Controllability of the one-dimensional fractional heat equation under positivity constraints

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-\Delta)^s$ ($0<s<1$) on the interval $(-1,1)$. We prove the existence of a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the action of a positive control, when $s>1/2$. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.

The controllability properties of the fractional heat equation on open subsets of R N (N ≥ 2) are still not fully understood by the mathematical community. The classical tools (see e.g. [41] and the references therein) like the Carleman estimates usually used to study the controllability for heat equations are still not available for the fractional Laplacian (except on the whole space R N ). For this reason, our analysis in the present article is limited to the one-dimensional case. Another difficulty for analyzing the system (1.1) by using some spectral properties is that contrarily to the local case s = 1 where the eigenvalues and eigenfunctions of the system are well known, for the fractional case, we just know an asymptotic for the eigenvalues and an explicit formula for the eigenfunction is not accessible.
In the absence of constraints, the fractional heat equation (1.1) is null-controllable in any positive time T > 0, provided s > 1/2. This has been proved in [3] by using the gap condition on the eigenvalues, and it has been validated through numerical experiments. In space dimension N ≥ 2, the best possible controllability result available for the fractional heat equation is the approximate controllability recently obtained in [39].
In this work, we have obtained the following specific results: (i) Firstly, we show that, if s > 1/2, then the system (1.1) is controllable from any given initial datum in L 2 (−1, 1) to zero (and, by translation, to trajectories) in any positive time T > 0 by means of L ∞ -controls. This extends the analysis of [3], where only the classical case of L 2 -controls was considered. The proof will use the canonical approach of reducing the question of controllability with an L ∞ -control to a dual observability problem in L 1 , and the use of Fourier series expansions to obtain a new result on the L 1 -observation of linear combinations of real exponentials.
(ii) Secondly, as a consequence of our first result, we prove the existence of a minimal (strictly positive) time T min such that the fractional heat dynamics (1.1) can be controlled to positive trajectories through the action of a positive control. Moreover, if the initial datum is supposed to be positive as well, then the maximum principle guarantees the positivity of the states too.
Fractional order operators (in particular the fractional Laplace operator) have recently emerged as a modeling alternative in various branches of science. They usually describe anomalous diffusion. A number of stochastic models for explaining anomalous diffusion have been introduced in the literature. Among them we quote the fractional Brownian motion, the continuous time random walk, the Lévy flights, the Schneider gray Brownian motion, and more generally, random walk models based on evolution equations of single and distributed fractional order in space (see e.g. [11,16,26,34]). In general, a fractional diffusion operator corresponds to a diverging jump length variance in the random walk.
In many PDEs models some constraints need to be imposed when considering practical applications. This is for instance the case of diffusion processes (heat conduction, population dynamics, etc.), where realistic models have to take into account that the state represents some physical quantity which must necessarily remain positive (see, e.g., [7]).
This topic is also related to some other relevant applications, like the optimal management of compressors in gas transportation networks, requiring the preservation of severe safety constraints (see [9,27,36]).
Finally, this issue is also important in other PDE problems based on scalar conservation laws, including the Lighthill-Whitham and Richards traffic flow models ( [8,21,32]) or the isentropic compressible Euler equation ( [14]).
Most of the existing controllability theory for PDEs has been developed in the absence of constraints on the controls and/or state. To the best of our knowledge, the literature on constrained controllability is currently very limited and the majority of the available results do not guarantee that controlled trajectories fulfill the physical restrictions of the processes under consideration.
In the context of the heat equation, the problem of constrained controllability has been firstly addressed in [23] in the linear case, and it has been later extended to semi-linear models in [30]. In particular, in the mentioned references, the authors proved that the linear and semi-linear heat equations are controllable to any positive steady state or trajectory by means of non-negative boundary controls, provided the control time is long enough. Moreover, for positive initial data, the maximum principle guarantees that also the positivity of the state is preserved. On the other hand, it was also proved that controllability by non-negative controls fails if time is too short, whenever the initial datum differs from the final target.
In addition to the results for heat-like equations, constrained controllability has been also analyzed in the context of population dynamics. In more detail, in [17,24] it has been shown that the controllability of Lotka-McKendrick type systems with age structuring can be obtained by preserving the positivity of the state, once again in a long enough time horizon. These results have been recently extended in [25] to general infinite-dimensional systems with age structure.
The study of the controllability properties under positivity constraints is a very reasonable question for scalar-valued parabolic equations, which are canonical examples where positivity is preserved for the free dynamics. Therefore, the issue of whether the system can be controlled in between two states by means of positive controls, by possibly preserving also the positivity of the controlled solution, arises naturally.
The existence of a minimal time for constrained controllability is in counter-trend with respect to the unconstrained case, in which linear and semi-linear parabolic systems are known to be controllable at any positive time. Notwithstanding, often times, norm-optimal controls allowing to reach the target at the final time are restrictions of solutions of the adjoint system. Accordingly these controls experience large oscillations in the proximity of the final time, which are enhanced when the time horizon of control is small. This eventually leads to control trajectories that go beyond the physical thresholds and fail to fulfill the positivity constraint (see [15]).
On the other hand, when the time interval is long, we expect the control property to be achieved with controls of small amplitude, thus ensuring small deformations of the state and, in particular, preserving its positivity. Roughly speaking, by imposing constraints to the control, we are somehow providing an impediment for the state to reach the target, unless the control time horizon is long enough. This behavior is then a warning that existing unconstrained controllability results, that are valid within arbitrarily short time, may be unsuitable in practical applications in which state-constraints need to be preserved along controlled trajectories.
In addition to the results for parabolic equations, analogous questions for the linear wave equation have been analyzed in [29]. There, the authors proved the controllability to steady states and trajectories through the action of a positive control, acting either in the interior or on the boundary of the domain considered. Nevertheless, in that case control and state positivity are not interlinked. Indeed, because of the lack of a maximum principle, the sign of the control does not determine the sign of the solution, whose positivity is no longer guaranteed.
The rest of the paper is organized as follows. We state the main result of the paper in Section 2. In Section 3, we start by presenting some preliminary technical results that are needed throughout the paper. Moreover, we give there the proof of the unconstrained controllability of the fractional heat equation (1.1) with L ∞ controls. Section 4 is devoted to the proof of the main result, namely, Theorem 2.1 below. The proof is divided in three parts. In Section 4.1, we prove the first part concerning the constrained controllability of (2.1). In Section 4.2, we obtain the strict positivity of the minimal controllability time T min . Section 4.3 is devoted to the proof of the controllability in minimal time by means of measure controls. In Section 5, we present some numerical simulations validating our theoretical results. Finally, in Section 6, we give some concluding remarks and propose some open problems.

