4). Let us come back to systems (1.1) and (1.2). First, observe that from the null controllability result stated in [25], if the function qsatisfies (1.6), with σ > 0 and ω0⊂ωan open interval, then system (1.2) is null controllable at any positive time T. Therefore, a natural question arises: what happens if Supp q∩ω=∅? The first and partial answer concerns the approximate controllability of this system. More precisely, in [29], the approximate controllability of system (1.2) at every time T > 0 is proved when q= 1Owith Oa nonempty open subset of Ω. Later, other partial 4 answers are given for the null controllability of systems (1.1) and (1.2) under sign conditions on the function q(see [2], [34], [1] and [17]): q6≡ 0 and q≥0 or q≤0 in (0, π).(1.8) These results have been obtained as a consequence of the corresponding hyperbolic results by using the transmutation strategy (see [32]). Of course in the N-dimensional case (N≥2), they assume the Geometric Control Condition (CGC) defined in [11] on both sets ωand Supp q. Clearly these assumptions are not necessary in the parabolic setting. The first satisfying answer without sign conditions on qconcerns the null controllability of systems (1.1) and (1.2) when qsatisfies the condition Zπ 0 q(x)dx 6= 0.(1.9) Under this condition, in [7] the authors give a necessary and sufficient condition for the approximate and null controllability at time T > 0 of system (1.1). As a consequence, they also obtain the null controllability property at any positive time Tfor (1.2) under the same conditions. The first general result for the distributed controllability of system (1.2) concerns the approximate controllability and is proved in [14]. For general open sets ω, the authors provide a necessary and sufficient condition for the approximate controllability of system (1.2) in terms of Supp qand the connected components of Ω \ω. Some results presented here have been announced in [9]. In fact, in [9] the controllability of system (1.1) and the controllability of system (1.2) when the function qand the control interval ω= (a, b) satisfy the geometrical condition Supp q⊂[0, a] or Supp q⊂[b, π] (1.10) are analyzed. Under the previous condition (1.10), a minimal time of boundary and distributed null controllability, T0(q)∈[0,+∞], arises in such a way that these systems are null controllable at time T if T∈(T0(q),∞) and are not when T∈(0, T0(q)). In this paper, we are going to provide a complete answer to the controllability problem of system (1.1) and system (1.2) without imposing condition (1.10) and when the control domain is an interval, ω= (a, b). More precisely, we will analyze the controllability properties of system (1.1), for a general function q∈L∞(0, π), and of system (1.2), when qsatisfies Supp q∩ω=∅,(1.11) i.e., when Supp q⊂[0, a]∪[b, π]. In the sequel, we set ϕkthe normalized eigenvectors of the Dirichlet laplacian in (0, π), i.e., ϕk(x) = r2 πsin(kx),∀x∈(0, π), k ≥1. On the other hand, the corresponding eigenvalues are given by k2,k≥1. For any k≥1, we associate with the function q∈L∞(0, π) satisfying (1.11) the sequences {Ik(q)}k≥1and {Ii,k(q)}k≥1,i= 1,2, given by        I1,k(q) := Za 0 q(x)|ϕk(x)|2dx, I2,k(q) := Zπ b q(x)|ϕk(x)|2dx, Ik(q) := I1,k(q) + I2,k(q) = Zπ 0 q(x)|ϕk(x)|2dx. (1.12) Let us present our boundary control results, that is, our main result related to system (1.1). Theorem 1.1. Let us consider A0and Bgiven by (1.3)and q∈L∞(0, π), a given function. Then, one has: 5 1. System (1.1)is approximately controllable at time T > 0if and only if Ik(q)6= 0 ∀k≥1.(1.13) 2. Assume that condition (1.13)holds and define e T0(q) := lim sup −log |Ik(q)| k2∈[0,∞].(1.14) Then, if T > e T0(q)system (1.1)is null controllable at time T. On the other hand, if T < e T0(q)system (1.1)is not null controllable at time T. This result has been announced in [9]. Remark 1.2. The approximate controllability result stated in Theorem 1.1 does not depend on the final time T: approximate controllability of system (1.1) at a time T0>0 is equivalent to the approximate controllability of system (1.1) at any time T > 0. On the other hand, condition (1.13) characterizes the approximate controllability property of system (1.1). Thus, (1.13) is a necessary condition for the null controllability at time T > 0 of this system. Remark 1.3. Note that the sequences {Ii,k(q)}k≥1,i= 1,2, and {Ik(q)}k≥1are convergent and from a simple computation one has: lim Ik(q) = 1 πZπ 0 q(x)dx, lim I1,k(q) = 1 πZa 0 q(x)dx, lim I2,k(q) = 1 πZπ b q(x)dx. From this, it readily follows that the sequence {Ik(q)−1}k∈Λis bounded and e T0(q) = 0 whenever condition (1.9) holds (for the expression of the set Λ, see (1.16)). Observe that, under condition (1.8) on the function q, (1.13) holds and e T0(q) = 0. In particular, Theorem 1.1 generalizes the one-dimensional parabolic boundary controllability results obtained in [1] and [34]. Remark 1.4. We will see in Section 7that there are functions q∈L∞(0, π) such that e T0(q)>0 (in fact, e T0(q) may take any value in [0,∞]). In particular, Theorem 1.1 implies that, even in a parabolic setting, a positive time of control may appear and that, unlike the scalar case, boundary approximate and null controllability are not equivalent properties in the non-scalar case (see also [10] for a similar result). On the other hand, Theorem 1.1 also infers negative boundary controllability results for hyperbolic versions of system (1.1). Indeed, if q∈L∞(0, π) is such that e T0(q)>0, the transmutation strategy (see [32]) implies that the corresponding hyperbolic version of (1.1) is not controllable in the natural space associated to the system (see Theorem 3.6 in [1]) at any time T > 0. For the distributed control problem (system (1.2)), let us first recall a recent result on approximate controllability: Theorem 1.2 ([14]).Let us consider A0and Bgiven by (1.3)and q∈L∞(0, π), a function satisfying (1.11). Then, system (1.2)is approximately controllable at time T > 0if and only if |Ik(q)|+|I1,k(q)| 6= 0 ∀k≥1.