Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 26, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NULL CONTROLLABILITY FROM THE EXTERIOR OF FRACTIONAL PARABOLIC-ELLIPTIC COUPLED SYSTEMS CAROLE LOUIS-ROSE Abstract. We analyze the null controllability properties from the exterior of two parabolic-elliptic coupled systems governed by the fractional Laplacian (−d2x)s, s ∈ (0, 1), in one space dimension. In each system, the control is located on a non-empty open set of R \ (0, 1). Using the spectral theory of the fractional Laplacian and a unique continuation principle for the dual equation, we show that the problem is null controllable if and only if 1/2 < s < 1. 1. Introduction Let ω be a non-empty open set of R\(0, 1). We will denote by 1ω the characteris- tic function of ω. We consider the following linear parabolic-elliptic one-dimensional coupled systems ∂tu+ (−d2 x)su = au+ bv in (0, 1)× (0, T ), (−d2 x)sv = cu+ dv in (0, 1)× (0, T ), u = g1ω, v = 0 in [R \ (0, 1)]× (0, T ), u(·, 0) = u0 in (0, 1), (1.1) and ∂tu+ (−d2 x)su = au+ bv in (0, 1)× (0, T ), (−d2 x)sv = cu+ dv in (0, 1)× (0, T ), u = 0, v = h1ω in [R \ (0, 1)]× (0, T ), u(·, 0) = u0 in (0, 1), (1.2) where s ∈ (0, 1) and a, b, c, d are real numbers. In (1.1) and (1.2), we have that u = u(x, t), v = v(x, t), u0 is the initial state, g and h are the controls acting on the system through ω × (0, T ). The operator (−d2 x)s denotes the one-dimensional fractional Laplace operator which is defined, for smooth functions u that will be 2010 Mathematics Subject Classification. 93B05, 35R11, 93C05, 35C10, 93B60. Key words and phrases. Controllability; fractional partial differential equation; linear system; series solution; eigenvalue problem. c©2020 Texas State University. Submitted April 5, 2019. Published March 27, 2020. 1 2 C. LOUIS-ROSE EJDE-2020/26 specified in Section 3, by (−d2 x)su(x) = lim ε→0+ c1,s ∫ {y∈R:|x−y|>ε} u(x)− u(y) |x− y|1+2s dy = c1,sP.V. ∫ R u(x)− u(y) |x− y|1+2s dy, x ∈ R, (1.3) provided that the limit exists. The constant c1,s is c1,s = 22ssΓ ( 1+2s 2 ) √ πΓ(1− s) , (1.4) Γ being the Euler Gamma function. In this article, we study the controllability of the parabolic-elliptic systems (1.1) and (1.2) from the exterior. As far as we know, the null controllability of such fractional parabolic-elliptic systems involving the fractional Laplacian has not been studied yet. The null controllability of the systems (1.1) and (1.2) is the purpose of [6] in the semilinear case. Both equations considered in [6] are governed by the Laplacian and to reach the result, the authors proved some Carleman estimates for the solu- tion of the adjoint system. They also extended these results in [7] to a semilinear system of two parabolic PDEs and one elliptic PDE. In the fractional case, the authors of [13] showed the null controllability of a parabolic equation governed by the fractional power of the Dirichlet Laplacian. Thus they generalized the clas- sical null controllability results for the heat equation established for example in [5, 22, 2, 14]. In [15] the author analyzed controllability properties of the fractional diffusion equation involving the spectral fractional Laplacian when the control is localized in the domain under consideration. Then the authors of [1] focused on the null controllability of a similar parabolic problem governed by the one-dimensional fractional Laplacian (−d2 x)s defined by (1.3) for all s ∈ (0, 1). They proved that the interior approximate controllability holds for every s ∈ (0, 1) and that the in- terior null controllability of the equation holds if and only if s > 1/2. The latter result follows from the spectral method developed in [9, 10]. The null controlla- bility from the exterior of the interval (0, 1) of