Null controllability from the exterior of a one-dimensional nonlocal

arXiv:1811.10477v1 [math.AP] 21 Nov 2018

NULL CONTROLLABILITY FROM THE EXTERIOR OF A ONE-DIMENSIONAL NONLOCAL HEAT EQUATION ´ ZAMORANO MAHAMADI WARMA AND SEBASTIAN Abstract. We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval (0, 1) associated with the fractional Laplace operator (−∂x2 )s , where 0 < s < 1. We show that there is a control function which is localized in a non-empty open set O ⊂ (R \ (0, 1)), that is, at the exterior of the interval (0, 1), such that the system is null controllable at any time T > 0 if and only if 12 < s < 1.

1. Introduction and main results In this paper we are concerned with the controllability from the exterior of the one dimensional heat equation associated with the fractional Laplace operator. More precisely, we consider the system  2 s  ∂t u + (−∂x ) u = 0 in (0, 1) × (0, T ), (1.1) u = gχO×(0,T ) in (R \ (0, 1)) × (0, T ),   u(·, 0) = u0 in (0, 1),

where u = u(x, t) is the state to be controlled, 0 < s < 1 is a real number, (−∂x2 )s denotes the fractional Laplace operator (see (2.2)) and g = g(x, t) is the exterior control function which is localized in a subset O of (R \ (0, 1)). We mention that it has been shown in [19] that boundary control (the case where the control g is localized in a subset of the boundary) does not make sense for the fractional Laplace operator. That is, for the fractional Laplacian, the classical boundary control problem must be replaced by an exterior control problem. That is, the control function must be localized outside the open set as it is formulated in (1.1). We shall show that for every u0 ∈ L2 (0, 1) and g ∈ L2 ((0, T ); H s (R \ (0, 1))), the system (1.1) has a unique weak solution u ∈ C([0, 1]; L2 (0, 1)) (see Section 3). In that case the set of reachable states is given by R(u0 , T ) = u(·, T ) : g ∈ L2 ((0, T ); H s (R \ (0, 1))) .

We shall say that the system (1.1) is null controllable at time T > 0 if 0 ∈ R(u0 , T ). The system is said to be exact controllable at T > 0 if R(u0 , T ) = L2 (0, 1). We mention that as in the classical local case (s = 1) discussed in [21, Chapter 2], we have the following situation for the nonlocal case. Since the system is linear, without restriction one may assume that u0 = 0. In that case, solutions of (1.1) (with u0 = 0) are of class C ∞ far from (R \ (0, 1)) at time t = T . This shows that the elements of R(u0 , T ) are C ∞ functions in (0, 1). Thus exact controllability may not hold. For this reason we shall study the null controllability of the system. The following theorem is the main result of the paper. 2010 Mathematics Subject Classification. 35R11, 35S11, 35K20, 93B05, 93B07. Key words and phrases. Nonlocal heat equation, exterior control, null controllability. The work of the first author is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242 and the second author is supported by the Fondecyt Postdoctoral Grant No. 3180322. 1

2

´ ZAMORANO MAHAMADI WARMA AND SEBASTIAN

Theorem 1.1. The following assertions hold. (a) Let 21 < s < 1. Then the system (1.1) is null controllable at any time T > 0 and g ∈ D(O×(0, T )), where O ⊂ (R \ (0, 1)) is an arbitrary nonempty open set. (b) If 0 < s ≤ 21 , then the system (1.1) is not null controllable at time T > 0. We mention that in the proof of Theorem 1.1, we shall heavily exploit the fact that the eigenvalues {λn }n∈N of the realization of (−∂x2 )s in L2 (0, 1) with zero exterior Dirichlet condition (see Section 2) satisfy the following asymptotics (see e.g. [13]):

λn =

nπ (2 − 2s)π − 2 8

2s

+O

1 as n → ∞. n

(1.2)