(1.15) For the sake of completeness this result will be proved in Section 4. Remark 1.5. As in the boundary case, the approximate controllability result for system (1.2) does not depend on the final time T: system (1.2) is approximately controllable at a time T0>0 if and only if it is approximately controllable at any time T > 0. 6 To state our null controllability result for system (1.2) when q∈L∞(0, π) satisfies (1.11), we need some definitions and notations. First, let us define the sets      Λ := {k≥1 : Ik(q)6= 0}= Λ1∪Λ2, Λ1:= {k∈Λ : I1,k(q)6= 0},Λ2:= {k∈Λ : I1,k(q) = 0}and Λ3:= {k≥1 : Ik(q) = 0}, (1.16) where Ik(q) and I1,k(q) are given in (1.12). Observe that Λ1, Λ2and Λ3are disjoint sets and, of course, Λ1∪Λ2∪Λ3= Λ ∪Λ3=N∗. On the other hand, let us assume that the function q∈L∞(0, π) is such that condition (1.15) holds. Thus, we can introduce the quantities −log |I1,k(q)|and −log |Ik(q)|where we will use the notation −log |x|=∞when x= 0. With this notation and under assumption (1.15), we deduce min{− log |I1,k(q)|,−log |Ik(q)|} ∈ R,∀k≥1. One has: Theorem 1.3. Let us consider A0∈L(R2)and B∈R2, given by (1.3), and q∈L∞(0, π), a function satisfying (1.11). Let us also assume condition (1.15), and define T0(q) := lim sup min{− log |I1,k(q)|,−log |Ik(q)|} k2.(1.17) Then, given T > 0, one has: 1. Assume that T > T0(q). Then, system (1.2)is null controllable at time T. 2. If T < T0(q), then system (1.2)is not null controllable at time T. As in the boundary case, condition (1.15) characterizes the distributed approximate controllability of system (1.2). This implies that (1.15) is a necessary condition for the distributed null controllability at time T > 0 of (1.2). We end the presentation of our main results with some remarks. Remark 1.6. Under condition (1.15), the minimal time T0(q) is well-defined and, taking into account Remark 1.3, satisfies T0(q)∈[0,∞]. We will check in Section 7that, given the control interval ω= (a, b), there are functions q∈L∞(0, π), which fulfill condition (1.11), for which T0(q)>0 and even T0(q) = ∞. For such functions, system (1.2) is approximately controllable at all positive time Tbut it is not null controllable at time Tif T∈(0, T0(q)). Again and unlike the scalar case, the distributed approximate property is not equivalent, in general, to the distributed null controllability property in the non-scalar case. Also, following the reasoning in Remark 1.4, from Theorem 1.3, we deduce that when q∈L∞(0, π) satisfies (1.11) and T0(q)>0, the hyperbolic version of system (1.2) is not controllable in the natural space associated to the system (see Theorem 3.5 in [1] and Definition 1.1 in [17] for the definition of this space) at any time T > 0. Remark 1.7. Let us fix a function q∈L∞(0, π) and a control interval ωthat satisfies (1.11). Taking into account that condition (1.13) implies (1.15) and the inequality T0(q)≤e T0(q), the boundary controllability at time T > 0 of system (1.1) implies the distributed controllability at time T > 0 of system (1.2) provided the control interval ωsatisfies (1.11). But, it is interesting to note that there exist functions q∈L∞(0, π) and control intervals ωfulfilling condition (1.11) for which T0(q)<e T0(q) (see Example 7.3). This provides another difference with the scalar case: boundary and distributed controllability are not equivalent in the non-scalar parabolic setting. However, if ω= (a, b) and q∈L∞(0, π) are such that (1.10) holds, then T0(q) = e T0(q) and system (1.1) is null controllable at time T > 0 if and only if system (1.2) is also null controllable at time T. 7 Remark 1.8. The minimal time T0(q) (see (1.17)) depends on the function qbut also on the position of the control interval ω(satisfying (1.11)). This fact provides a new phenomenon in the framework of the distributed controllability of non-scalar parabolic problems: the dependence of the controllability result on the position of the control set. Indeed, we will also see in Section 7(see Example 7.3) that, given τ0∈(0,∞] (which could be τ0=∞), there exist a function q∈L∞(0, π) and control intervals ω1, ω2⊂(0, π), satisfying (1.11), such that (1.15) holds, for ω1and ω2, and T(1) 0(q) = 0 and T(2) 0(q) = τ0>0. In the previous equalities, T(i) 0(q) is the minimal time associated to the function qand to the interval ωi(see (1.17)). In conclusion, system (1.2) is null controllable at every positive time T, if the control is exerted on ω1, but it is not null controllable at time Tif T∈(0, τ0) and the control is exerted on ω2. This is another big difference with the scalar parabolic case. This dependence of the zone of control was highlighted in [14], in the case of the approximate controllability of system (1.2). Remark 1.9. Under assumption (1.8) on the function q, conditions (1.13) and (1.15) hold for every interval ω⊂(0, π) satisfying (1.11). This means that systems (1.1) and (1.2) are approximately controllable at any positive time T. In