the fractional heat equation was subsequently investigated in [20]. The null controllability from the exterior means that the control is located in R \ (0, 1), precisely in a non-empty open subset of R \ (0, 1). This type of controllability problem was introduced first by Warma for space-time fractional diffusion equations associated with the fractional Laplacian and the Caputo derivative of order α ∈ (0, 1] (see [19]). The approximate control- lability by means of a unique continuation property for the dual equation is the purpose of Warma’s paper. The same work was extended in [12] to the fractional wave equation with Dirichlet or Robin type exterior conditions and in [21] to the strong damping nonlocal wave equation. Systems such as (1.1) (resp. (1.2)) can arise in chemistry to describe the behav- ior in systems of interacting components. The fractional Laplacian is a nonlocal operator modelling the multi-scale behavior. Applications with models containing fractional diffusion operators such as phase field models, are possible. The rest of the paper is organized as follows. In Section 2 we state the main result dealing with null controllability properties of (1.1) (resp. (1.2)). The main result is based on a unique continuation property for the realization of (−d2 x)s in L2(0, 1) with the homogeneous exterior Dirichlet condition (see Lemma 3.2). Then EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 3 we give in Section 3 some preliminary results including the spectral properties of the fractional Laplacian. In Section 4 a representation in the form of convergent series of the solution of (1.1) (resp. (1.2)) and of the dual equation, is proved. Finally Section 5 is devoted to the proof of the main result stated in Section 2. 2. Main result In this section the main result of this work is stated. Let s be a real number which will be fixed in (0, 1) until the end of the article. First we give the dual system associated with (1.1) (resp. (1.2)). Using the integration by parts (3.3) (see Lemma 3.1), we have the following dual system −∂tϕ+ (−d2 x)sϕ = aϕ+ cσ in (0, 1)× (0, T ), (−d2 x)sσ = bϕ+ dσ in (0, 1)× (0, T ), ϕ = σ = 0 in [R \ (0, 1)]× (0, T ), ϕ(·, T ) = ϕT in (0, 1), (2.1) where ϕT ∈ L2(0, 1). Then we define the notion of weak solution of (1.1) (resp. (1.2)). Let 〈·, ·〉 be the duality pair between the fractional order Sobolev space Hs(R) and its dual H−s(R). We shall give the definition of these spaces in Section 3. Definition 2.1. Let a, b, c, d be real constants, g ∈ L2(R \ (0, 1)) (resp. h ∈ L2(R \ (0, 1))) and u0 ∈ L2(0, 1). A function (u, v) is said to be a weak solution of (1.1) (resp. (1.2)) if the following properties hold. • Regularity: (u, v) ∈ ( C([0, T ];L2(0, 1)) )2 . • Variational identity: for every (ϕ, σ) ∈ (Hs(R)) 2 and a.e. t ∈ (0, T ), 〈∂tu, ϕ〉+ 〈(−d2 x)su, ϕ〉 = a〈u, ϕ〉+ b〈v, ϕ〉, 〈(−d2 x)sv, σ〉 = c〈u, σ〉+ d〈v, σ〉. • Initial and exterior conditions: u = g and v = 0 in [R\ (0, 1)]× (0, T ) (resp. u = 0 and v = h in [R \ (0, 1)]× (0, T )), and u(·, 0) = u0 in (0, 1). System (1.1) (resp. (1.2)) is well posed in the sense that for each u0 ∈ L2(0, 1) and each g ∈ D([R \ (0, 1)]× (0, T )) (resp. h ∈ D([R \ (0, 1)]× (0, T ))), it possesses exactly one solution (u, v) such that (u, v) ∈ ( C([0, T ];L2(0, 1)) )2 (2.2) and there is a constant C > 0 such that for any t ∈ [0, T ), ‖(u, v)(·, t)‖2(L2(0,1))2 6 C ( ‖u0‖2L2(0,1) + T 3‖gt‖2L∞(0,T ;L2(R\(0,1))) ) ( resp. ‖(u, v)(·, t)‖2(L2(0,1))2 6 C ( ‖u0‖2L2(0,1) + T 3‖ht‖2L∞(0,T ;L2(R\(0,1))) )) . (2.3) The assertions (2.2) and (2.3) are proved in Theorems 4.4 and 4.5. The notion of null controllability of the system (1.1) (resp. (1.2)) is the following. Definition 2.2. System (1.1) (resp. (1.2)) is null controllable at time T > 0 if for any