Recall that by Theorem 1.1, the system (1.1) is not null controllable at time T > 0 if 0 < s ≤ 12 . It has been recently shown in [19] that the system is indeed approximately controllable at any time T > 0 and g ∈ D(O × (0, T )). The result obtained in [19] is more general since it included the N -dimensional case and the fractional diffusion equation, that is, the case where ∂t u is replaced by the Caputo time fractional derivative of u, namely Dα t u (0 < α ≤ 1). Of course the case α = 1 corresponds to (1.1). The null controllability from the interior (that is, the case where the control function is localized in a non-empty subset ω of (0, 1)) of the one-dimensional heat equation has been recently investigated in [2] where the authors have shown that the system is null controllable if and only if 12 < s < 1. They have also shown that the associated system is indeed approximately controllable if 0 < s ≤ 21 . The interior null controllability of the Schr¨odinger and wave equations have been studied in [1]. The approximate controllability from the exterior of the super-diffusive system (that is, the case where utt is replaced by the Caputo time fractional derivative Dα t , 1 < α < 2) has been very recently considered in [14]. The case of the (possible) strong damping nonlocal wave equation has been studied in [20]. The study of nonlocal operators and nonlocal PDEs is nowadays a topic with interest to the mathematics and scientist communities due to the numerous applications that nonlocal PDEs provide. A motivation for this growing interest relies in the large number of possible applications in the modeling of several complex phenomena for which a local approach turns out to be inappropriate or limiting. Indeed, there is an ample spectrum of situations in which a nonlocal equation gives a significantly better description than a local PDE of the problem one wants to analyze. Among others, we mention applications in turbulence, anomalous transport and diffusion, elasticity, image processing, porous media flow, wave propagation in heterogeneous high contrast media. Also, it is well known that the fractional Laplacian is the generator of s-stable L´evy processes, and it is often used in stochastic models with applications, for instance, in mathematical finance. One of the main differences between nonlocal models and classical PDEs is that the fulfillment of a nonlocal equation at a point involves the values of the function far away from that point. We refer to [4, 5, 6] and their references for more applications and information on the topic. The rest of the paper is structured as follows. In Section 2 we introduce the function spaces needed to study our problem and we give some intermediate known results that are needed in the proof of our main results. In Section 3 we show the well-posedness of the system (1.1) and its associated dual system and we give the explicit representation of the solutions in terms of series for both problems. Finally, in Section 4 we give the proof of our main result.

NONLOCAL HEAT EQUATION

3

2. Preliminary results In this section we give some notations and recall some known results as they are needed in the proof of our main results. We start with fractional order Sobolev spaces. Given 0 < s < 1, we let Z 1Z 1 |u(x) − u(y)|2 s 2 dxdy < ∞ , H (0, 1) := u ∈ L (0, 1) : |x − y|1+2s 0 0 and we endow it with the norm defined by Z 1 Z kukH s (0,1) := |u(x)|2 dx + 0

0

We set

1

Z

1 0

|u(x) − u(y)|2 dxdy |x − y|1+2s

12

.

n o H0s ((0, 1)) := u ∈ H s (R) : u = 0 in R \ (0, 1) .

We shall denote by H −s ((0, 1)) the dual of H0s ((0, 1)), that is, H −s ((0, 1)) = (H0s ((0, 1)))⋆ and we shall let h·, ·i−s,s denote their duality pairing. For more information on fractional order Sobolev spaces, we refer to [7, 18] and their references. Next, we give a rigorous definition of the fractional Laplace operator. Let Z |u(x)| dx < ∞ . L1s (R) := u : R → R measurable, 1+2s R (1 + |x|)

For u ∈ L1s (R) and ε > 0 we set

(−∂x2 )sε u(x)

:= Cs

Z

{y∈R: |x−y|>ε}

u(x) − u(y) dy, x ∈ R, |x − y|1+2s

where Cs is a normalization constant given by Cs :=

s22s Γ 1

2s+1 2

π 2 Γ(1 − s)

.