Problem formulation and main result
In this section, we formulate precisely the problem we would like to investigate and we state our main results.
Let T > 0 be a real number, and define Q := (−1, 1) × (0, T ) and Q c := (−1, 1) c × (0, T ) where (−1, 1) c := R \ (−1, 1). Consider the following controllability problem for the fractional heat equation: in (−1, 1). (2.1) In (2.1), ω ⊂ (−1, 1) is the control region, u is the control function and z is the state to be controlled, while for s ∈ (0, 1), the operator (−d 2 x ) s is the fractional Laplacian, defined for any function v sufficiently smooth as the following singular integral: with c s an explicit normalization constant (see e.g. [10]). It is known (see [3]) that the fractional heat equation (2.1) is null controllable in any time T > 0 by means of a control u ∈ L 2 (ω × (0, T )), if and only if s > 1/2. In other words, given any z 0 ∈ L 2 (−1, 1) and T > 0, there exists a control function u ∈ L 2 (ω × (0, T )) such that the corresponding unique solution z of (2.1) satisfies z(x, T ) = 0 a.e. in (−1, 1). If s ≤ 1/2, instead, null-controllability cannot be achieved and the system turns out to be only approximately controllable. Besides, the equation being linear, by translation the same result holds if the final target is a trajectory z.
Moreover, it is also known (see Lemma 3.1 below) that the fractional heat equation preserves positivity. More precisely, if z 0 is a given non-negative initial datum in L 2 (−1, 1) and u is a non-negative function, then so it is for the solution z of (2.1). Hence, the following question arises naturally: Can we control the fractional heat dynamics (2.1) from any initial datum z 0 ∈ L 2 (−1, 1) to any positive trajectory z, under positivity constraints on the control and/or the state? In other words we want to analyze whether it is possible to choose a control u ≥ 0 steering the solution of (2.1) from z 0 ∈ L 2 (−1, 1) to a positive trajectory z(·, T ) = z(·, T ) > 0, while possibly maintaining this solution non-negative along the whole time interval, i.e., z(x, t) ≥ 0 for every (x, t) ∈ (−1, 1) × (0, T ).
Clearly, we are only interested in the case z(·, T ) = z 0 . Otherwise, the trajectory z ≡ z 0 = z(·, T ) trivially solves the problem.
As we will see, the answer to the above question is positive, provided that the controllability time is large enough. In particular, our main result in the present paper is the following.
This result is in the same spirit of the ones obtained in [23,30] in the context of the linear and semi-linear local heat equations under the action of a boundary control. Following the methodology presented in the mentioned references, the first ingredient for proving Theorem 2.1 is to show that, in absence of constraints, (2.1) is controllable by means of an L ∞ -control. This will be given by the following: Theorem 2.2. For any z 0 ∈ L 2 (−1, 1), s > 1/2 and T > 0, there exists a control function u ∈ L ∞ (ω ×(0, T )) such that the corresponding unique weak solution z of (2.1) with initial datum z(x, 0) = z 0 (x) satisfies z(x, T ) = 0 a.e. in (−1, 1). Moreover, there is a constant C > 0 (depending only on T ) such that By means of a classical duality argument (see [12,13,28]), Theorem 2.2 is equivalent to the following observability result for the adjoint equation associated to (2.1). Proposition 2.3. For any T > 0 and p T ∈ L 2 (−1, 1), let p ∈ L 2 ((0, T ); H s 0 (−1, 1)) ∩ C([0, T ]; L 2 (−1, 1)) with p t ∈ L 2 ((0, T ); H −s (−1, 1)) be the weak solution of the adjoint system in (−1, 1).