given u0 ∈ L2(0, 1), there exists a control g ∈ L2(ω × (0, T )) (resp. h ∈ 4 C. LOUIS-ROSE EJDE-2020/26 L2(ω × (0, T ))) such that the corresponding solution (u, v) satisfies (u, v) ∈ ( C([0, T ];L2(0, 1)) )2 u(x, T ) = 0 in (0, 1), lim sup t→T− ‖v(·, t)‖L2(0,1) = 0. (2.4) Let (λn)n∈N be the sequence of eigenvalues of the fractional Laplacian (−d2 x)s. We shall recall their properties in Section 3. The main result of this article reads as follows. Theorem 2.3. Let ω be a non-empty open set of R \ (0, 1). Assume that c = b, b > 0, max{a, d} 6 −b and b2 < (λ1 − a)(λ1 − d). (2.5) (1) If 1/2 < s < 1 then system (1.1) (resp. (1.2)) is null controllable at any time T > 0 for any g ∈ L2(ω × (0, T )) (resp. h ∈ L2(ω × (0, T ))). (2) If 0 < s 6 1 2 then system (1.1) (resp. (1.2)) is not null controllable at time T > 0. 3. Preliminary results In this section, we define the functional framework, we present the spectral theory of the fractional Laplace operator, and we give some preliminary results that will be useful in the rest of the paper. First we define the space L1 s(R) = { u : R→ R, ∫ R |u(x)| (1 + |x|)1+2s dx <∞ } . Then Definition (1.3) is valid for any u ∈ L1 s(R) because the integral∫ {y∈R:|x−y|>ε} u(x)− u(y) |x− y|1+2s dy exists for any ε > 0. The fractional Sobolev space Hs(0, 1) is defined for s ∈ (0, 1), as Hs(0, 1) = { u ∈ L2(0, 1) : |u(x)− u(y)| |x− y| 12 +s ∈ L2((0, 1)× (0, 1)) } . Endowed with the norm ‖u‖Hs(0,1) = (∫ 1 0 |u|2 dx+ ∫ 1 0 ∫ 1 0 |u(x)− u(y)|2 |x− y|1+2s dx dy )1/2 . Then Hs(0, 1) is a Hilbert space. The dual of Hs(0, 1) is denoted by H−s(0, 1). The space of test functions on (0, 1), that is C∞ functions compactly supported in (0, 1), is denoted by D(0, 1). The fractional space Hs 0(0, 1) is defined as the closure of D(0, 1) in Hs(0, 1) with respect to the norm ‖ · ‖Hs(0,1) (i.e. Hs 0(0, 1) = D(0, 1) ‖·‖Hs(0,1) ). We refer the reader to [3] for a precise definition of these spaces. If E ⊂ R, the scalar product in L2(E) is denoted by (·, ·)L2(E). Let (ψk)k∈N be the orthonormal basis of eigenfunctions of the fractional Laplace operator associated with the eigenvalues (λk)k∈N. Then (ψk)k∈N is total in L2(0, 1) and is solution of the system (−d2 x)sψk = λkψk, x ∈ (0, 1), ψk = 0 in R \ (0, 1). (3.1) We set Hs 0((0, 1)) = {u ∈ Hs(R) : u = 0 in R \ (0, 1)}. EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 5 Let L be the bilinear form L(u, v) = c1,s 2 ∫ R ∫ R (u(x)− u(y))(v(x)− v(y)) |x− y|1+2s dx dy, u, v ∈ Hs 0((0, 1)) and let (−d2 x)sD be the selfadjoint operator on L2(0, 1) associated with L in the sense that D((−d2 x)sD) = {u ∈ Hs 0((0, 1)) : ∃f ∈ L2(0, 1),L(u, v) = (f, v)L2(0,1) ∀v ∈ Hs 0((0, 1))} and (−d2 x)sDu = f . We have that for u ∈ Hs 0((0, 1)), ‖u‖2 Hs0 ((0,1)) = ∑ n∈N ∣∣λ1/2 n (u, ψn)L2(0,1) ∣∣2 defines an equivalent norm on Hs 0((0, 1)). We also have that D((−d2 x)sD) = {u ∈ L2(0, 1) : ∑ n∈N |λn(u, ψn)L2(0,1)|2 <∞}. If u ∈ D((−d2 x)sD), then ‖u‖2D((−d2x)sD) = ‖(−d2 x)sDu‖2L2(0,1) = ∑ n∈N |λn(u, ψn)L2(0,1)|2. We know that ψk ∈ D((−d2 x)sD) and, using [18], that the operator (−d2 x)sD has a compact resolvent and its eigenvalues are real numbers such that 0 < λ1 6 λ2 6 · · · 6 λk 6 · · · and lim n→∞ λn =∞. Moreover, the eigenvalues of (−d2 x)sD possess the following asymptotic behavior (see [10, Theorem 1]) λk = (kπ 2 − (2− 2s)π 8 )2s +O (1 k ) as k →∞. (3.2) For u ∈ Hs(R), the nonlocal normal derivative is Nsu(x) = c1,s ∫ 1 0 u(x)− u(y) |x− y|1+2s dy, x ∈ R \ (0, 1), where c1,s is the positive constant given by (1.4). From [8, Lemma 3.2] we have that for every u ∈ Hs(R), Nsu ∈ L2(R \ (0, 1)). The following integration by parts formula will be useful in the remainder of the article (see [20, Lemma 2.2], [19, Proposition 13], [4, Lemma 3.3]). Lemma 3.1. Let u ∈ Hs 0((0, 1)) be