The fractional Laplacian (−∂x2 )s is defined by the following singular integral: Z u(x) − u(y) 2 s dy = lim(−∂x2 )sε u(x), x ∈ R, (−∂x ) u(x) := Cs P.V. 1+2s ε↓0 R |x − y|

(2.1)

(2.2)

provided that the limit exists. We notice that L1s (R) is the right space for which v := (−∂x2 )sε u exists for every ε > 0, v being also continuous at the continuity points of u. For more details on the fractional Laplace operator we refer to [6, 7, 10, 18] and their references. Next, we consider the realization of (−∂x2 )s in L2 (0, 1) with the condition u = 0 in R \ (0, 1). More precisely, we consider the closed and bilinear form Z Z Cs (u(x) − u(y))(v(x) − v(y)) F(u, v) := dxdy, u, v ∈ H0s ((0, 1)). 2 R R |x − y|1+2s Let (−∂x2 )sD be the selfadjoint operator in L2 (0, 1) associated with F in the sense that n o ( D((−∂x2 )sD ) = u ∈ H0s ((0, 1)), ∃ f ∈ L2 (0, 1), F(u, v) = (f, v)L2 (0,1) ∀ v ∈ H0s ((0, 1)) , (−∂x2 )sD u = f.

More precisely, we have that o n D((−∂x2 )sD ) = u ∈ H0s ((0, 1)), (−∂x2 )s u ∈ L2 (0, 1) , (−∂x2 )sD u = (−∂x2 )s u.

(2.3)


4

Then (−∂x2 )sD is the realization of (−∂x2 )s in L2 (0, 1) with the condition u = 0 in R \ (0, 1). It has been shown in [17] that (−∂x2 )sD has a compact resolvent and its eigenvalues form a non-decreasing sequence of real numbers 0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · satisfying limn→∞ λn = ∞. In addition, the eigenvalues are of finite multiplicity. Let (ϕn )n∈N be the orthonormal basis of eigenfunctions associated with the eigenvalues (λn )n∈N . Then ϕn ∈ D((−∂x2 )sD ) for every n ∈ N, (ϕn )n∈N is total in L2 (0, 1) and satisfies ( (−∂x2 )s ϕn = λn ϕn in (0, 1), (2.4) ϕn = 0 in R \ (0, 1). Next, for u ∈ H s (R) we introduce the nonlocal normal derivative Ns given by Z 1 u(x) − u(y) dy, x ∈ R \ (0, 1), Ns u(x) := Cs 1+2s 0 |x − y|

(2.5)

where Cs is the constant given in (2.1). We notice that since equality is to be understood a.e., then (2.5) is the same as for a.e. x ∈ R \ (0, 1). By [11, Lemma 3.2], for every u ∈ H s (R), we have that Ns u ∈ L2 (R \ (0, 1)). The following unique continuation property which shall play an important role in the proof of Theorem 1.1 has been recently obtained in [19, Theorem 3.10]. Lemma 2.1. Let λ > 0 be a real number and O ⊂ (R \ (0, 1)) an arbitrary non-empty open set. If ϕ ∈ D((−∂x2 )sD ) satisfies (−∂x2 )sD ϕ = λϕ in (0, 1) and Ns ϕ = 0 in O, then ϕ = 0 in R. For more details on the Dirichlet problem associated with the fractional Laplace operator we refer the interested reader to [3, 12, 15, 16, 19] and their references. We conclude this section with the following integration by parts formula. Lemma 2.2. Let u ∈ H0s ((0, 1)) be such that (−∂x2 )s ∈ L2 (0, 1). Then for every v ∈ H s (R) the following identity Cs 2

Z Z R

R

(u(x) − u(y))(v(x) − v(y)) dxdy = |x − y|1+2s

Z

0

1

v(x)(−∂x2 )s u(x) dx +

Z

v(x)Ns u(x) dx,

(2.6)

R\(0,1)

holds. We refer to [8, Lemma 3.3] (see also [19, Proposition 3.7] or [20, Remark 3.6]) for the proof and for more details. 3. Well-posedness of the parabolic problem This section is devoted to the well posedness and the explicit representation in terms of series for solutions to the system (1.1) and its associated dual system. The series representation of solutions shall play a crucial role in the proof of our main result. Throughout the remainder of the article, (ϕn )n∈N denotes the orthonormal basis of eigenfunctions of the operator (−∂x2 )sD associated with the eigenvalues (λn )n∈N . Furthermore, for a given u ∈ L2 (0, 1) and n ∈ N, we shall let un := (u, ϕn )L2 (0,1) and for a given set E ⊆ R, we shall denote by (·, ·)L2 (E) the scalar product in L2 (E).