(2.5)
Then, for any s > 1/2, there is a constant C = C(T ) > 0 such that We refer to Section 3 for the definition of the spaces H s 0 (−1, 1) and H −s (−1, 1). We shall prove Proposition 2.3 by employing spectral techniques and with the help of the following L 1 observability result for linear combinations of real exponentials.
Theorem 2.4. Let {µ k } k≥1 be a sequence of real numbers satisfying the following conditions: Then, for any T > 0, there is a positive constant C = C(T ) > 0 such that, for any finite sequence {c k } k≥1 it holds the inequality .
(2.8) Theorem 2.4 will follow from the classical Müntz Theorem for families of real exponentials and from the results of [35].
We stress that Theorem 2.4, of independent interest on its own, is not specific to the equation (2.1) we are considering but it allows for more general results. Indeed, it yields the immediate knowledge of L ∞ -controls for one-dimensional problems simply by knowing the explicit spectrum of the equation.

Preliminary results
We present here some preliminary results which are needed for the proof of Theorem 2.1. We start by introducing the appropriate function spaces needed to study our problem. For any s ∈ (0, 1) we denote by and we let Moreover, we let H −s (−1, 1) := (H s 0 (−1, 1)) be the dual space of H s 0 (−1, 1) with respect to the pivot space L 2 (−1, 1). Then we have the following continuous embeddings: If s ≥ 1, then the above spaces are defined as in [4] and their references. For more details on fractional order Sobolev spaces we refer to [10,38] and their references.
We claim that (−d 2 x ) s D is a resolvent positive operator. Indeed, let λ > 0 be a real number, f ∈ L 2 (−1, 1) and set Then, φ belongs to H s 0 (−1, 1) and is a weak solution of the Dirichlet problem It is clear that that there is a constant C > 0 such that for all v ∈ H s 0 (−1, 1). Now, assume that f ≤ 0 a.e. in (−1, 1) and define φ + := max{φ, 0}. It follows from [38] Then by (3.1), we obtain that By (3.2), the preceding estimate implies that φ + = 0, that is, φ ≤ 0 almost everywhere. We have shown that the resolvent (λ + (−d 2 x ) s D ) −1 is a positive operator. Now it follows from the corresponding result on abstract Cauchy problems [2, Theorem 3.11.11] that, if u ≥ 0 and z 0 ≥ 0, then the solution z of the fractional heat equation (2.1) is also non-negative. We notice that this can be also seen from the representation of the solution z. More precisely, we have that the solution z of (2.1) is given by where (T (t)) t≥0 is the submarkovian (positivity-preserving and L ∞ -contractive) semigroup on The proof is finished. We can now prove Theorem 2.2 ensuring that, without imposing any constraint on the control, it is possible to obtain the null-controllability of (2.1) by means of a control u ∈ L ∞ (ω × (0, T )).
To this end, we shall first give the proof of Theorem 2.4, and then use this result to obtain the observability inequality (2.6).
Proof of Theorem 2.4. The proof of (2.8) is based on some results presented in [35]. Let us define the function According to [ with C(t) a positive constant, depending only on t and uniformly bounded for all t > 0. Then from (3.4) and (3.5), the estimate (2.8) is immediately obtained as follows: .
The proof is finished.
We can no employ (2.8) to prove Proposition 2.3. To this end, we will first need the following technical result.
Proof. The proof is based on the asymptotic results on the spectrum of the fractional Laplacian contained in the papers [18,19]. Let us introduce the following auxiliary function: Moreover, for any α > 0, let us consider the function where G is the Laplace transform of the function γ(y) := √ 4s sin(sπ) 2π According to [18], G is a completely monotone function satisfying Then, if we define according to [18,Example 6.1] we have that F µ k is the solution to the equation In other words, {F µ k } k≥1 are the eigenfunctions of the fractional Laplacian on the half-line with the zero Dirichlet exterior condition, and {µ k } k≥1 are the corresponding eigenvalues.
Let us now define From this fact, and from the definition (3.8) of the function q, it immediately follows that, for all k ≥ 1, k (x) = 0 for x ∈ (−1, 1) c . In addition, it has been proved in [19] that there is a constant C > 0 such that In particular, the family Hence, instead of looking for L 1 -estimates of the eigenfunctions φ k , we can consider the functions k defined in (3.12).
First of all, we can trivially check that where we have denoted Moreover, since the open set ω ⊂ (−1, 1), then for all x ∈ ω, we have that 1 ± x > 0. Therefore, In addition, for x ∈ ω we have From here, if we denote b k := ω | k (x)| dx, we can easily show that Indeed, from (3.10), (3.13) and (3.14) we have that Moreover, since lim k→+∞ µ −1−2s k = 0, we have that there exists k 0 ∈ N such that dx, for all k > k 0 .
It follows that B 1 := inf k>k0 b k > 0. Hence, B > 0 since b k > 0 for all k (the integrand being positive except possibly for a set of zero measure in which it is zero). The proof is then concluded. Remark 3.3. We mention that it has been shown in [6,Equation (5.3)] that the first positive eigenfunction φ 1 of the fractional Laplacian satisfies φ 1 (x) dist(x, (−1, 1) c ) s = (1 − |x|) s , in the sense that, there are two positive constants 0 < C 1 ≤ C 2 such that Proof of Proposition 2.3. The proof of the estimate (2.6) is based on standard spectral techniques. First of all, it is immediate to show that the solution of (2.5) can be expressed in the basis of the eigenfunctions {φ k } k≥1 of the operator (−d 2 x ) s D as follows: From (3.16), by using the orthonormality of the eigenfunctions in L 2 (−1, 1) and the change of variables T − t → t, it is easy to see that the observability inequality (2.6) can be rewritten as Moreover, employing the lower bound (3.7) for the L 1 (ω)-norm of φ k , we immediately get .
Hence, in order to obtain (2.6) it will be enough to show that it holds the inequality .