such that (−d2 x)su ∈ L2(0, 1). Then for every v ∈ Hs(R) we have c1,s 2 ∫ R ∫ R (u(x)− u(y))(v(x)− v(y)) |x− y|1+2s dx dy = ∫ 1 0 v(x)(−d2 x)su(x) dx+ ∫ R\(0,1) v(x)Nsu(x) dx . (3.3) The following unique continuation property which has been proved in [19, The- orem 16], is one of the main tool in this work. 6 C. LOUIS-ROSE EJDE-2020/26 Lemma 3.2. Let ω ⊂ [R \ (0, 1)] an arbitrary non-empty open set and λ > 0 be a real number. If ϕ ∈ D((−d2 x)sD) satisfies (−d2 x)sD = λϕ in (0, 1) and Nsϕ = 0 in ω then ϕ = 0 in R. 4. Well-posedness results This section is devoted to the proof of the well-posedness of system (1.1) (resp. (1.2)) and its dual system (2.1). Also an explicit representation of their solution is given. Let (ψn)n∈N be the sequence of eigenfunctions of the fractional Laplacian operator associated with the eigenvalues (λn)n∈N. 4.1. Well-posedness of systems (1.1) and (1.2). We begin with system (1.1). First of all, we consider the following elliptic Dirichlet problem for the fractional Laplacian (−d2 x)sα = aα+ bβ in (0, 1), (−d2 x)sβ = cα+ dβ in (0, 1), α = g, β = 0 in R \ (0, 1). (4.1) Definition 4.1. Let g ∈ Hs(R \ (0, 1)), a, b, c, d ∈ R and let G ∈ Hs(R) be such that G|R\(0,1) = g. A function (α, β) is said to be a weak solution of (4.1) if (α−G, β) ∈ ( Hs 0((0, 1)) )2 and for every (ϕ, σ) ∈ ( Hs 0((0, 1)) )2 it holds c1,s 2 ∫ R ∫ R (U(x)− U(y)) · (V (x)− V (y)) |x− y|1+2s dx dy = ∫ 1 0 AU · V dx , (4.2) where U = ( α β ) , V = ( ϕ σ ) , and A = ( a b c d ) . We prove the existence and the uniqueness of a solution of (4.1) in the following proposition. Proposition 4.2. Let g ∈ Hs(R \ (0, 1)). Assume that the real numbers a, b, c, d satisfy c = b, b > 0, max{a, d} 6 −b, b2 < (λ1 − a)(λ1 − d). (4.3) Then there exists a unique solution (α, β) in (Hs(R)) 2 of (4.1) in the sense of Definition 4.1, given by α(x) = − ∑ n∈N 1 λn − a− bc λn−d (g,Nsψn)L2(R\(0,1))ψn(x), (4.4) β(x) = − ∑ n∈N c λn − d 1 λn − a− bc λn−d (g,Nsψn)L2(R\(0,1))ψn(x). (4.5) Moreover there is a constant C > 0 such that ‖(α, β)‖(Hs(R))2 6 C‖g‖Hs(R\(0,1)). (4.6) Remark 4.3. Note that the assumption max{a, d} 6 −b with b > 0, implies that max{a, d} < λ1 since λ1 > 0. EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 7 Proof. We prove the proposition in two steps. Step 1. We show that there is a unique weak solution of (4.1) in ( Hs(R) )2 . Let us consider the bilinear form defined for every U, V ∈ ( Hs(R) )2 by F(U, V ) = c1,s 2 ∫ R ∫ R (U(x)− U(y)) · (V (x)− V (y)) |x− y|1+2s dx dy − ∫ 1 0 AU · V dx. We have that F is symmetric since the matrix A is symmetric in view of the assumption (4.3). Moreover using the Cauchy-Schwarz inequality, we can write |F(U, V )| 6 c1,s 2 ∫ R ∫ R |(U(x)− U(y)) · (V (x)− V (y))| |x− y| 12 +s|x− y| 12 +s dx dy + ∫ 1 0 |AU · V | dx 6 c1,s 2 (∫ R ∫ R (U(x)− U(y)) · (U(x)− U(y)) |x− y|1+2s dx dy )1/2 × (∫ R ∫ R (V (x)− V (y)) · (V (x)− V (y)) |x− y|1+2s dx dy )1/2 + (a2 + b2 + c2 + d2)1/2 (∫ 1 0 U · U dx )1/2(∫ 1 0 V · V dx )1/2 6 max {c1,s 2 , (a2 + 2b2 + d2)1/2 } ‖U‖(Hs(R))2‖V ‖(Hs(R))2 , then the bilinear form F is continuous. In addition, we have F(U,U) = c1,s 2 ∫ R ∫ R (U(x)− U(y)) · (U(x)− U(y)) |x− y|1+2s dx dy − ∫ 1 0 AU · U dx > c1,s 2 ∫ R ∫ R (U(x)− U(y)) · (U(x)− U(y)) |x− y|1+2s dx dy + min{−(a+ b),−(d+ b)} ∫ 1 0 U · U dx > 0 (4.7) in view of assumption (4.3), and F is coercive. We conclude using the Lax-Milgram Theorem that there exists a unique weak solution U ∈ ( Hs(R) )2 such that F(U, V ) = 0 (4.8) for every V ∈ ( Hs(R) )2 . Then we show the inequality (4.6). Taking V = U in (4.8) and using (3.3), we obtain c1,s 2 ∫ R ∫ R (U(x)− U(y)) · (U(x)− U(y)) |x− y|1+2s dx dy + ∫ R\(0,1) ( g 0 ) · NsU dx− ∫ 1 0 AU · U dx = 0. It follows from (4.7) that min {c1,s 2 ,−(a+ b),−(d+ b) } ‖U‖2(Hs(R))2 6 − ∫ R\(0,1) ( g 0 ) · NsU dx 6 C‖g‖L2(R\(0,1))‖U‖(Hs(R))2 . In the last inequality, we have used the fact that the operator Ns : Hs(R) → L2(R \ (0, 1)) is bounded. So we have shown (4.6). 