5

3.1. Representation of solution to the system (1.1). We have the following existence and explicit representation of solutions in terms of series. Theorem 3.1. For every u0 ∈ L2 (0, 1) and g ∈ D((R \ (0, 1)) × (0, T )), the system (1.1) has a unique weak solution u ∈ C([0, T ]; L2 (0, 1)) given by ∞ ∞ Z t X X −λn ((t−τ ) −λn t (g(·, τ ), Ns ϕn )L2 (R\(0,1)) e dτ ϕn (x). (3.1) u(x, t) = u0,n e ϕn (x) + n=1

n=1

0

Proof. Let g ∈ D((R \ (0, 1)) × (0, T )) and let v be the unique weak solution of the Dirichlet problem ( (−∂x2 )s v = 0 in (0, 1), (3.2) v=g in R \ (0, 1). It has been shown in [19] that the Dirichlet problem (3.2) has a unique weak solution v ∈ H s (R) and there is a constant C > 0 such that kvkH s (R) ≤ CkgkH s ((R\(0,1))) .

(3.3) ∂tm g,

Since g depends on (x, t), then v also depends on (x, t). If in (3.2) one replaces g by m ∈ N0 , then the associated unique solution is given by ∂tm v for every m ∈ N0 . From this, we can deduce that v ∈ C ∞ ([0, T ]; H s (RN )). Now let u be a solution of (1.1) and set w := u − v. Then a simple calculation gives wt + (−∂x2 )s w = ut − vt + (−∂x2 )s u − (−∂x2 )s v = −vt in Ω × (0, T ). In addition w = u − v = g − g = 0 in (R \ (0, 1)) × (0, T ) and w(·, 0) = u(·, 0) − v(·, 0) = u0 − v(·, 0)

in Ω.

Since g ∈ D((R \ (0, 1)) × (0, T )), we have that v(·, 0) = 0 in Ω. We have shown that a solution u of (1.1) can be decomposed as u = v + w, where w solves the system  2 s  in Ω × (0, T ), wt + (−∂x ) w = −vt (3.4) w=0 in (R \ (0, 1)) × (0, T ),   w(·, 0) = u0 in Ω. ∞ s We notice that vt ∈ C ([0, T ]; H (R)). The problem (3.4) can be rewritten as the following Cauchy problem ( wt + (−∂x2 )sD w = −vt in (0, 1) × (0, T ), (3.5) w(·, 0) = u0 in (0, 1). It follows from semigroup theory that the Cauchy problem (3.5) (hence, the system (3.4)) has a unique weak solution w ∈ C([0, T ], L2 (Ω) ∩ C ∞ ((0, T ); L2 (Ω)) given by ∞ ∞ Z t X X (vτ (·, τ ), ϕn )L2 ((0,1) e−λn (t−τ ) dτ ϕn (x). (3.6) w(x, t) = u0,n e−λn t ϕn (x) − n=1

n=1

0

Integrating (3.6) by part, we obtain that ∞ ∞ Z t X X −λn (t−τ ) −λn t (v(·, τ ), λn ϕn )L2 ((0,1) e dτ ϕn (x). w(x, t) = u0,n e ϕn (x) − v(x, t) + n=1

n=1

0

(3.7)


6

Using the integration by parts formula (2.6), we get that (v(·, τ ), λn ϕn )L2 ((0,1) = v(·, τ ), (−∂x2 )sD ϕn L2 ((0,1) = − (g, Ns ϕn )L2 (R\(0,1)) .