fact, taking into account Remark 1.3, we get that e T0(q) = T0(q) = 0 and systems (1.1) and (1.2) are also null controllable at any positive time T. Thus, our results recover the one-dimensional parabolic version of the results in [29], [34], [1] and [17], with less restrictive assumptions on q. The rest of the paper is organized as follows: In Section 2we set and analyze some preliminary results related to the spectrum and the (generalized) eigenspaces of the operator associated with systems (1.1) and (1.2). Section 3is devoted to studying the boundary controllability problem for system (1.1), namely to the proof of Theorem 1.1. For clarity, this section has been divided into two subsections; in the first one it can be found the proofs concerning the approximate controllability of system (1.1). In Subsection 3.2, the null-controllability property of this system is proved. The distributed approximate controllability problem is considered in Section 4. Theorem 1.3 is proved in Sections 5(the positive null-controllability part) and 6(negative null-controllability part). The last section contains some complementary results and some examples that illustrate the different situations. 2 Some preliminary results In this section we will give some properties which will be used below. Let us consider the vectorial operator L:= −d2 dx2Id +q(x)A0:D(L)⊂L2(0, π;R2)−→ L2(0, π;R2) (2.1) with domain D(L) = H2(0, π;R2)∩H1 0(0, π;R2) and also its adjoint L∗. We will always denote by h·,·i the standard scalar product of either L2(0, π;R) or L2(0, π;R2), by h·,·iX0,X the duality pairing between the Hilert space Xand its dual X0. We are interested in studying the spectrum of the operators Land L∗. To this end, given a function q∈L∞(0, π), we consider the quantity Ik(q) given by (1.12), k≥1. With this notation, one has: Proposition 2.1. Let A0be given by (1.3)and consider the operator Lgiven by (2.1)and its adjoint L∗. Then, 1. The spectra of Land L∗are given by σ(L) = σ(L∗) = {k2:k≥1}. 2. Given k≥1, if Φ1,k =ϕk 0,Φ2,k =ψk ϕk, 8 (resp., if Φ∗ 1,k := ϕk ψk,Φ∗ 2,k := 0 ϕk), where ψkis the unique solution of the non-homogeneous Sturm-Liouville problem:          −ψxx −k2ψ= [Ik(q)−q(x)] ϕkin (0, π), ψ(0) = 0, ψ(π)=0, Zπ 0 ψ(x)ϕk(x)dx = 0, (2.2) then, (L−k2Id)Φ1,k = 0 and (L−k2Id)Φ2,k =Ik(q)Φ1,k (2.3) (resp., L∗−k2IdΦ∗ 1,k =Ik(q)Φ∗ 2,k and L∗−k2IdΦ∗ 2,k = 0).(2.4) In particular, if k∈Λthen k2is a simple eigenvalue and Φ1,k and Φ2,k (resp., Φ∗ 2,k and Φ∗ 1,k) are, respectively, an eigenfunction and a generalized eigenfunction of the operator L (resp., L∗) associated to k2, while if k∈Λ3then Φ1,k and Φ2,k are both eigenfunctions of L (resp., L∗) associated to k2. Proof. First, Lcan be written L=−∆q 0−∆ where ∆ = d2 dx2:L2(0, π)−→ L2(0, π) with domain D(∆) = H2(0, π)∩H1 0(0, π) is, as is wellknown, boundedly invertible with compact inverse. We can check that: L−1=(−∆)−1−(−∆)−1◦q◦(−∆)−1 0 (−∆)−1 which readily implies that L−1is a compact operator on L2(0, π;R2).Thus, the spectrum of L reduces to its point spectrum. We have now to solve the eigenvalue problem:      −y00 1+qy2=λy1in (0, π), −y00 2=λy2in (0, π), y1(0) = y2(0) = 0, y1(π) = y2(π) = 0. If y2≡0, then, λ=k2is an eigenvalue of Land taking y1=ϕkwe obtain Φ1,k as associated eigenfunction of L. If we now assume that y26≡ 0, then, again λ=k2and y2=ϕkis a (normalized) solution to the second o.d.e. Observe that the first equation admits a solution if and only if k∈Λ3, i.e., Ik(q) = 0. In this case, Φ2,k is a second associated eigenfunction of L. In conclusion, if k∈Λ3, then k2is a double eigenvalue of L. From the above considerations, it is clear that if Ik(q)6= 0, then the eigenvalue k2of Lis simple and Φ1,k is an associated eigenfunction. Observe that, taking Φ2,k = (y1, y2), the equation (L−k2Id)Φ2,k =cΦ1,k writes:      −y00 1−k2y1=cϕk−qy2in (0, π), −y00 2−k2y2= 0 in (0, π), y1(0) = y2(0) = 0, y1(π) = y2(π) = 0. Thus, again, choosing y2=ϕkand inserting this expression in the first equation, we get for y1: (−y00 1−k2y1= [c−q]ϕk, y1(0) = y2(0) = 0 9 3.2.1 Positive boundary controllability result Let us assume that T > e T0(q)∈[0,∞) (see (1.14)). Our objective is to prove that system (1.1) is exactly controllable to zero at time T. To this end, for y0∈H−1(0, π;R2), we will reformulate the null controllability problem as a moment problem. Using Propositions 3.1 and 3.2, we deduce that the control u∈L2(0, T) drives the solution of (1.1) to zero at time Tif and only if u∈L2(0, T) satisfies ZT 0 u(t)B∗θx(0, t)dt =−hy0, θ(·,0)iH−1,H1 0,∀θ0∈H1 0(0, π;R2), where θ∈C0([0, T]; H1 0(0, π;R2)) is the solution to the adjoint problem (3.1) associated with θ0. Since B∗is a basis of H1 0(0, π;R2) (see Corollary 2.5), the null controllability property at time T for system (1.1) is equivalent to find u∈L2(0, T ) such that ZT 0 u(t)B∗θi,k x(0, t)dt =−hy0, θi,k(·,0)iH−1,H1 0,∀k≥1,∀i= 1,2,(3.5) where θi,k is the solution of system (3.1) associated with θ0= Φ∗ i,k (for the expression of the function Φ∗ i,k, see Proposition 2.1). Let us take u(t) = v(T−t), t ∈(0, T). Developing the equality (3.5), one has: 1. If we take θ0= Φ∗ 2,k, the solution of the adjoint problem is θ2,k(·, t) = e−k2(T−t)Φ∗ 2,k and (3.5) becomes, ZT 0 e−k2tv(t)dt =−1 krπ 2e−k2Thy0,Φ∗ 2,kiH−1,H1 0:= e−k2Tf M(k) 1(y0),∀k≥1. It is easy to see that f M(k) 1(y0)≤Cky0kH−1(0,π;R2),∀k≥1,(3.6) for a positive constant Cindependent of kand y0. 