(3.17)
Notice that (3.17) is nothing more than (2.8) with c k = a k and µ k = λ k . Hence, we only need to show that the eigenvalues of the fractional Laplacian satisfy (2.7a) and (2.7b).
For the gap condition (2.7a), we can use [19,Proposition 3] ensuring that the eigenvalues λ k are all simple if s ≥ 1/2.
Concerning instead the summability in (2.7b), according to [19,Theorem 1] we have the following asymptotic behavior: Hence, by means of (3.18), it is immediate to see that (2.7b) holds if and only if s > 1/2. If s ≤ 1/2, instead, k≥1 λ −1 k behaves as the harmonic series and therefore, it is divergent. Summarizing, since we are assuming s > 1/2, we have that the two conditions (2.7a) and (2.7b) are both satisfied and, consequently, (3.17) holds. The proof is finished.
Proof of Theorem 2.2. The proof is based on a duality argument. This approach being classical in PDE control theory, for brevity we are going to present here only the principal ideas. The interested reader may found the complete details in [28].
Then, according to [28,Proposition 2.6], for any z 0 ∈ L 2 (−1, 1) the corresponding solution of (2.1) is null controllable at time T by means of a control function u ∈ L ∞ (ω × (0, T )) if and only if the observability inequality (2.6) holds. Moreover, following step by step the proof of that proposition it is easy to obtain the inequality (2.4) for the L ∞ -norm of the control. Finally, according to [28,Proposition 2.7], the control function u is such that u L ∞ (ω×(0,T )) = p L 1 (ω×(0,T )) . (

3.19)
The proof is then concluded.

Proof of the main result
In this section, we give the proof of our main result, namely, Theorem 2.1. The proof will be divided in three parts.

4.1.
Constrained controllability of the system (2.1). We present here the proof of the first part of Theorem 2.1 concerning the controllability of (2.1) through a non-negative control u ∈ L ∞ (ω × (0, T )).
Proof of Theorem 2.1(I). First of all, observe that, since the equation is linear, by subtracting z, it is enough to show that there exist a time T > 0 and a control v ∈ L ∞ (ω × (0, T )) fulfilling the constraint v > −ν a.e. in ω × (0, T ), such that the solution ξ of the system satisfies ξ(x, T ) = 0 a.e. in (−1, 1). According to Proposition 2.3, the null-controllability of (4.1) with v ∈ L ∞ (ω × (0, T )) is equivalent to the observability inequality (2.6). Actually, the controllability (and therefore the observability of the adjoint system) being true for any time interval (τ, T ), we also have Using (3.16), the fact that the eigenvalues {λ k } k≥1 of the operator (−d 2 x ) s D form a non-decreasing sequence, and the dissipativity of the fractional heat semigroup ensuring exponential stability, we can readily check that for every 0 < τ < T and therefore, By duality, this means that the control v can be chosen such that 1) . Furthermore, we recall that the observability constant C(T ) is uniformly bounded for any T > 0. Hence, for T large enough we have v 2 L ∞ (ω×(0,T )) < ν. This immediately implies that v > −ν. Therefore, we have shown the existence of a control v > −ν steering the solution of (4.1) from z 0 − z 0 to zero in time T > 0, provided T is large enough. Consequently, the solution z of (2.1) is controllable to the trajectory z in time T .