Substituting this identity into (3.7) gives w(x, t) = −v(x, t) +

∞ X

u0,n e

−λn t

ϕn (x) +

n=1

∞ Z X

n=1

0

t

(g(·, τ ), Ns ϕn )L2 (R\(0,1) e

−λn (t−τ )

dτ

ϕn (x),

and we have shown (3.1). It is straightforward to verify that the series in (3.1) converges in L2 (0, 1) uniformly in [0, T ]. In addition, using (3.6) and (3.3), we get that there is a constant C > 0 such that for every t ∈ [0, T ], ku(·, t)k2L2 (0,1) ≤C ku0 k2L2 (0,1) + t2 kvt kL∞ ((0,T );L2 (0,1)) ≤C ku0 k2L2 (0,1) + t2 kvt kL∞ ((0,T );H s (R)) ≤C ku0 k2L2 (0,1) + t2 kgt kL∞ ((0,T );L2 (R\(0,1))) . We have shown that u ∈ C([0, T ]; L2 (0, 1)) and the proof is finished.

3.2. Representation of solution to the dual system. we have that the following backward system  2 s  in −∂t ψ + (−∂x ) ψ = 0 ψ=0 in   ψ(·, T ) = ψ0 in

Using the classical integration by parts formula, (0, 1) × (0, T ), (R \ (0, 1)) × (0, T ), Ω,

(3.8)

can be view as the dual system associated with (1.1). We have the following existence result.

Theorem 3.2. Let ψ0 ∈ L2 (0, 1). Then (3.8) has a unique weak solution ψ ∈ C([0, T ]; L2 (0, 1)) which is given by ψ(x, t) =

∞ X

ψ0,n e−λn (T −t) ϕn (x).

(3.9)

n=1

In addition the following assertions hold.

(a) There is a constant C > 0 such that for all t ∈ [0, T ], kψ(·, t)kL2 (0,1) ≤ Ckψ0 kL2 (0,1) .

(3.10)

(b) For every t ∈ [0, T ) fixed, Ns ψ(·, t) exists, belongs to L2 (R \ (0, 1)) and is given by Ns ψ(x, t) =

∞ X

ψ0,n e−λn (T −t) Ns ϕn (x).

(3.11)

n=1

Proof. Using the spectral theorem for selfadjoint operators with compact resolvent we are reduced to look for a solution ψ of the form ψ(x, t) =

∞ X

(ψ(·, t), ϕn )L2 (0,1) ϕn (x).

n=1


7

Replacing this expression in (3.8) and letting ψn (t) := (ψ(·, t), ϕn )L2 (0,1) , we get that ψn (t) solve the following ODE −ψn′ (t) + λn ψn (t) = 0 and ψn (T ) = ψ0,n . It is straightforward to show that ψ is give by (3.9). The estimate (3.10) and the identity (3.11) can be also easily justified. 4. Proof of the main result In this section we give the proof of the main result of this work, namely Theorem 1.1. For this purpose, we need first to establish some auxiliaries results that will be used in the proof. Lemma 4.1. The system (1.1) is null controllable in time T > 0 if and only if for each initial data u0 ∈ L2 (0, 1), there exists a control function g ∈ L2 ((0, T ); H s (R \ (0, 1))) such that the solution ψ of the dual system (3.8) satisfies Z

1

u0 (x)ψ(x, 0) dx =

0

for each ψ0 ∈ L2 (0, 1).

Z

T 0

Z

g(x, t)Ns ψ(x, t)dxdt,

(4.1)

O

Proof. Let g ∈ L2 (0, T ; H s (R \ (0, 1))). Multiplying (1.1) by the solution ψ of (3.8) and integrating by parts by using the integration by parts formula (2.6) we obtain that Z TZ 1 ut + (−∂x2 )s u ψ dxdt 0= 0

=

Z

0

1

(u(x, T )ψ(x, T ) − u(x, 0)ψ(x, 0)) dx −

0

+ Z

T

0

T

Z

0

=

Z

Z

0

1

1

Z

u(−∂x2 )s ψ dxdt +

T 0

Z

0

1

u(x, 0)ψ(x, 0) dx −

Z

0

uψt dxdt

0

(u(x, t)Ns ψ(x, t) − ψ(x, t)Ns u(x, t)) dxdt Z

0

This implies that Z

1

R\(0,1)

(u(x, T )ψ(x, T ) − u(x, 0)ψ(x, 0)) dx +

0

Z

T

Z

u(x, t)Ns ψ(x, t) dxdt.