2. Let us now take θ0= Φ∗ 1,k. In this case the solution of the adjoint system (3.1) is θ1,k(·, t) = e−k2(T−t)Φ∗ 1,k −(T−t)Ik(q)e−k2(T−t)Φ∗ 2,k and, then, the equality (3.5) transforms into (u(t) = v(T−t), t∈(0, T))          ψ0 k(0) ZT 0 e−k2tv(t)dt −Ik(q)ϕ0 k(0) ZT 0 te−k2tv(t)dt =−e−k2Thhy0,Φ∗ 1,kiH−1,H1 0−TIk(q)hy0,Φ∗ 2,kiH−1,H1 0i. Thus, the control u=v(T− ·) must also satisfy ZT 0 te−k2tv(t)dt =e−k2T Ik(q)f M(k) 2(y0),∀k≥1, where f M(k) 2(y0) := 1 krπ 2nψ0 k(0)f M(k) 1(y0) + hhy0,Φ∗ 1,kiH−1,H1 0−TIk(q)hy0,Φ∗ 2,kiH−1,H1 0io. Using the properties of the function ψkstated in Proposition 2.2 (see (2.6)), one has f M(k) 2(y0)≤Cky0kH−1(0,π;R2),∀k≥1.(3.7) for a new positive constant Cindependent of kand y0. 16 Summarizing, we have proved that u∈L2(0, T) is such that the solution yof system (1.1) satisfies y(·, T) = 0 in (0, π) if and only if v=u(T− ·)∈L2(0, T) satisfies          ZT 0 e−k2tv(t)dt =e−k2Tf M(k) 1(y0), ZT 0 te−k2tv(t)dt =e−k2T Ik(q)f M(k) 2(y0),∀k≥1, (3.8) with f M(k) 1(y0) and f M(k) 2(y0) satisfying (3.6) and (3.7). From the results in [21] (see also [6]), we can conclude that the sequence ne1,k := e−k2t, e2,k := te−k2tok≥1 admits a biorthogonal family {q1,k, q2,k}k≥1in L2(0, T ), i.e., a family {q1,k, q2,k}k≥1in L2(0, T) satisfying ZT 0 er,kqs,j(t)dt =δkj δrs,∀k, j ≥1,1≤r, s ≤2,(3.9) which moreover satisfies that for every ε > 0 there exists a constant Cε,T >0 such that kqi,kkL2(0,T )≤Cε,T eεk2,∀k≥1, i = 1,2.(3.10) Using the formulas in (3.8) and the property (3.9), we infer that an explicit formal solution of the moment problem (3.5) is given by u(T−t) = v(t) = X k≥1 e−k2Tf M(k) 1(y0)q1,k(t) + 1 Ik(q)f M(k) 2(y0)q2,k(t). Let us see that this series defines an element of L2(0, T) when T > e T0(q), i.e., the previous series converges in L2(0, T) if T > e T0(q). Indeed, from the definition of the minimal time e T0(q) (see (1.14)) and for any fixed ε > 0, we can infer that there exists a positive constant Cεsuch that 1 |Ik(q)|≤Cεek2(e T0(q)+ε),∀k≥1. On the other hand, we can use the bound (3.10) and get a new positive constant Cε,T for which          e−k2Tf M(k) 1(y0)q1,k +1 Ik(q)f M(k) 2(y0)q2,kL2(0,T ) ≤Cε,T e−k2Teεk2 |Ik(q)| ≤Cε,T e−k2(T− e T0(q)−2ε). This last inequality proves the absolute convergence of the series which defines the control usince εmay be chosen arbitrarily small. This proves the null controllability of system (1.1) at time T when T > e T0(q). 3.2.2 Negative boundary controllability result In order to finish the proof of Theorem 1.1, let us prove that if 0 < T < e T0(q), then system (1.1) is not null controllable at time T. Recall that condition (1.13) holds. We argue by contradiction. Assume that system (1.1) is null controllable at time T < e T0(q). By means of Proposition 3.3, this last fact is equivalent to the existence of a positive constant Csuch that the observability inequality (3.3) holds for every solution θof the adjoint problem (3.1). Let us work with the particular solutions associated with initial data θ0=akΦ∗ 1,k +bkΦ∗ 2,k, with ak, bk∈R, to be 17 determined, and Φ∗ 1,k and Φ∗ 2,k given in Proposition 2.1. With this choice, the solution θkof (3.1) is given by θk(·, t) = ake−k2(T−t)Φ∗ 1,k −(T−t)Ik(q) Φ∗ 2,k+bke−k2(T−t)Φ∗ 2,k,∀k≥1. Thus, the observability inequality (3.3) becomes A1,k ≤CA2,k,∀k≥1, with A1,k := e−2k2Tnk2|ak|2+h|ak|2kψ0 kk2 L2(0,π)+k2(bk−TIk(q)ak)2io≥e−2k2Tk2|ak|2,∀k≥1, and A2,k := ZT 0 e−2k2t|akψ0 k(0) + (bk−tIk(q)ak)ϕ0 k(0)|2dt, ∀k≥1. Taking ak= 1 and bk=−ψ0 k(0)/ϕ0 k(0) = −1 kpπ 2ψ0 k(0), the inequality observability transforms into e−2k2Tk2≤A1,k ≤CA2,k =C2 π|Ik(q)|2k2ZT 0 t2e−2k2tdt, ∀k≥1, that is to say, for a new constant C > 0 not depending on k, one has, 1≤Ce2k2T|Ik(q)|2,∀k≥1.(3.11) From the definition of e T0(q), we obtain the existence of an increasing unbounded subsequence {kn}n≥1such that e T0(q) = lim n→∞ −log |Ikn(q)| k2 n ∈(0,∞]. Assume that 0 <e T0(q)<∞(the case e T0(q) = ∞is much simpler and the details are left to the reader). In this case, for every ε > 0, there exits a positive integer nεsuch that e T0(q)−ε≤−log |Ikn(q)| k2 n ,∀n≥nε. This last inequality together with (3.11) provide the new inequality 1≤Ce−2k2 n(e T0(q)−T−ε),∀n≥nε. The previous inequality gives a contradiction if we take 0 <ε<e T0(q)−T/2. This ends the proof. 4 Distributed approximate controllability In this section we will address the problem of the approximate controllability at time T > 0 of system (1.2), i.e, we will prove Theorem 1.2. As said above, Theorem 1.2 is a direct consequence of the results on approximate controllability stated in [14]. For the sake of completeness we will provide a direct proof of the result. As in Section 3, we will first establish the relation between system (1.2) and (3.1). On the other hand, we will also give a general characterization of the controllability properties of system (1.2). One has: 18 Proposition 4.1. Let us consider A0and Bgiven by (1.3)and q∈L∞(0, π), a given function. Then, for any y0∈L2(0, π;R2),v∈L2(QT)and θ0∈L2(0, π;R2), one has ZZQT v(x, t)1ωB∗θ(x, t)dx dt =hy(·, T), θ0i−hy0, θ(·,0)i, where y, θ ∈L2(0, T;H1 0(0, π;R2))∩C0([0, T]; L2(0, π;R2)) are, resp., the solutions to (1.2)and (3.1) associated to (y0, v)and θ0. For a proof of the previous result see for instance [15], [36] or [21]. Proposition 4.2. Under assumptions of Proposition 4.1, one has: 1. System (1.2)is approximately controllable at time T > 0if and only if the following unique continuation property holds: “Let θ0∈L2(0, π;R2)be given and let θbe the corresponding solution of the adjoint problem (3.1). Then, if B∗θ= 0 in ω×(0, T), one has θ0≡0in (0, π).” 