4.2.
Positivity of the minimal time for constrained controllability. This section is devoted to the proof of the second part of Theorem 2.1 that shows that the minimal time T min needed for controlling the system (2.1) with a non-negative control u ∈ L ∞ (ω × (0, T )) is necessarily strictly positive.
Proof of Theorem 2.1(II). Let us start by writing the solution of (2.1) in the basis of the eigenfunctions {φ k } k≥1 , that is, Derivating (4.3) and using (2.1) (or multiplying (2.1) by φ k and integrating over (−1, 1)), we can readily check that the coefficients z k (t) satisfy the following first order ODE: Hence, employing the variation of constants formula we easily get that On the other hand, since z(x, T ) = z(x, T ) a.e. in (−1, 1), we have that and from (4.4) we immediately obtain that Now, for every 0 ≤ τ ≤ T , we have Therefore (notice that u k ≥ 0), from which we obtain that Assume by contradiction that, for every T > 0, there exists a non-negative control function u T steering z 0 to z(·, T ) in time T , and that z(·, T ) = z 0 (otherwise, the trajectory z ≡ z 0 = z(·, T ) trivially solves the problem). Then, (4.5) ensures that Since z 0 ∈ L 2 (−1, 1), we clearly have Moreover, since {φ k } k≥1 is an orthonormal complete system in L 2 (−1, 1), it follows that φ k 0 (weak convergence) in L 2 (−1, 1) as k → +∞. This implies that This identity and (4.6) yield γ = 0. Thus, we immediately have that This is possible if and only if z 0 (x) = z(x, T ) a.e. in (−1, 1), which contradicts our previous assumption. The proof is then concluded.
We recall that solutions of (2.1) with controls in M(ω × (0, T min )) are defined by transposition (see [22]). Definition 4.1. Given z 0 ∈ L 2 (−1, 1), T > 0, and u ∈ M(ω × (0, T )), we shall say that the function z ∈ L 1 (Q), is a solution of (2.1) defined by transposition if it satisfies the identity where, for every p T ∈ L ∞ (−1, 1), the function p ∈ L ∞ (Q) is the unique solution of the adjoint system The existence of a unique transposition solution z ∈ L 1 (Q) of (2.1) is a consequence of the maximum principle for parabolic equations together with duality and approximation arguments. These arguments being classical, we omit here the technical details.
Proof of Theorem 2.1(III). Let us now prove the existence of a measure-valued non-negative control u ∈ M(ω × (0, T min )) realizing the controllability of the system (2.1) exactly in time T min . Let us denote In view of the definition (2.3) of the minimal control time, there exists a sequence of non-negative controls {u T k } k≥1 ⊂ L ∞ (ω × (0, T k )) such that the corresponding solution z k of (2.1) with z k (x, 0) = z 0 (x) a.e. in (−1, 1) satisfies z k (x, T k ) = z(x, T k ) a.e. in (−1, 1). We extend these controls by u on (T k , T min + 1) to get a new sequence (that we shall denote again by u T k ) in L ∞ (ω × (0, T min+1 )).
We want to prove that this sequence is bounded in L 1 (ω × (0, T min+1 )). To this end, let us assume that the initial datum p T in (4.8) is positive which, thanks to Lemma 3.1, implies that the corresponding solution p satisfies p(x, t) ≥ θ > 0 for all (x, t) ∈ (−1, 1) × (0, T min + 1) and for some positive constant θ. Then, (4.7) and the positivity of u T k yield where the last inequality is due to the continuous dependence of the solutions of the direct and adjoint problems on the initial data. We have shown that the sequence {u T k } k≥1 is bounded in L 1 (ω × (0, T min+1 )), hence, it is bounded in the space M(ω × (0, T min+1 )). Thus, extracting a sub-sequence if necessary, we have that u T k * u weakly- * in M(ω × (0, T min+1 )) as k → +∞.
Clearly, the limit control u satisfies the non-negativity constraint. Now, for any k large enough and T min < T 0 < T min+1 , by (4.7) and the definition of the control u T k we have Letting p T be the first non-negative eigenfunction φ 1 of (−d 2 x ) s D (see (3.15)), or generally any non-negative function in the domain D((−d 2 x ) s D ) of the operator (−d 2 x ) s D , we get that the solution p of the system (4.8) 1]). Therefore, by definition of weak * limit, letting k → +∞ in (4.9) we obtain which in turn implies that z(x, T 0 ) = z(x, T 0 ) a.e. in (−1, 1). Finally, taking the limit as T 0 → T min and using the fact that | u|(ω × (T min , T 0 )) = |û|(ω × (T min , T 0 )) = 0, as T 0 → T min , we can deduce that z(x, T min ) = z(x, T min ) a.e. in (−1, 1). This concludes the proof.