R\(0,1)

1

u(x, T )ψ(x, T ) dx =

Z

0

T

Z

uNs ψ dxdt.

(4.2)

R\(0,1)

R1 Now if (4.1) holds, it follows from (4.2) that 0 u(x, T )ψ(x, T ) dx = 0 for every ψ0 ∈ L2 (0, 1). Thus u(·, T ) = 0 in (0, 1) and the system (1.1) is null controllable. Conversely, if the system (1.1) is null controllable, that is, u(·, T ) = 0 in (0, 1), then (4.1) follows from (4.2) and the proof is finished. Finally, for the proof of Theorem 1.1 we also need the following fact. Lemma 4.2. Let {ϕn }n∈N be the orthogonal basis of normalized eigenfunctions associate with the eigenvalues {λn }n∈N . Then, there exists a scalar η > 0 such that for every n ∈ N, and for every non-empty open set O ⊂ R \ (0, 1) the functions Ns ϕn are uniformly bounded from below by η in L2 (O). Namely, ∃η > 0, ∀n ∈ N, kNs ϕn kL2 (O) ≥ η.

(4.3)


8

Proof. The proof is a consequence of the unique continuation property for the fractional Laplacian operator given in Lemma 2.1. Indeed, assume that for every η > 0 there exists O ⊂ R \ (0, 1) and k ∈ N such that kNs ϕk kL2 (O) < η.

(4.4)

Since (4.4) is valid for every η > 0, we can deduce that Ns ϕk = 0 in O. The unique continuation property implies that ϕk = 0 in R, which is a contradiction with the fact that {ϕn }n∈N is a basis for L2 (0, 1). Proof of Theorem 1.1. Let u be the unique weak solution of (1.1) and ψ the unique weak solution of the dual problem (3.8). Recall that by Lemma 4.1, the system (1.1) is null controllable if and only if the identity (4.1) holds. It is a well know result that (4.1) is equivalent to the following observability inequality for the dual system: there exists a constant C > 0 such that Z TZ 2 |Ns ψ(x, t)dx| dt. (4.5) kψ(·, 0)k2L2 (0,1) ≤ C 0

O

From Section 3, the solution ψ of (3.8) is given by ∞ X ψ(·, t) = ψ0,n e−λn (T −t) ϕn (x),

(4.6)

n=1

and the nonlocal normal derivative is given by ∞ X ψ0,n e−λn (T −t) Ns ϕn (x). Ns ψ(·, t) =

(4.7)

n=1

Therefore, letting an := ψ0,n Ns ϕn , we obtain Z Z TZ 2 |Ns ψ(x, t)dx| dt = 0

O

T 0

2 ∞

X

an e−λn (T −t)

2 n=1

dt.

(4.8)

L (O)

Thus, to obtain the observability inequality (4.5), it is enough to prove the following estimate:

2 Z T ∞ ∞

X X

−λn (T −t) (4.9) dt ≥ C kan k2L2 (O) . an e

0 2 n=1

Indeed, if (4.9) holds, then

2 Z T ∞

X

−λn (T −t) an e

2 0 n=1

L (O)

L (O)

dt ≥ C

∞ X

n=1

n=1

kan k2L2 (O) = C

∞ X

|ψ0,n |2 kNs ϕn k2L2 (O) .

n=1

By Lemma 4.2, the norm of kNs ϕn kL2 (O) is uniformly bounded from below by η > 0. Therefore we can deduce from the preceding estimate that there is a constant C1 > 0 such that