2. System (1.2)is null controllable at time T > 0if and only if there exists a positive constant Csuch that the observability inequality kθ(·,0)k2 L2(0,π;R2)≤CZZω×(0,T ) |B∗θ(x, t)|2dx dt (4.1) holds for every θ0∈L2(0, π;R2). In (4.1),θis the adjoint state associated to θ0, i.e., the solution of (3.1)associated to θ0. Again, this result is very well known. For a proof see, for instance, [37], [15] or [36]. We can already prove Theorem 1.2. The arguments will be similar to those used in Section 3. We recall that q∈L∞(0, π) is a function satisfying (1.11), where ω= (a, b). Necessary condition: Again, we argue by contradiction. Let us suppose that condition (1.15) does not hold, i.e., that there exists k0≥1 such that Ik0(q) = I1,k0(q) = 0. We will see that the distributed unique continuation property for the adjoint system (3.1) fails to be true. First, from Proposition 2.6, the function ψk0is given by: ψk0(x) = τk0ϕk0(x),∀x∈ω, (4.2) (since Ik0(q) = I1,k0(q) = 0) where τk0is given in Proposition 2.6. On the other hand, let us take θ0=aΦ∗ 1,k0+bΦ∗ 2,k0∈L2(0, π;R2), with a, b ∈Rto be determined. Again, the functions Φ∗ 1,k0and Φ∗ 2,k0are eigenfunctions of the operator L∗(see Proposition 2.1). Thus, the solution of the adjoint problem (3.1) is given by (3.4), so that: B∗θ(x, t) = e−k2 0(T−t)(aψk0(x) + bϕk0(x)) = e−k2 0(T−t)(aτk0+b)ϕk0(x),∀(x, t)∈ω×(0, T), thanks to (4.2). Just taking a= 1 and b=−τk0we obtain B∗θ≡0 in ω×(0, T) and θ6≡ 0. This contradicts the distributed unique continuation property for system (3.1). So, system (1.2) is not approximately controllable at time T > 0. This proves the necessary part of Theorem 1.2. Sufficient condition: Let us assume that condition (1.15) holds. The objective is to show that system (1.2) is approximately controllable at time T, when q∈L∞(0, π) satisfies (1.11). This amounts to prove the distributed unique continuation property for system (3.1) stated in Proposition 4.2. Let us fix θ0∈L2(0, π;R2) and assume that the corresponding solution θof (3.1) satisfies B∗θ≡0 in ω×(0, T). Since B∗is a basis for L2(0, π;R2) (for the expression of B∗, see (2.7)), we can write θ0=X k≥1akΦ∗ 1,k +bkΦ∗ 2,k, 19 where the coefficients are given by ak=hθ0,Φ1,kiand bk=hθ0,Φ2,kifor any k≥1. As it has been already observed, we have: θ(·, t) = X k≥1 e−k2(T−t)akΦ∗ 1,k −(T−t)Ik(q) Φ∗ 2,k+bkΦ∗ 2,kin QT. In fact, following the ideas in Lemma 2.3, it is not difficult to prove the convergence of this series in C0([0, T]; L2(0, π;R2)). Thus,        B∗θ(·, t)|ω=X k≥1 e−k2(T−t)akB∗Φ∗ 1,k|ω+bkB∗Φ∗ 2,k|ω−X k≥1 (T−t)e−k2(T−t)akIk(q)B∗Φ∗ 2,k|ω =X k≥1 e−k2(T−t)[akψk|ω+bkϕk|ω]−X k≥1 (T−t)e−k2(T−t)akIk(q)ϕk|ω for any t∈(0, T). Using again that the family {e−k2t, te−k2t}k≥1⊂L2(0, T) is minimal in L2(0, T) and the assumption B∗θ≡0 in ω×(0, T) we get akψk|ω+bkϕk|ω≡0 and akIk(q)ϕk|ω≡0∀k≥1 It is clear that from the previous identities that ak=bk= 0 for all k∈Λ. On the other hand, taking into account the expression of the ψkin ω(see Proposition 2.6), the last equality becomes (akτk+bk)ϕk(x)−rπ 2 I1,k(q) kakcos(kx)=0 ∀x∈ω, ∀k∈Λ3. Using the independence of ϕkand the function cos(k·) in ω, we conclude that ak=bk= 0 for every k∈Λ3. This proves that θ0≡0. Therefore, we have proved the distributed continuation property for the solutions to the adjoint problem (3.1) and the approximate controllability of system (1.2) at any positive time T. 5 Proof of Theorem 1.3: The positive null controllability result This section will be devoted to proving the null controllability of system (1.2) at time T > 0, when this time satisfies T > T0(q) (T0(q), given by (1.17), is assumed to be finite in this section). In order to make the proof clearer, we will divide it into several steps. 5.1 The moment problem We start the proof of the first point of Theorem 1.3 by reformulating the null controllability property for system (1.2) as a moment problem. To this end, let us consider T > T0(q) (T0(q) is given by (1.17)). The aim is to prove that for any y0∈L2(0, π;R2) there exists a control v∈L2(QT) such that the corresponding solution yof system (1.2) satisfies y(·, T) = 0 in (0, π). Let us fix an initial datum y0∈L2(0, π;R2). Thanks to Proposition 3.1 and 4.1, it is easy to see that the solution y∈C0([0, T]; L2(0, π;R2)) of system (1.2) associated with y0and a control v∈L2(QT) satisfies y(·, T) = 0 in (0, π) if and only if the control v∈L2(QT) satisfies ZZQT v(x, t)1ωB∗θ(x, t)dx dt =− hy0, θ(·,0)i,∀θ0∈L2(0, π;R2), where θis the solution of the adjoint problem (3.1) corresponding to θ0. Using that B∗is a basis of L2(0, π;R2) (see Lemma 2.3) this last property is equivalent to v∈L2(QT) and satisfies ZZQT v(x, t)1ωB∗θi,k(x, t)dx dt =− hy0, θi,k(·,0)i,∀k≥1,∀i= 1,2,(5.1) 20 where θi,k denotes the solution of system (3.1) associated with θ0= Φ∗ i,k. By means of the previous problem we have reformulated the null controllability property for system (1.2) as a moment problem. In order to solve the moment problem (5.1), the first main idea is to search controls under the particular form v(x, t) = f1(x)v1(T−t) + f2(x)v2(T−t),(x, t)∈QT,(5.2) where v1, v2∈L2(0, T) are new controls, only depending on t, and f1, f2∈L2(0, π) are appropriate functions satisfying the condition Supp f1,Supp f2⊆ω= (a, b). This choice will be made clearer a little further in the text. For k≥1 and θ0= Φ∗ 2,k, the solution to (3.1) is given by θ2,k(·, t) = e−k2(T−t)Φ∗ 2,k. Thus, after a change of variables, the moment problem (5.1) with controls vgiven by (5.2) reads as follows: f1,k ZT 0 v1(t)e−k2tdt +f2,k ZT 0 v2(t)e−k2tdt =−e−k2Ty0,Φ∗ 2,k, where f1,k,f2,k are, respectively, the Fourier coefficients with respect to ϕkcorresponding to f1, f2: fi,k := Zπ 0 fi(x)ϕk(x)dx, i = 1,2,∀k≥1.