8 C. LOUIS-ROSE EJDE-2020/26 Step 2. We give the series solution of (4.1), that will be useful in the sequel. Let αn := (α,ψn)L2(0,1), βn := (β, ψn)L2(0,1). (4.9) Let also g ∈ Hs(R \ (0, 1)). Multiplying (4.1)1 and (4.1)2 by ψn then integrating by parts over (0, 1) while using (3.3), we obtain∫ 1 0 α(x)(−d2 x)sψn(x) dx+ ∫ R\(0,1) α(x)Nsψn(x) dx = a ∫ 1 0 ψn(x)α(x) dx+ b ∫ 1 0 ψn(x)β(x) dx,∫ 1 0 β(x)(−d2 x)sψn(x) dx = c ∫ 1 0 ψn(x)α(x) dx+ d ∫ 1 0 ψn(x)β(x)dx. Then we find that (λn − a)αn − bβn = −(g,Nsψn)L2(R\(0,1)) (λn − d)βn − cαn = 0 (4.10) which has a solution αn = − 1 λn − a− bc λn−d (g,Nsψn)L2(R\(0,1)), (4.11) βn = − c λn − d 1 λn − a− bc λn−d (g,Nsψn)L2(R\(0,1)). (4.12) Finally we find (4.4) and (4.5) using (4.9). If g is replaced by ∂mt g := ∂mg ∂tm , m ∈ N, in (4.1), then (∂mt α, ∂ m t β) is the unique associated solution. It follows that (α, β) ∈( C∞([0, T ];L2(0, 1)) )2 . � We adopt the following notation: the scalar product (u0, ψn)L2(0,1) is denoted by u0 n. Then we have the following existence result. Theorem 4.4. Assume that the real numbers a, b, c, d satisfy c = b, b > 0, max{a, d} 6 −b, b2 < (λ1 − a)(λ1 − d). (4.13) For each u0 ∈ L2(0, 1) and g ∈ D([R \ (0, 1)] × (0, T )), system (1.1) possesses a unique solution (u, v) in ( C([0, T ];L2(0, 1)) )2 given by u(x, t) = ∑ n∈N u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N (∫ t 0 ( g(·, τ),Nsψn ) L2(R\(0,1)) × e−(λn−a− bc λn−d )(t−τ)dτ ) ψn(x), (4.14) v(x, t) = ∑ n∈N c λn − d u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N c λn − d (∫ t 0 ( g(·, τ),Nsψn ) L2(R\(0,1)) × e−(λn−a− bc λn−d )(t−τ)dτ ) ψn(x). (4.15) EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 9 Proof. Let g ∈ D([R \ (0, 1)]× (0, T )). We prove the theorem in 3 steps. Step 1. We prove (4.14) and (4.15). Assume that (u, v) is a solution of (1.1). Setting un(t) := (u(·, t), ψn)L2(0,1), vn(t) := (v(·, t), ψn)L2(0,1), (4.16) we can write u(x, t) = ∑ n∈N un(t)ψn(x), v(x, t) = ∑ n∈N vn(t)ψn(x). (4.17) Multiplying (1.1)1 and (1.1)2 by ψn and integrating by parts over (0, 1), we obtain∫ 1 0 ψn(x)(∂tu(x, t) + (−d2 x)su(x, t)) dx = ∫ 1 0 ψn(x)(au(x, t) + bv(x, t)) dx,∫ 1 0 ψn(x)(−d2 x)sv(x, t) dx = ∫ 1 0 ψn(x)(cu(x, t) + dv(x, t)) dx. Using the integration by parts formula (3.3), we obtain ∂ ∂t ∫ 1 0 ψn(x)u(x, t) dx+ ∫ 1 0 u(x, t)(−d2 x)sψn(x) dx+ ∫ R\(0,1) u(x, t)Nsψn(x) dx = a ∫ 1 0 ψn(x)u(x, t) dx+ b ∫ 1 0 ψn(x)v(x, t) dx,∫ 1 0 v(x, t)(−d2 x)sψn(x) dx = c ∫ 1 0 ψn(x)u(x, t) dx+ d ∫ 1 0 ψn(x)v(x, t) dx. Hence using the notation (4.16) it follows that u′n(t) + (λn − a)un(t)− bvn(t) = − ∫ R\(0,1) g(·, t)Nsψn dx, (4.18) (λn − d)vn(t)− cun(t) = 0 (4.19) un(0) = u0 n. Then (4.19) gives vn(t) = c λn − d un(t). (4.20) Replacing it in (4.18), we find that un(t) is solution of the Cauchy problem u′n(t) + ( λn − a− bc λn − d ) un(t) = − ∫ R\(0,1) g(·, t)Nsψn dx un(0) = u0 n. (4.21) Now we show that the solution of (4.21) is un(t) = un0 e −(λn−a− bc λn−d )t + ∫ t 0 e−(λn−a− bc λn−d )(t−τ) ( g(·, τ),Nsψn ) L2(R\(0,1)) dτ. Inserting this in (4.20) yields vn(t) = c λn − d un0 e −(λn−a− bc λn−d )t + c λn − d ∫ t 0 e−(λn−a− bc λn−d )(t−τ)(g(·, τ),Nsψn)L2(R\(0,1)) dτ. (4.22) Finally we obtain the explicit forms of u and v using (4.17). 