4.4.
Lower bounds for the minimal controllability time. Theorem 2.1 shows that the fractional heat equation (2.1) is controllable to positive trajectories by means of a non-negative control u, provided that the controllability time is large enough. Moreover, in the minimal controllability time T min defined by (2.3), we proved that the non-negative controls are in the space M(ω × (0, T min )) of Radon measures. Notwithstanding, our result does not provide a precise estimate for T min . This is indeed a delicate issue. In [30], it has been addressed for the case of the classical linear and semi-linear heat equations by means of a quite general approach. Nevertheless, this methodology does not immediately apply to the fractional heat equation (2.1).
In order to clarify this point, in what follows we present an abridged description of how the techniques developed in [30] should be applied in our case and we highlight the difficulties we encounter.
The starting point is to notice that, by a simple translation argument, we can reduce ourselves to consider the case in which the fractional heat equation (2.1) has zero initial datum. Then, from the definition of transposition solution (4.7) we have Following the procedure of [30], the idea is now to find T 0 > 0 and p T ∈ L 2 (−1, 1) such that the corresponding solution of the adjoint system (2.5) satisfies in ω × (0, T 0 ), z(·, T ), p T < 0, for all T ∈ [0, T 0 ). (4.10) Then, an explicit lower bound of the controllability minimal time is obtained by analyzing sharply the conditions required for (4.10) to hold. See [30, Sections 5.1 and 6.1] for more details.
The choice of a suitable initial datum for the adjoint equation (2.5) is not at all obvious. In [30], in the case of the linear and semi-linear heat equations with boundary control, the authors propose to consider p T in the form where φ 1 and φ 3 are respectively the first and third eigenfunction of the Dirichlet Laplacian, α and β are suitable positive constants, and ζ is a cut-off function supported outside the control region.
With these choices, a lower estimate for T min is obtained by employing the positivity of φ 1 and the explicit knowledge of the eigenfunctions φ 1 and φ 3 .
Nevertheless, the above proposals for p T do not seem to be appropriate for our fractional heat equation (2.1). This for at least two main reasons. On the one hand, we cannot ensure that with p T in the form (4.11) or (4.12) the corresponding solution of the adjoint equation (2.5) remains positive in ω. On the other hand, even if we were able to overcome this first difficulty, for the eigenfunctions of the Dirichlet fractional Laplacian we do not have a nice expression in terms of sinusoidal functions, as in the case of the classical local operator. Therefore, to perform explicit estimates is a much more difficult issue.
In view of this discussion, we can conclude that the methodology just presented is not immediately applicable to the context of the present paper. For this reason, we are not able to provide explicit analytic lower estimates for the minimal time. This will be done, instead, in Section 5 through a numerical approach.