(5.3) For θ0= Φ∗ 1,k, the corresponding solution of (3.1) is given by θ(·, t) = e−k2(T−t)Φ∗ 1,k −(T−t)Ik(q) Φ∗ 2,k. From the expression of functions Φ∗ i,k (see the statement of Proposition 2.1), for k≥1 and i= 1, the equality (5.1) with controls vgiven by (5.2) changes into              e f1,k ZT 0 v1(t)e−k2tdt +e f2,k ZT 0 v2(t)e−k2tdt −Ik(q)f1,k ZT 0 v1(t)te−k2tdt −Ik(q)f2,k ZT 0 v2(t)te−k2tdt =−e−k2Ty0,Φ∗ 1,k−TIk(q)y0,Φ∗ 2,k, where, for k≥1, e f1,k,e f2,k are given by e fi,k := Zπ 0 fi(x)ψk(x)dx, i = 1,2.(5.4) Let us point out that, thanks to the properties of the function ψk(see (2.6)), one has e fi,k≤C k, i = 1,2,if k≥1,(5.5) for some positive constant C. Summarizing, we have transformed the null-controllability problem at time T > 0 for system (1.2) into the following moment problem: Find v∈L2(QT) under the form (5.2) such that v1, v2∈L2(0, T) satisfy                        f1,k ZT 0 v1(t)e−k2tdt +f2,k ZT 0 v2(t)e−k2tdt =−e−k2Ty0,Φ∗ 2,k e f1,k ZT 0 v1(t)e−k2tdt +e f2,k ZT 0 v2(t)e−k2tdt −Ik(q)f1,k ZT 0 v1(t)te−k2tdt −Ik(q)f2,k ZT 0 v2(t)te−k2tdt =−e−k2Ty0,Φ∗ 1,k−TIk(q)y0,Φ∗ 2,k, k≥1,(5.6) 21 with the notations in (5.3) and (5.4). Our objective is to solve the previous moment problem under the assumption (1.15) and when T > T0(q) (see (1.17)). To this end, we will construct appropriate functions f1, f2∈L2(0, π) satisfying Supp f1,Supp f2⊆ω= (a, b). Let us remark that, if we fix k≥1, (5.6) is a linear system of two equations and four unknown quantities: ZT 0 v1(t)e−k2tdt, ZT 0 v2(t)e−k2tdt, ZT 0 v1(t)te−k2tdt and ZT 0 v2(t)te−k2tdt. The moment problem (5.6) can be written as AkVk+e Ake Vk=Fk∀k≥1,(5.7) whith for k≥1 : Ak= f1,k f2,k e f1,k e f2,k !,e Ak= 0 0 −Ik(q)f1,k −Ik(q)f2,k !(5.8) Vk:=     ZT 0 v1(t)e−k2tdt ZT 0 v2(t)e−k2tdt     ,e Vk:=     ZT 0 v1(t)te−k2tdt ZT 0 v2(t)te−k2k2tdt     ,(5.9) and Fk= −e−k2TDy0,Φ∗ 2,kE −e−k2TDy0,Φ∗ 1,kE−TIk(q)Dy0,Φ∗ 2,kE .(5.10) Remind that fi,k is the Fourier coefficient of fiwith respect to ϕkand e fi,k is given by (5.4). 5.2 Construction of the functions f1and f2 In this subsection we will construct appropriate functions f1, f2∈L2(0, T) satisfying Supp f1,Supp f2⊆ω, which will allow us to solve the moment problem (5.7) . One has: Lemma 5.1. There exist functions f1, f2∈L2(0, π)satisfying Supp f1,Supp f2⊆ωand such that      min {|f1,k|,|f2,k|} ≥ C k3,∀k≥1, |Bk|:= f1,k b f2,k −f2,k b f1,k≥C k5,∀k≥1. (5.11) In (5.11)Cis a positive constant only depending on f1and f2,fi,k (i= 1,2) is the Fourier coefficient of the function fiwith respect to ϕkand b fi,k is given by b fi,k =Zπ 0 fi(x) cos(kx)dx, k ≥1, i = 1,2.(5.12) Proof. Let us consider the functions f1:= 1(a1,b1)and f2:= 1(a2,b2)with a1, b1, a2, b2∈ωand ai< bi,i= 1,2. Then,          fi,k =Zπ 0 fi(x)ϕk(x)dx =2 kr2 πsin kai+bi 2sin kbi−ai 2, b fi,k =Zπ 0 fi(x) cos(kx)dx =2 kcos kai+bi 2sin kbi−ai 2. 22 Direct computations show that |Bk|=4 k2r2 πsin kb1−a1 2sin kb2−a2 2sin ka1+b1−a2−b2 2. Let us now take b1=a1+ 2`,a2=a1+`and b2=a1+ 3`, with a1∈(a, (3a+b)/4) and `∈(0,(b−a)/4) such that a1/π is a rational number and `/π is an irrational algebraic number of order 2. In this case, we have that (a1+`)/π and (a1+ 2`)/π are also irrational algebraic numbers of order 2. Thus, a1, b1, a2, b2∈ωand ai< bi,i= 1,2. On the other hand, let us admit the following property which will be proved below: if ξ/π ∈(0,∞) is an irrational algebraic number of order 2, then inf k≥1(k|sin(kξ)|)≥C, (5.13) for a positive constant Conly depending on ξ. Coming back to the expressions of f1,k,f2,k and |Bk|and taking into account the previous property, one obtains                  |f1,k|=2 kr2 π|sin (k(a1+`))| |sin (k`)| ≥ C1 k3, |f2,k|=2 kr2 π|sin (k(a1+ 2`))| |sin (k`)| ≥ C2 k3, |Bk|=4 k2r2 π|sin (k`)|3≥C3 k5,∀k≥1, with C1,C2and C3positive constants only depending on a1and `. This proves (5.11). Let us finalize the proof showing inequality (5.13). This inequality is a consequence of Liouville’s theorem on diophantine approximation: Lemma 5.2 ([30]).Let νbe an irrational algebraic number of degree n≥2, i.e., νis an irrational number which is the root of a polynomial of degree nwith integer coefficients. Then, there exists a positive number C, depending on ν, such that ν−p q>C qn,∀p, q ∈N∗, q > 0. Let us consider ξ > 0 such that ξ/π is an irrational algebraic number of degree 2 and let us see inequality (5.13). First, for any k≥1 there exists hk∈N∗such that kξ π−hk≤1 2,∀k≥1. Indeed, we can take hk=bkξ/πcif kξ/π − bkξ/πc ≤ 1/2 or hk=bkξ/πc+ 1 otherwise (b·c is the floor function, i.e., for x∈R,bxcgives the largest integer less than or equal to x). If we now apply Lemma 5.2 with ν=ξ/π,n= 2, q=kand p=hkwe get Cπ k≤ |kξ −hkπ| ≤ π 2,∀k≥1, and k|sin (kξ)|=k|sin (kξ −hkπ)|=ksin |kξ −hkπ| ≥ ksin Cπ k≥2C, ∀k≥1. In the last inequality we have used sin x x≥2 π,∀x∈(0, π/2]. This proves inequality (5.13). 23 As a consequence of the previous result, we also have: Corollary 5.3. Let us consider the functions f1and f2provided by Lemma 5.1 and the associated matrix Akgiven in (5.8). Then, there exists positive constants C1and C2(only depending on f1 and f2) such that |det Ak| ≥ C1 |I1,k(q)| k6−C2 |Ik(q)| k,∀k≥1.