10 C. LOUIS-ROSE EJDE-2020/26 Step 2. We prove the existence and the uniqueness solution of an intermediary problem. Let (α, β) be the solution of the Dirichlet problem (4.1). Let also (µ, ν) be the solution of ∂tµ+ (−d2 x)sµ = −αt + aµ+ bν in (0, 1)× (0, T ), (−d2 x)sν = cµ+ dν in (0, 1)× (0, T ), µ = ν = 0 in [R \ (0, 1)]× (0, T ), µ(x, 0) = u0(x) in (0, 1). (4.23) Note that αt ∈ C∞([0, T ];Hs(R)) depends on (x, t). We see that (α, β) + (µ, ν) is solution of (1.1). Thus, using (4.14) and (4.15), the unique weak solution of (4.23) is µ(x, t) = −α(x, t) + ∑ n∈N u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N (∫ t 0 ( g(·, τ),Nsψn ) L2(R\(0,1)) e−(λn−a− bc λn−d )(t−τ)dτ ) ψn(x) (4.24) ν(x, t) = −β(x, t) + ∑ n∈N c λn − d u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N c λn − d (∫ t 0 (g(·, τ),Nsψn)L2(R\(0,1)) × e−(λn−a− bc λn−d )(t−τ)dτ ) ψn(x). (4.25) From (4.11) we obtain ∫ t 0 (g(·, τ),Nsψn)L2(R\(0,1))e −(λn−a− bc λn−d )(t−τ)dτ = − ∫ t 0 ( λn − a− bc λn − d ) (α(·, τ), ψn)L2(0,1)e −(λn−a− bc λn−d )(t−τ)dτ = −(α(x, t), ψn)L2(0,1) + ∫ t 0 (ατ (·, τ), ψn)L2(0,1)e −(λn−a− bc λn−d )(t−τ)dτ. (4.26) We obtained the last equality integrating by parts over [0, t]. Substituting (4.26) in (4.24) yields µ(x, t) = −2α(x, t) + ∑ n∈N u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N (∫ t 0 (ατ (·, τ), ψn)L2(0,1)e −(λn−a− bc λn−d )(t−τ)dτ ) ψn(x). (4.27) EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 11 Now using (4.12) we obtain∫ t 0 (g(·, τ),Nsψn)L2(R\(0,1))e −(λn−a− bc λn−d )(t−τ)dτ = −λn − d c ∫ t 0 ( λn − a− bc λn − d ) (β(·, τ), ψn)L2(0,1) × e−(λn−a− bc λn−d )(t−τ)dτ = λn − d c ( − (β(x, t), ψn)L2(0,1) + ∫ t 0 (βτ (·, τ), ψn)L2(0,1) × e−(λn−a− bc λn−d )(t−τ)dτ ) . (4.28) Substituting (4.28) in (4.25) gives ν(x, t) = −2β(x, t) + ∑ n∈N c λn − d u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N (∫ t 0 (βτ (·, τ), ψn)L2(0,1)e −(λn−a− bc λn−d )(t−τ)dτ ) ψn(x). (4.29) Step 3. We prove that (4.14) and (4.15) are convergent in C([0, T ];L2(0, 1)). We first observe that there exists a constant C > 0 such that n∑ k=1 |u0 k|2e −2(λk−a− bc λk−d )t 6 C n∑ k=1 |u0 k|2. (4.30) For the second part of the series in (4.14), we get in view of (4.26) that it is sufficient to show the convergence of the series∑ n∈N (∫ t 0 (ατ (·, τ), ψn)L2(0,1)e −(λn−a− bc λn−d )(t−τ)dτ ) ψn(x) in C([0, T ];L2(0, 1)). For any t ∈ [0, T ], we have that there exists a constant C > 0 such that n∑ k=1 (∫ t 0 (ατ (·, τ), ψk)L2(0,1)e −(λk−a− bc λk−d )(t−τ) dτ )2 6 t2 n∑ k=1 (∫ t 0 |(ατ (·, τ), ψk)L2(0,1)|2e −2(λk−a− bc λk−d )(t−τ) dτ ) 6 t2 ∫ t 0 ( n∑ k=1 |(ατ (·, τ), ψk)L2(0,1)|2e −2(λk−a− bc λk−d )(t−τ) ) dτ 6 Ct2‖αt‖2L∞(0,T ;L2(0,1)) ∫ t 0 dτ 6 Ct3‖αt‖2L∞(0,T ;Hs(R)) 6 Ct3‖gt‖2L∞(0,T ;L2(R\(0,1))), (4.31) where we used (4.6). We deduce that the series in u is uniformly convergent in L2(0, 1) in [0, T ]. In addition, using (4.30) and (4.31), we deduce that there exists 12 C. LOUIS-ROSE EJDE-2020/26 a constant C > 0 such that for any t ∈ [0, T ], ‖u(·, t)‖2L2(Ω) 6 C ( ‖u0‖2L2(0,1) + t3‖gt‖2L∞(0,T ;L2(R\(0,1))) ) . We use the following inequality to prove the convergence of series (4.15): ‖v(·, t)‖2L2(0,1) 6 ∣∣ c λ1 − d ∣∣2‖u(·, t)‖2L2(0,1). (4.32) We deduce that the series in v is also uniformly convergent in L2(0, 1) in [0, T ]. We conclude that (u, v) ∈ ( C([0, T ];L2(0, 1)) )2 . Then the inequality ‖v(·, t)‖2L2(Ω) 6 C ( ‖u0‖2L2(0,1) + t3‖gt‖2L∞(0,T ;L2(R\(0,1))) ) is a consequence of (4.32). We also have that the solution (µ, ν) of (4.23) given by (u, v)− (α, β), is in ( C([0, T ];L2(0, 1)) )2 . The proof is complete.. � For system (1.2) we have the following existence result. Theorem 4.5. Assume that the real numbers a, b, c, d satisfy c = b, b > 0, max{a, d} 6 −b, b2 < (λ1 − a)(λ1 − d). (4.33) For each u0 ∈ L2(0, 1) and h ∈ D([R \ (0, 1)] × (0, T )), system (1.2) possesses a unique solution (u, v) in ( C([0, T ];L2(0, 1)) )2 , given by u(x, t) = ∑ n∈N u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N (∫ t 0 (h(·, τ),Nsψn)L2(R\(0,1))e −(λn−a− bc λn−d )(t−τ)dτ ) ψn(x), v(x, t) = ∑ n∈N c λn − d u0 ne −(λn−a− bc λn−d )tψn(x) + ∑ n∈N c λn − d (∫ t 0 (h(·, τ),Nsψn)L2(R\(0,1))e −(λn−a− bc