The problems considered
We obtain a finite element (FE) scheme for the numerical approximation of the solution to the following non-local Poisson equation
Here, is an explicit normalization constant.
The main novelty of this work relies on the fact that, since we are dealing with a one-dimensional problem, each entry of the stiffness matrix can be computed explicitly, without requiring numerical integration, which is instead needed for the multidimensional case (see ). This has the great advantage of reducing the computational cost and allows to be used on more general applications.
A natural example is given by the numerical resolution of the following control problem: given any , find a control function such that the corresponding solution to the parabolic problem
The approach emploied for solving this control problem is based on the penalized Hilbert Uniqueness Method ().
Variational formulation of the elliptic problem
Find such that
for all . Here, denotes the space
while is the classical fractional Sobolev space of order s. Since the bilinear form a is continuous and coercive, Lax-Milgram Theorem immediately implies that, if , the dual space of , then (1) admits a unique weak solution .
Definition 1. System (3) is said to be null-controllable at time if, for any , there exists such that the corresponding solution satisfies
Definition 2. System (3) is said to be approximately controllable at time if, for any and any , there exists such that the corresponding solution satisfies
There are few results in the literature on the null-controllability of the fractional heat equation, and they all deal with the spectral definition of the fractional Laplace operator, which is introduced, e.g., in ,. In this framework, it is known that the fractional heat equation is null-controllable, provided that . For , instead, null controllability does not hold, not even for large.
However, also for the parabolic equation involving the integral fractional Laplacian (2), at least in the one space dimension, these properties are easily achievable. In more detail, we prove the following.
Proposition 1. For all the parabolic problem (3) is null-controllable with a control function if and only if .
Even if for null controllability for (3) fails, we still have the following result of approximate controllability.
Proposition 2. Let . For all , there exists a control function such that the unique solution to the parabolic problem (3) is approximately controllable.
Development of the numerical scheme
Finite element approximation of the elliptic problem
Let us introduce a partition of the interval as follows:
with , , . Having defined we consider the discrete space
where is the space of the continuous and piece-wise linear functions. If we indicate with the classical basis of in which each is the tent function with and verifying , at the end we are reduced to solve the linear system
while the vector is given by with
The construction of the stiffness matrix is done in three steps, since its elements can be computed differentiating among three well defined regions: the upper triangle, corresponding to indices , the upper diagonal corresponding to , and the diagonal corresponding to . In each of these regions, the intersections among the support of the basis functions are different, thus generating different values of the bilinear form.
After several computations, we obtain the following expressions for the elements of the stiffness matrix : for
For , instead, we have
Control problem for the fractional heat equation
where is the solution to the adjoint problem
Then, the approximate and null controllability properties of the system, for a given initial datum , can be expressed in terms of the behavior of the penalized HUM approach. In particular, according to [2,Theorem 1.7] we have:
- The equation is approximately controllable at time from the initial datum if and only if
- The equation is null-controllable at time from the initial datum if and only if
In this case, we have
Here, we present the numerical simulations corresponding to our algorithms. We provide a complete discussion of the results obtained.
In figure 1 and figure 2, we show a comparison for different values of between the exact solution (8) and the computed numerical approximation. Here we consider and . One can notice that, when , the computed solution is to a certain extent different from the exact solution. However, one should be careful with such result and a more precise analysis of the error should be carried.
In figure 3, we present the computational errors evaluated for different values of and .
The rates of convergence shown are of order (in ) of . This is in accordance with the known theoretical results (see, e.g., [1,Theorem 4.6]). One can see that the convergence rate is maintained also for small values of . Since it is well-known that the notion of trace is not defined for the spaces with , it is somehow natural that we cannot expect a point-wise convergence in this case.
We present some numerical results for the controllability of the fractional heat equation (3). We use the finite-element approximation of for the space discretization and the implicit Euler scheme in the time variable. We choose the penalization term ε as a function of h. A practical rule () is to choose where is the order of accuracy in space of the numerical method used for the discretization of the spatial operator involved. In this case, we take .
In figure 4, we observe the time evolution of the uncontrolled solution as well as the controlled solution. Here, we set , and , and choose . The control domain is the highlighted zone on the plane . As expected, we observe that the uncontrolled solution decays with time, but does not reach zero at time , while the controlled solution does.
In figure 5, for we present the computed values of various quantities of interest when the mesh size goes to zero. More precisely, we observe that the control cost and the optimal energy remain bounded as . On the other hand, we see that
We know that, for , system (3) is null controllable. This is now confirmed by (9), according to [2,Theorem 1.7]. In fact, the same experiment can be repeated for different values of , obtaining the same conclusions.
In figure 6, we illustrate the numerical results for the case . We observe that the cost of the control and the optimal energy increase in both cases, while the target tends to zero with a slower rate than . This seems to confirm that a uniform observability estimate for (3) does not hold and that we can only expect to have approximate controllability.
 Acosta, G., Bersetche, F. M., and Borthagaray, J. P. A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian Comput. Math. Appl. (2017).
 Boyer, F On the penalized hum approach and its applications to the numerical approximation of null-controls for parabolic problems ESAIM: Proceedings vol. 41 (2013), pp. 15–58.
 Micu, S., and Zuazua, E. On the controllability of a fractional order parabolic equation. SIAM J. Control Optim., vol. 44, no. 6 (2006), pp. 1950–1972.
 Miller, L. On the controllability of anomalous diffusions generated by the fractional Laplacian. Math. Control Signals Systems, vol. 18, no. 3 (2006), pp. 260–271.