(5.14) Proof. Let k≥1. We have (see (5.8)) det Ak=f1ke f2,k −f2,k e f1,k, where f1kand f2kare the Fourier coefficients of f1and f2and where e f1,k and e f2,k are given by (5.4). Using Proposition 2.6 and taking into account that Supp fi⊂ω, one gets e fi,k =τkfi,k +Zπ 0 fi(x)gk(x)dx. So det Ak=f1,k Zπ 0 f2(x)gk(x)dx −f2,k Zπ 0 f1(x)gk(x)dx. Using again Proposition 2.6,gkcan be written as gk(x) = −Ik(q) kZx 0 sin(k(x−ξ))ϕk(ξ)dξ −rπ 2 I1,k(q) kcos(kx),∀x∈ω, ∀k≥1. We deduce then that det Ak=−rπ 2 I1,k(q) kf1,k b f2,k −f2,k b f1,k−Ik(q) k(f1,kG2,k −f2,kG1,k), k ≥1, where b fi,k is given in (5.12) and Gi,k =Zπ 0Zx 0 fi(x) sin(k(x−ξ))ϕk(ξ)dξ dx, for i= 1,2 and k≥1. Finally, from (5.11) and using that the sequence {Gi,k}k≥1(i= 1,2) is bounded, we deduce (5.14) for k≥1. This ends the proof. 5.3 Solving the moment problem We will devote this subsection to solving the moment problem (5.7) when T > T0(q) (T0(q), given by (1.17), is assumed to be finite in this section). To this end, we will work with the functions f1 and f2provided by Lemma 5.1 and Corollary 5.3. Theorem 5.4. Let y0∈L2(0, π;R2)be given and let us consider the moment problem (5.7). Then, we can find a solution of this problem under the form          ZT 0 vi(t)e−k2tdt =e−k2TM(k) 1,i (y0), ZT 0 vi(t)te−k2tdt =e−k2TM(k) 2,i (y0), (5.15) where the quantities M(k) i,j (y0)∈R, with k≥1and 1≤i, j ≤2, satisfy the following property: for any ε > 0there exists a positive constant Cε(only depending on ε) such that M(k) i,j (y0)≤Cεek2(T0(q)+2ε)ky0kL2(0,π;R2),∀k≥1,1≤i, j ≤2.(5.16) 24 In the sequel, let us fix ε > 0. From the definition of the minimal time T0(q), we can infer the existence of a positive integer kεfor which min {− log |I1,k(q)|,−log |Ik(q)|} k2< T0(q) + ε, ∀k > kε.(5.17) In order to find a solution of the moment problem (5.7) under the form (5.15), we are going to distinguish if kbelongs to the set Λ1, the set Λ2or the set Λ3(see (1.16)). 5.3.1 The case k∈Λ1 Let us start solving the moment problem (5.7) when k∈Λ1(for the definition of Λ1, see (1.16)). 1. Let us first consider k∈Λ1with k≤kε. Thanks to Lemma 5.1 (see (5.11)) we can deduce that f1,kf2,k 6= 0 for any k≥1. In this case, we solve the moment problem (5.7) as follows. Take ZT 0 v2(t)e−k2tdt =ZT 0 v2(t)te−k2tdt = 0,∀k∈Λ1, k ≤kε. With this choice, system (5.7) is equivalent to          f1,k ZT 0 v1(t)e−k2tdt =F(1) k, e f1,k ZT 0 v1(t)e−k2tdt −Ik(q)f1,k ZT 0 v1(t)te−k2tdt =F(2) k, (5.18) with k∈Λ1,k≤kεand where F(i) k,i= 1,2, are the components of Fk(see (5.10)). Observe that in the set Λ1one has Ik(q)6= 0. Therefore, the previous problem can be solved as in the boundary case (see Section 3.2) obtaining a solution under the form (5.15), for any k∈Λ1with k≤kε. In particular, M(k) 1,2(y0) = M(k) 2,2(y0) = 0. Using the properties of e fi,k (see (5.5)) and taking into account that k∈Λ1and k≤kε, we deduce the existence of a positive constant Cεsuch that M(k) i,j (y0)≤Cεky0kL2(0,π;R2),∀k∈Λ1, k ≤kε,1≤i, j ≤2.(5.19) As a consequence, we get inequality (5.16) for any k∈Λ1, with k≤kε. 2. Let us now deal with the case k∈Λ1and k > kε. As before, our objective is to solve the moment problem (5.7). To this end, for k > kε, let us split the set Λ1into two subsets        Λ? 1,ε := k∈Λ1:k > kεand −1 k2log |Ik(q)| ≤ T0(q) + 3 2ε, Λ1,ε := k∈Λ1:k > kεand −1 k2log |Ik(q)|> T0(q) + 3 2ε. If k∈Λ? 1,ε, then we reason as in the previous case. We take ZT 0 v2(t)e−k2tdt =ZT 0 v2(t)te−k2tdt = 0,∀k∈Λ? 1,ε, and the moment problem (5.7) is equivalent to (5.18), with k∈Λ? 1,ε. Again, we can compute the solution of this system, which is given by (5.15) (k∈Λ? 1,ε), where M(k) 1,2(y0) = M(k) 2,2(y0) = 0 and          M(k) 1,1(y0) = −1 f1,k y0,Φ∗ 2,k, M(k) 2,1(y0) = −1 f1,kIk(q) y0,Φ∗ 1,k−TIk(q)y0,Φ∗ 2,k+e f1,k f1,k y0,Φ∗ 2,k!. 25 Let us also introduce the function q: q(x) := (1 if x∈[a1, a1+`π], −1 if x∈[a2+`π, a2+`π]. With this function q, the objective is to analyze the dependence of the minimal time of null controllability for system (1.2) on the position of the control open set ω= (a, b)⊂(0, π). To this end, we will consider three different situations: 1. Supp q∩ω6=∅: In this case, system (1.2) is a particular case of system (1.4) (C≡0) where the coefficient a12 =qsatisfies condition (1.5) with σ= 1 and ω0could be a connected component of the interior of the set Supp q∩ω6=∅. From very well-known results (see for instance [35], [25] or [26]), we deduce that system (1.2) is null controllable at time Tfor any T > 0, that is to say, the minimal time of distributed null controllability is zero: T0(q) = 0. 2. a1+`≤a<b≤a2: In this case, condition (1.11) holds and it is easy to show (see (1.12)) I1,k(q) = 1 π`−1 ksin (k`π) cos (k(2a1+`π)), I2,k(q) = −1 π`−1 ksin (k`) cos k2a1+3 2`π, Ik(q) = I1,k(q) + I2,k(q) = −2 kπ sin (k`π) sin (k(a1+a2+`π)) sin (k(a2−a1)) =−2 kπ sin (k`π) sin (kα1π) sin (kα2π). Thanks to the assumption on α1,α2and `, we deduce that Ik(q)6= 0 for any k≥1 and qfulfills condition (1.15). Since ` > 0, we also obtain the existence of k0≥1 such that |I1,k(q)|>|Ik(q)|for all k≥k0. Therefore (see (1.17)), T0(q) = lim sup −log |I1,k(q)| k2= 0. In conclusion, under the previous geometrical situation, one obtains that system (1.2) is approximately and null controllable at any positive time T. Observe that the null controllability property of system (1.2) is independent of the diophantine approximation properties of the irrational number `. 3. 