In this paper we introduce new characterizations of the spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order s ∈ (0, 1) with nonzero Dirichlet and Neumann boundary condition. Here the domain Ω is assumed to be a bounded, quasi-convex Lipschitz domain. To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.For the numerical computation of solutions of (1.5), we rely on well established techniques, see for instance [12,8,7]. It is even possible to apply a standard finite element method especially if the boundary datum g is regular enough. However, the numerical realization of the nonlocal operator (−∆ D,0 ) s in (1.6) is more challenging. Several approaches have been advocated, for instance, computing the eigenvalues and eigenvectors of −∆ D,0 (cf. [39]), Dunford-Taylor integral representation [13], or numerical schemes based on the Caffarelli-Silvestre (or the Stinga-Torrea) extension, just to name a few. In our work, we choose the latter even though the proposed ideas directly apply to other approaches where (−∆ D,0 ) s appears, for instance [13]. Notice that the aforementioned extension of Caffarelli-Silvestre (or the Stinga-Torrea) is only applicable to (−∆ D,0 ) s and not directly to the operator (−∆ D ) s in (1.1).The extension approach was introduced in [17] for R n , see its extensions to bounded domains [19,40]. It states that (−∆ D,0 ) s can be realized as an operator that maps a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi-infinite cylinder C = Ω × (0, ∞), i.e., a Dirichlet-to-Neumann operator. A first finite element method to solve (1.6) based on the extension approach is given in [37]. This was applied to semilinear problems in [4]. In the context of fractional distributed optimal control problems, the extension approach was considered in [3] where related discretization error estimates are established as well.An additional advantage is that our characterization allows for imposing other types of nonhomogeneous boundary conditions such as Neumann boundary conditions (see sections 2.4 and 5) and that it immediately extends to general second order fractional operators (see Section 8).We remark that the diffic...
We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the Balakrishnan formula that is based on a finite element discretization in space and a sinc quadrature approximation of the additionally involved integral. A tailored approach for the numerical solution of the resulting linear systems is proposed.Concerning the discretization of the optimal control problem we consider two schemes. The first one is the variational approach, where the control set is not discretized, and the second one is the fully discrete scheme where the control is discretized by piecewise constant functions. We derive finite element error estimates for both methods and illustrate our results by numerical experiments.
Due to the upward trend in the aviation industry, the existing approaches for air traffic control need to be improved to achieve efficient schedules. This paper deals with the aircraft landing problem, which consists of determining a landing time for each aircraft within the radar range of an airport and allocating it to a runway. We propose an exact solution approach that involves mixed-integer linear programming. The objective is hereby to minimize the sum of weighted deviations from the target landing times under consideration of different safety, efficiency and fairness constraints. Despite of the problem’s NP-hardness, our method exhibits low execution times thanks to a modified modeling strategy and provides near-optimal results. Numerical experiments prove efficiency of the approach for different large airports.
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