Numerical simulations
Our main result, Theorem 2.1, states that the fractional heat equation (2.1) is controllable from any initial datum z 0 ∈ L 2 (−1, 1) to any positive trajectory z by means of the action of a non-negative control u ∈ L ∞ (ω × (0, T )), provided that s > 1/2 and the controllability time T is large enough.
In this section, we present some numerical simulations that confirm these theoretical results. In particular, we will focus on two specific situations: • Case 1: we choose as initial datum z 0 (x) = 2 cos π 2 x , and we set as a target z(·, T ) the solution at time T of (2.1) with initial datum z 0 (x) = 1 20 cos π 2 x and right-hand side u ≡ 1/5. In this case, as it can be observed in figure 2, we have z 0 > z(·, T ) a.e. in (−1, 1). • Case 2: we choose as initial datum and we set as a target z(·, T ) the solution at time T of (2.1) with initial datum z 0 (x) = 6 cos π 2 x and right-hand side u ≡ 1. In this case, as it can be observed in figure 8, we have z 0 < z(·, T ) a.e. in (−1, 1). In both cases, we first estimate numerically T min by formulating the minimal-time control problem as an optimization one. In a second moment, we will show that for T ≥ T min the fractional heat equation (2.1) is controllable from z 0 ∈ L 2 (−1, 1) to the given trajectories z(·, T ) (see above) by means of a non-negative control u, while for T < T min this controllability result is not achieved.
To simplify the presentation, we always choose the interval ω = (−0.3, 0.8) ⊂ (−1, 1) as the control region. Moreover, we focus on the case s > 1/2, in which we know that (2.1) is controllable. We recall that, if s ≤ 1/2, it has been shown in [3], both on a theoretical and numerical level, that the fractional heat equation (2.1) is not controllable, not even in the absence of constraints. 5.1. Case 1: z 0 > z(·, T ). Let us start by analyzing the case of an initial datum z 0 above the final target z(·, T ). As we mentioned, we first estimate the minimal controllability time T min through a suitable optimization problem and we address the numerical constrained controllability of (2.1) in a time horizon T ≥ T min . In a second moment, we will consider the case T < T min . 5.1.1. Numerical approximation of the minimal controllability time. To obtain an approximation of the minimal controllability time T min , we solve the following constrained optimization problem: To solve this problem numerically, we employ the expert interior-point optimization routine IpOpt (see [37]) combined with automatic differentiation and the modeling language AMPL ( [31]).
To perform our simulations, we apply a FE method for the space discretization of the fractional Laplacian on a uniform space-grid [3]). Moreover, we use an explicit Euler scheme for the time integration on the time-grid t j = T j Nt , j = 0, . . . , N t , with N t satisfying the Courant-Friedrich-Lewy condition. In particular, we will choose here N t = 300.
The minimal time that we obtain from our simulations is T min 0.8285 and we can see in figure 2 that, in this time horizon, we are able to steer the initial datum z 0 to the desired target by maintaining the positivity of the solution. We have to stress here that the minimal time T min we have obtained is just an approximation computed by solving numerically the optimization problem (5.1), (5.2). The validity of this approximation shall be confirmed by a convergence result as the mesh-sizes tend to zero. We will present more details on this issue in Section 6.
In figures 3 and 4, we show the behavior of the minimal-time control corresponding to the dynamics of figure 2. As we can see, the control is initially inactive and leaves the equation evolving under the dissipative effect of the heat semigroup. When the state finally approaches the target, the control prevents it to pass below by means of an impulsional action localized in certain specif points of the control region. Moreover, we have to mention that, since the range of amplitudes of the impulses of our minimal-time control is quite large, the plot in figure 4 (and in figure 10 below) is in logarithmic scale. In this way, also the impulses with smaller amplitude can be appreciated.  This behavior of the control is not surprising. Indeed, as it was already observed in [23,30], the minimaltime controls are expected to be atomic measures, in particular linear combinations of Dirac deltas. Our simulations are thus consistent with the aforementioned papers. Nevertheless, a more complete analysis of the positions and amplitudes of these impulses shall be addressed. This will be the subject of a future work.
The impulsional behavior of the control is then lost when extending the time horizon beyond T min . In figure 5, we show the evolution of the solution of the fractional heat equation (2.1) from the initial datum z 0 to the target z(·, T ) in the time horizon T = 0.9. As we can observe, in accordance with our theoretical results, the equation is still controllable in time T . Nevertheless, the action of the control is now more distributed in ω (see figure 6).

5.1.2.
Lack of controllability in short time. In this section, we conclude our discussion on Case 1 by showing the lack of controllability of (2.1) when the time horizon is too short.
As before, we apply a FE method for the space discretization of the fractional Laplacian on a uniform space-grid with N x = 20 points and we use an explicit Euler scheme for the time integration on a time-grid with N t = 300 points. Furthermore, we choose a time horizon T = 0.7, which is below the minimal controllability time T min .
Our simulation then show that the solution of (2.1) fails to be controlled (see figure 7). In fact, since we want to reach a final target which is below the initial datum z 0 , the natural approach would be to push down the state with a ''negative'' action. Since, however, the control is not allowed to do this because of the non-negativity constraint, its best option is to remain inactive for the entire time interval and to let the solution diffuses under the action of the fractional heat semigroup. Nevertheless, this is not enough to reach the target in the time horizon provided. 5.2. Case 2: z 0 < z(·, T ). Let us now consider the case of an initial datum z 0 which is smaller than the final target z(·, T ). As before, we start by using IpOpt for solving the optimization problem (5.1), (5.2) and computing the minimal controllability time. Also in this case, we apply a FE method for the space discretization of the fractional Laplacian on a uniform space-grid with N x = 20 points and we use an explicit Euler scheme for the time integration on a time-grid with N t = 100 points.
This time, we obtain T min 0, 2101 and, once again, our simulations displayed in figure 8 show that in this time horizon the fractional heat equation (2.1) is controllable from the initial datum z 0 to the desired trajectory z(·, T ). Nevertheless, notice that, this time, we want to reach a target which is above the initial datum z 0 . This means that the control needs to countervail also the dissipation of the solution of (2.1), by acting on it from the very beginning with a positive force. Moreover, also in this case, the minimal-time control has an atomic nature, as it is shown in figures 9 and 10.
Moreover, when extending the time horizon beyond T min we can observe once again how the solution of (2.1) is still controlled but, this time, the control is distributed in the a larger part of the control region ω and not anymore localized in specific points (see Figures 11 and 12) Finally, when considering a time horizon T < T min we can notice once more that the solution of (2.1) fails to be controlled to the desired trajectory z(·, T ). In fact, Figure 13 shows that the numerical target (displayed in green) computed by employing the tolls of the DyCon computational toolbox for solving the optimization problems (5.3), (5.4) does not totally match the desired target in blue.