0 ≤a < b ≤a1or a2+`≤a<b≤π: In this case, condition (1.11) also holds. Let us analyze the case 0 ≤a<b≤a1. An analogous result can be obtained in the case a2+`≤ a < b ≤π. With the previous choice, I1,k(q) = 0, Ik(q) = −2 kπ sin (k`π) sin (kα1π) sin (kα2π), and T0(q) = lim sup −log |Ik(q)| k2=e T0(q). Again, we will use the properties of irrational algebraic numbers proved before. To be precise, as a consequence of inequality (5.13) applied to α1πand α2π, we deduce the existence of two positive constants C1and C2such that −log 2 kπ −log |sin (k`π)|≤−log |Ik(q)|≤−log 2C1C2 k3π−log |sin (k`π)|,∀k≥1. As a consequence, T0(q) = lim sup −log |sin (k`π)| k2, 32 and the minimal time of null controllability for system (1.2) depends on the diophantine approximation properties of the irrational number `. Thanks to Lemma 7.1, given τ0∈ [0,∞], there is `∈(0,1/5) satisfying (7.1), that is to say, there is `∈(0,1/5) such that T0(q) = τ0. In contrast with the geometrical situation in item 2, in the current case, the null controllability property of system (1.2) strongly depends on the diophantine approximation property of the irrational number `. Summarizing, with this example we have shown that, given a function q∈L∞(0, π), the null controllability property of system (1.2) is different when the function qand the control interval ωsatisfy Supp q∩ω6=∅or Supp q∩ω=∅. But even in this last case, i.e., in the case in which condition (1.11) holds, the distributed null controllability result depends on the relative position of the set Supp qand the control interval ω. 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[35] L. de Teresa,Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations 25 (2000), no. 1–2, 39–72. [36] M. Tucsnak, G. Weiss,Observation and Control for Operator Semigroups, Birkh¨auser Advanced Texts: Basler Lehrb¨ucher, Birkh¨auser Verlag, Basel, 2009. [37] J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications, Birkh¨auser Boston, Inc., Boston, MA, 1992. A Proof of Lemma 7.1 We will obtain the proof of Lemma 7.1 as a consequence of Lemma 5.2 and the result: Lemma A.1. 1. Let us fixed τ0∈(0,∞),x0∈[0,∞)and ε > 0. Then, there exist an irrational number ν > 0and a sequence of rational numbers {pk/qk}k≥1such that pkand qk are co-prime positive integers, the sequences {pk}k≥1and {qk}k≥1are strictly increasing, |ν−x0| ≤ εand lim eτ0q2 kν−pk qk= 1.(A.1) Moreover, for any k≥1one has 0<|qkν−pk|≤|qν −p|,∀p, q ∈N∗,with q < qk+1.(A.2) 2. For any σ∈(0,∞),x0∈[0,∞)and ε > 0, there exists an irrational number ν > 0and a sequence of rational numbers {pk/qk}k≥1such that pkand qkare co-prime positive integers, the sequences {pk}k≥1and {qk}k≥1are strictly increasing and |ν−x0| ≤ εand lim eq2+σ kν−pk qk= 0.(A.3) The previous result has been proved in [10] (see Lemma 6.22, Corollary 6.25 and Appendix A). Let us fix x0≥0 and ε > 0. In order to prove Lemma 7.1 we will use some ideas from [10]. We will divide the proof of Lemma 7.1 into three different cases: 35 Case τ0= 0. Given x0≥0 and ε > 0, let us take ν∈[x0−ε, x0+ε] a positive irrational algebraic number of order 2. From (5.13) applied to ξ=νπ, we deduce the existence of a positive constant Csuch that |sin (kνπ)| ≥ C k,∀k≥1. Thus, lim sup −log |sin (kνππ)| k2≤lim sup −log (C/k) k2= 0. Taking into account that the previous limit superior is always nonnegative, we deduce (7.1). This proves the result for τ0= 0. Case τ0∈(0,∞). Given x0≥0, ε > 0 and τ0, we can apply the first item in Lemma A.1 and conclude the existence of an irrational number ν∈[x0−ε, x0+ε] satisfying (A.1) and (A.2) for the sequences of positive integers {pk}k≥1and {qk}k≥1. With this choice we deduce that νsatisfies lim pk/qk=ν,   lim 1 qk eτ0q2 k|νqk−pk|= 1 and 0<|νqk−pk|≤|νq −p|,∀p, q ∈N∗, q < qk+1. (A.4) Let us see that the previous number νsatisfies (7.1). From the first equality in (A.4) we deduce lim |νqk−pk|= 0 and e T0(q) = lim sup −log |sin (νkπ)| k2≥lim sup −log |sin (νqkπ)| q2 k = lim sup −log |sin [π(νqk−pk)]| q2 k = lim −log [π|νqk−pk|] q2 k = lim −log πqke−τ0q2 k q2 k =τ0. Then e T0(q)≥τ0. Observe that the previous reasoning also implies the existence of the following limit: lim −log |sin [π(νqk−pk)]| q2 k =τ0. Let us now prove the inequality e T0(q)≤τ0. To this end, let us fix ε > 0. From the previous property, there exists k0(ε)≥1 such that −log |sin [π(νqk−pk)]| q2 k ≤τ0+ε, ∀k≥k0(ε).(A.5) As in the proof of Lemma 5.2, for every n≥1 there is hn∈N∗for which |νn −hn| ≤ 1 2,∀n≥1. Let us take n0(ε) = qk0(ε)≥1. Thus, using that the sequence {qk}k≥1is strictly increasing, if n≥n0(ε), we infer the existence of k≥k0(ε) such that qk≤n<qk+1. These last properties together with the second formula in (A.4) allow us to write 0<|νqk−pk| ≤ |νn −hn| ≤ 1 2, and (|sin (νnπ)|=|sin (νnπ −hnπ)|= sin |νnπ −hnπ| ≥sin |νqkπ−pkπ|=|sin (νqkπ−pkπ)|,∀n≥n0(ε) : qk≤n<qk+1. Since qk≤n<qk+1, the previous inequality and (A.5) give −log |sin (νnπ)| n2≤−log |sin [π(νqk−pk)]| q2 k ≤τ0+ε, ∀n≥n0(ε) : qk≤n<qk+1, 36 and lim sup −log |sin (νnπ)| n2≤τ0+ε∀ε > 0. In conclusion, we have obtained (7.1). This proves Lemma A.1 when τ0∈(0,∞). Case τ0=∞. For τ0=∞,x0≥0 and ε > 0, we apply the second item in Lemma A.1 with, for instance, σ= 1/2. We deduce the existence of a positive irrational number νwhich fulfills property (A.2). Repeating the arguments of the previous point we deduce lim |νqk−pk|= 0 and (σ= 1/2) e T0(q) = lim sup −log |sin (νkπ)| k2≥lim sup −log |sin (νqkπ)| q2 k = lim sup −log |sin [π(νqk−pk)]| q2 k = lim −log [π|νqk−pk|] q2 k = lim −log πqke−q2+σ k q2 k =∞. This shows that e T0(q) = ∞and finishes the third case and the proof of Lemma 7.1. 37