λn−d )(t−τ)dτ ) ψn(x). Proof. We consider the elliptic Dirichlet problem (−d2 x)sα = aα+ bβ in (0, 1), (−d2 x)sβ = cα+ dβ in (0, 1), α = 0, β = h in R \ (0, 1), (4.34) and prove this theorem as above. � 4.2. Well-posedness of the dual system. Now we prove existence and regularity for the dual system (2.1) associated with (1.1) (resp. (1.2)). Theorem 4.6. Assume that the real numbers a, b, c, d satisfy c = b, b > 0, max{a, d} 6 −b, b2 < (λ1 − a)(λ1 − d). (4.35) EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 13 Let ϕT ∈ L2(0, 1), then system (2.1) has a unique solution (ϕ, σ) in (C([0, T ]; L2(0, 1)))2 given by ϕ(x, t) = ∑ n∈N ϕTne −(λn−a− bc λn−d )(T−t)ψn(x), σ(x, t) = ∑ n∈N ϕTn b λn − d e−(λn−a− bc λn−d )(T−t)ψn(x). (4.36) Moreover there is a constant C > 0 such that for all t ∈ [0, T ], ‖(ϕ, σ)(x, t)‖(L2(0,1))2 6 C‖ϕT (x)‖L2(0,1), (4.37) and for any t ∈ [0, T ), there exist Nsϕ(·, t) and Nsσ(·, t) in L2(R \ (0, 1)), given by Nsϕ(x, t) = ∑ n∈N ϕTne −(λn−a− bc λn−d )(T−t)Nsψn(x), Nsσ(x, t) = ∑ n∈N ϕTn b λn − d e−(λn−a− bc λn−d )(T−t)Nsψn(x). (4.38) Proof. Let ϕn(t) := (ϕ(·, t), ψn)L2(0,1), σn(t) := (σ(·, t), ψn)L2(0,1), (4.39) and let ϕ(x, t) = ∑ n∈N ϕn(t)ψn(x), σ(x, t) = ∑ n∈N σn(t)ψn(x). (4.40) Multiplying (2.1) by ψn and integrating by parts over (0, 1), and using (3.3) we obtain − ∂ ∂t ∫ 1 0 ψn(x)ϕ(x, t) dx+ ∫ 1 0 ϕ(x, t)(−d2 x)sψn(x) dx = a ∫ 1 0 ψn(x)ϕ(x, t) dx+ c ∫ 1 0 ψn(x)σ(x, t) dx,∫ 1 0 σ(x, t)(−d2 x)sψn(x) dx = b ∫ 1 0 ψn(x)ϕ(x, t) dx+ d ∫ 1 0 ψn(x)σ(x, t) dx. Hence using (4.39) it follows that (ϕn(t), σn(t)) is solution of the eigenvalues prob- lem −ϕ′n(t) + λnϕn(t) = aϕn(t) + cσn(t), λnσn(t) = bϕn(t) + dσn(t), ϕn(T ) = ϕTn , (4.41) where ϕTn = (ϕT , ψn)L2(0,1). Then system (4.41) can be rewritten in the form −ϕ′n(t) + ( λn − a− bc λn − d ) ϕn(t) = 0, σn(t) = b λn − d ϕn(t), ϕn(T ) = ϕTn . (4.42) We show that ϕn(t) is solution of (4.42)1 with ϕn(t) = ϕTne −(λn−a− bc λn−d )(T−t), (4.43) 14 C. LOUIS-ROSE EJDE-2020/26 and using this in (4.42)2, we have σn(t) = ϕTn b λn − d e−(λn−a− bc λn−d )(T−t). (4.44) Using (4.43) and (4.44) in (4.40) gives the series solution of (2.1). The estimation (4.37) follows from the fact that there exists a constant C > 0 such that for all t ∈ [0, T ], n∑ k=1 |ϕTk |2e −2(λk−a− bc λk−d )(T−t) 6 C n∑ k=1 |ϕTk |2 = C‖ϕT ‖2L2(0,1) and ‖σ(·, t)‖2L2(0,1) 6 ∣∣ b λ1 − d ∣∣2‖ϕ(·, t)‖2L2(0,1). The convergence of the series involved in ϕ and σ follows from the preceding es- timates. The estimation (4.38) is easy to prove. The proof of the theorem is complete. � 5. Null controllability of (1.1) and (1.2) In this section, we prove Theorem 2.3, the main result of the article. It is a consequence of the following lemma which gives a necessary and sufficient condition for system (1.1) (resp. (1.2)) to be null controllable. Lemma 5.1. The system (1.1) (resp. (1.2)) is null controllable at time T > 0 if and only if for any given u0 ∈ L2(0, 1), there exists a control g ∈ L2(0, T ;Hs(R\(0, 1))) (resp. h ∈ L2(0, T ;Hs(R \ (0, 1)))) such that the solution (ϕ, σ) of (2.1) satisfies∫ 1 0 u0(x)ϕ(x, 0) dx = ∫ ω ∫ T 0 gNsϕdx dt, (5.1)( resp. ∫ 1 0 u0(x)σ(x, 0) dx = ∫ ω ∫ T 0 hNsσ dx dt ) . (5.2) Proof. Let g ∈ L2(0, T ;Hs(R\(0, 1))). We prove only (5.1), the proof of (5.2) being similar. Let us multiply (1.1)1 and (1.1)2 respectively by ϕ and σ, where (ϕ, σ) is the solution of (2.1). Integrating over (0, 1)× (0, T ), we obtain∫ 1 0 ∫ T 0 (∂tu+ (−d2 x)su)ϕdx dt = ∫ 1 0 ∫ T 0 (au+ bv)ϕdx dt,∫ 1 0 ∫ T 0 σ(−d2 x)sv dx dt = ∫ 1 0 ∫ T 0 (cu+ dv)σ dx dt. Using the integration by parts formula (3.3), we obtain∫ 1 0 (u(x, T )ϕ(x, T )− u(x, 0)ϕ(x, 0))dx− ∫ 1 0 ∫ T 0 u∂tϕdx dt + c1,s 2 ∫ R2 ∫ T 0 (u(x)− u(y))(ϕ(x)− ϕ(y)) |x− y|1+2s dx dy dt− ∫ R\(0,1) ∫ T 0 ϕNsu dx dt = a ∫ 1 0 ∫ T 0 uϕdx dt+ b ∫ 1 0 ∫ T 0 vϕ dx dt, EJDE-2020/26 NULL CONTROLLABILITY OF PARABOLIC-ELLIPTIC SYSTEMS 15 c1,s 2 ∫ R2 ∫ T 0 (v(x)− v(y))(σ(x)− σ(y)) |x− y|1+2s dx dy dt− ∫ R\(0,1) ∫ T 0 σNsv dx dt = c ∫ 1 0 ∫ T 0 uσ dx dt+ d ∫ 1 0 ∫ T 0 vσ dx dt. Using again the integration by parts formula (3.3), we have∫ 1 0 (u(x, T )ϕ(x, T )− u(x, 0)ϕ(x, 0))dx− ∫ 1 0 ∫ T 0 u∂tϕdx dt + ∫ 1 0 ∫ T 0 u(−d2 x)sϕdx dt+ ∫ R\(0,1) ∫ T 0 (uNsϕ− ϕNsu) dx dt = a ∫ 1 0 ∫ T 0 uϕdx dt+ b ∫ 1 0 ∫ T 0 vϕ dx dt, (5.3) ∫ 1 0 ∫ T 0 v(−d2 x)sσ dx dt+ ∫ R\(0,1) ∫ T 0 (vNsσ − σNsv) dx dt = c ∫ 1 0 ∫ T 0 uσ dx dt+ d ∫ 1 0 ∫ T 0 vσ dx dt. (5.4) Then (5.3) and (5.4) reduce to∫ 1 0 (u(x, T )ϕT (x)− u0(x)ϕ(x, 0))dx+ c ∫ 1 0 ∫ T 0 uσ dx dt + ∫ ω ∫ T 0 gNsϕdx dt = b ∫ 1 0 ∫ T 0 vϕ dx dt, (5.5) b ∫ 1 0 ∫ T 0 vϕ dx dt = c ∫ 1 0 ∫ T 0 uσ dx dt. (5.6) Combining (5.5) and (5.6), it follows that∫ 1 0 (u(x, T )ϕT (x)− u0(x)ϕ(x, 0))dx+ ∫ ω ∫ T 0 gNsϕdx dt = 0. Then we can check that u(x, T ) = 0 if and only if∫ 1 0 u0(x)ϕ(x, 0) dx = ∫ ω ∫ T 0 gNsϕdx dt which completes the proof. � We are now able to prove the main result. Proof of Theorem 2.3. Let (u, v) be the unique solution of (1.1) and (ϕ, σ) the unique solution of the adjoint problem (2.1). We prove that identity (5.1) is equiv- alent to the following inequality for the adjoint system: there is a constant C > 0 such that ‖ϕ(x, 0)‖2L2(0,1) 6 C ∫ ω ∫ T 0 |Nsϕ(x, t)|2 dx dt. (5.7) 16 C. LOUIS-ROSE EJDE-2020/26 Following Theorem 4.6, the function ϕ, where (ϕ, σ) is the solution of (2.1), is given by ϕ(x, t) = ∑ n∈N ϕTne −(λn−a− bc λn−d )(T−t)ψn(x), and its nonlocal normal derivative is Nsϕ(x, t) = ∑ n∈N ϕTne −(λn−a− bc λn−d )(T−t)Nsψn(x). Then∫ ω ∫ T 0 |Nsϕ(x, t)|2 dx dt = ∫ T 0 ∥∥∑ n∈N ϕTnNsψn(x)e−(λn−a− bc λn−d )(T−t)∥∥2 L2(ω) dt. Hence to obtain (5.7) it suffices to prove the estimate∫ T 0 ∥∥∥∑ n∈N ϕTnNsψn(x)e−(λn−a− bc λn−d )(T−t) ∥∥∥2 L2(ω) dt > C ∑ n∈N ‖ϕTnNsψn(x)‖2L2(ω). (5.8) Indeed, if (5.8) holds, then∫ T 0 ∥∥∥∑ n∈N ϕTnNsψn(x)e−(λn−a− bc λn−d )(T−t) ∥∥∥2 L2(ω) dt > C ∑ n∈N |ϕTn |2‖Nsψn(x)‖2L2(ω). In view of [20, Lemma 4.2], the norm of Nsψn in L2(ω) is uniformly bounded from below by a strictly positive constant η. This results follows from the unique continuation property stated in Lemma 3.2. In addition using (4.37), the estimation ‖(ϕ, σ)(x, t)‖(L2(0,1))2 6 C‖ϕT (x)‖L2(0,1) holds. Then it follows from the preceding inequality the existence of a strictly positive constant C1 such that∫ T 0 ∥∥∥∑ n∈N ϕTnNsψn(x)e−(λn−a− bc λn−d )(T−t) ∥∥∥2 L2(ω) dt > Cη2 ∑ n∈N |ϕTn |2 = Cη2‖ϕT (x)‖2L2(0,1) > C1‖(ϕ, σ)(x, 0)‖2(L2(0,1))2 > C1‖ϕ(x, 0)‖2L2(0,1). Then if (5.8) is true, so does (5.7). We must now prove (5.8). Using the Müntz Theorem (see [16, 17]), an inequality of type (5.8) holds if and only if the series∑ n∈N 1 λn − a− bc λn−d (5.9) is convergent. This series is convergent if and only if the series ∑ n∈N 1 λn is con- vergent. The eigenvalues of the operator (−d2 x)s satisfying (3.2), the series (5.9) converges if and only if s > 1/2. We proceed similarly for the null controllability results of (1.2). Now it remains to show the third part of (2.4). Let (u, v) be a solution of (1.1). 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[22] Zuazua, E.; Finite dimensional null controllability for the semilinear heat equation. J. Maths. Pures Appl., 76 (1997), 237-264. 18 C. LOUIS-ROSE EJDE-2020/26 Carole Louis-Rose Université des Antilles, Département de Mathématiques et Informatique, laboratoire LAMIA, campus de Fouillole, 97157 Pointe-à-Pitre, Guadeloupe Email address: carole.louis-rose@univ-antilles.fr 1. Introduction 2. Main result 3. Preliminary results 4. Well-posedness results 4.1. Well-posedness of systems (??) and (??) 4.2. Well-posedness of the dual system 5. Null controllability of (??) and (??) Conclusion References