Concluding remarks
In this paper, we have studied the controllability to trajectories for a one-dimensional fractional heat equation under nonnegativity state and control constraints.  For s > 1/2, when controllability for the unconstrained fractional heat equation holds in any positive time T > 0 by means of an L 2 -distributed control, we have shown that the introduction of state or control constraints creates a positive minimal time T min for achieving the same result. Moreover, we have also proved that, in this minimal time, constrained controllability holds with controls in the space of Radon measures.
Our results, which are in the same spirit of the analogous ones obtained in [23,30] for the linear and semi-linear heat equations in one and several space dimensions, are supported by the numerical simulations in Section 5.
We present hereafter a non-exhaustive list of open problems and perspectives related to our work. 1. Extension to the multi-dimensional case. Our analysis is based on spectral techniques, and applies only to a one-dimensional fractional heat equation. The extension to multi-dimensional problems on bounded domains Ω ⊂ R N , N ≥ 1, requires different tools such as Carleman estimates. Nevertheless, obtaining Carleman estimates for the fractional Laplacian is a very difficult issue which has been addressed only partially, and only for problems defined on the whole Euclidean space R N (see, e.g., [33]). The case of bounded domains remains open and it is quite challenging. As one expects, the main difficulties come from the non-local nature of the fractional Laplacian, which makes classical PDEs technique more delicate or even impossible to use.
2. Bang-bang nature of the controls. In Theorem 2.2 we showed that the fractional heat equation (2.1) is controllable to trajectories by means of L ∞ -controls satisfying u L ∞ (ω) = p L 1 (ω) , p being the solution of the adjoint equation (2.5). It is then a natural and interesting question to analyze whether these controls have a bang-bang nature, that is, u = p L ∞ (ω×(0,T )) sign(p).
To this end, the first step would be to show that the zero set of the solutions of the adjoint equation is of null measure, so that the sign of the adjoint state is well defined. This is true in the case of the classical heat equation, as a consequence of the space-time analyticity properties of the solutions. Nevertheless, we do not know whether the same holds also for the fractional heat equation (2.1) since, in this case, as far as we know no space-time analytic regularity results are available (in fact, it is known that solutions are analytic in time, but the space analyticity is still an open problem). Then, this becomes a very challenging problem, both in PDE analysis and control theory.
3. Constrained controllability from the exterior for the fractional heat equation. In [40], the null-controllability from the exterior for the one-dimensional fractional heat equation has been treated. In particular, it has been proved there that, if s > 1/2, null controllability is achievable by means of a control function g acting on a subset O ⊂ (R \ (0, 1)). Hence, to address constrained controllability in this framework becomes a very interesting issue.

4.
Lower bounds for the minimal constrained controllability time. In Section 5, we gave some numerical lower bound for the minimal constrained controllability time. Nevertheless, the bounds we presented are not optimal. This raises two very important issues. On the one hand, we shall obtain analytical lower bounds for the controllability time. This question was already addressed in [23,30] for the local heat equation but, as we discussed in Section 4.4, the methodology developed in those works does not apply immediately to our case. Therefore, there is the necessity to adapt the techniques of [23,30], or to develop new ones. On the other hand, we should develop a complete analysis of the efficiency of the numerical method we used for estimating this minimal time, in order to determine the accuracy of our approximation. 5. Convergence result for the minimal time. The minimal time T min in the simulations of Section 5 is just an approximation computed by solving numerically the optimization problem (5.1), (5.2). The validity of these computational result should be confirmed by showing that this minimal time of control for the discrete problem converges towards the continuous one as the mesh-sizes tend to zero. This could be done by adapting the procedure presented in [23,Section 5.3]. Nevertheless, we have to mention that, in order to corroborate this procedure, it is required the knowledge of an analytic lower bound for T min which, at the present stage, it is unknown (see point 4 above).