External optimal control of fractional parabolic PDEs

Abstract

In this paper we introduce a new notion of optimal control, or source identification in inverse, problems with fractional parabolic PDEs as constraints. This new notion allows a source/control placement outside the domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the Robin cases. For the fractional elliptic PDEs this has been recently investigated by the authors in [5]. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the source/control either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control in the exterior. We introduce the notions of weak and very-weak solutions to the parabolic Dirichlet problem. We present an approach on how to approximate the parabolic Dirichlet solutions by the parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.2010 Mathematics Subject Classification. 49J20, 49K20, 35S15, 65R20, 65N30.

Numerical methods for nonlocal and fractional models

“…The theoretical results are illustrated by two-dimensional numerical experiments. The main contribution of Antil et al (2019b) is to show the ability of nonlocal models to take advantage of information outside the domain and not only on the boundary, which is one of the limitations of control problems for PDEs. Even if not entirely focused on operators such as that in (3.1), we mention that Antil and Warma (2020) consider control problems for both a spectral fractional semilinear operator and for the integral fractional Laplacian.…”

“…Antil, Khatri and Warma (2019 a ) and Antil, Verma and Warma (2019 b ) chose the control as the data g in the volume constraint for the steady-state and time-dependent cases, respectively. In (16.1), nonlocal Dirichlet, Neumann and Robin volume constraints are considered for the operator .…”

“…(16.10) for the time-dependent version. In the more general formulation, for the time-dependent case, Antil et al (2019 b ) mostly focus on nonlocal Dirichlet and Robin optimal control problems. Well-posedness and regularity are discussed and a discretization scheme is proposed.…”

What is the fractional Laplacian? A comparative review with new results

“…\ Ω (28) has been extensively studied in the g ≡ 0 case [69,70,71,72,73], but literature based on the nonzero case is more recent and limited [73,22,74,75]. For the case of zero exterior Dirichlet condition g ≡ 0 in (28), finite element algorithms have been developed in [69], [71], and in particular the adaptive finite element scheme of [72] has been used for the computations of this paper.…”

“…In [76], a finite element method for nonzero exterior Dirichlet conditions was introduced in which the exterior condition was enforced through a Lagrange multiplier formulation. Another approach for the case of nonzero exterior condition, with application to exterior control problems, is that of [75], which implemented nonzero exterior Robin conditions and used this to approximate the solution to the nonzero exterior Dirichlet problem.…”