2 Z T ∞ ∞

X

X

an e−λn (T −t) dt ≥ Cη 2 |ψ0,n |2 = Cη 2 kψ0 k2L2 (0,1) ≥ C1 kψ(·, 0)k2L2 (0,1) ,

2 0 n=1

L (O)

n=1

where in the last inequality we have used the estimate (3.10). Now we have to show that the inequality (4.9) holds. It is a well known result for parabolic equations (see e.g. [9]), that an inequality of the type (4.9) holds if and only if the series ∞ X 1 (4.10) λ n=1 n


9

is convergent. As we have mentioned in the introduction, the eigenvalues of the operator (−∂x2 )sD satisfy (1.2). The asymptotic behavior (1.2) implies that the series (4.10) converges if and only if s > 12 . The proof is finished. References [1] U. Biccari. Internal control for non-local Schr¨ odinger and wave equations involving the fractional Laplace operator. ESAIM: Control Optimization and Calculus of Variations, 2018, to appear. [2] U. Biccari and V. Hern´ andez-Santamarıa. Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects. HAL Preprint-hal-01562358, 2017. [3] U. Biccari, M. Warma, and E. Zuazua. Local elliptic regularity for the Dirichlet fractional Laplacian. Adv. Nonlinear Stud., 17(2):387–409, 2017. [4] K. Bogdan, K. Burdzy, and Z-Q. Chen. Censored stable processes. Probab. Theory Related Fields, 127(1):89–152, 2003. [5] L. A. Caffarelli, J-M. Roquejoffre, and Y. Sire. Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc., 12(5):1151–1179, 2010. [6] L. A. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007. [7] E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012. [8] S. Dipierro, X. Ros-Oton, and E. Valdinoci. Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam., 33(2):377–416, 2017. [9] H. O Fattorini and D. Russell. Exact controllability theorems for linear parabolic equations in one space dimension. Archive for Rational Mechanics and Analysis, 43(4):272–292, 1971. [10] C. G. Gal and M. Warma. Nonlocal transmission problems with fractional diffusion and boundary conditions on nonsmooth interfaces. Comm. Partial Differential Equations, 42(4):579–625, 2017. [11] T. Ghosh, M. Salo, and G. Uhlmann. The Calder´ on problem for the fractional Schr¨ odinger equation. arXiv:1609.09248. [12] G. Grubb. Fractional Laplacians on domains, a development of H¨ ormander’s theory of µ-transmission pseudodifferential operators. Adv. Math., 268:478–528, 2015. [13] M. Kwa´snicki. Eigenvalues of the fractional laplace operator in the interval. Journal of Functional Analysis, 262(5):2379–2402, 2012. [14] C. Louis-Rose and M. Warma. Approximate controllability from the exterior of space-time fractional wave equations. Applied Mathematics & Optimization, pages 1–44, 2018. [15] X. Ros-Oton and J. Serra. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9), 101(3):275–302, 2014. [16] X. Ros-Oton and J. Serra. The extremal solution for the fractional Laplacian. Calc. Var. Partial Differential Equations, 50(3-4):723–750, 2014. [17] R. Servadei and E. Valdinoci. On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A, 144(4):831–855, 2014. [18] M. Warma. The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal., 42(2):499–547, 2015. [19] M. Warma. Approximate controllabilty from the exterior of space-time fractional diffusion equations with the fractional Laplacian. arXiv:1802.08028, 2018. [20] M. Warma and S. Zamorano. Analysis of the controllability from the exterior of strong damping nonlocal wave equations. arXiv preprint arXiv:1810.08060, 2018. [21] E. Zuazua. Controllability of partial differential equations. 3` eme cycle. Castro Urdiales, Espagne, 2006. M. Warma, University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 University AVE. STE 1701 San Juan PR 00925-2537 (USA) E-mail address: [email protected], [email protected] ´ tica y Ciencia de la Computacio ´ n, S. Zamorano, Universidad de Santiago de Chile, Departamento de Matema Facultad de Ciencia, Casilla 307-Correo 2, Santiago, Chile. E-mail address: [email protected]