Analysis and approximations of an optimal control problem for the Allen–Cahn equation

• Published in: Numerische Mathematik Vol. 155; no. 1-2; pp. 35 - 82
• Main Authors: Chrysafinos, Konstantinos, Plaka, Dimitra
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2023; Springer Nature B.V
• Subjects: Approximation; Estimates; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Optimal control; Parameters; Simulation; Theoretical; Thickness; 65M15; 49M41; 49K20; 49J20; 65M60; 49M25; 35K55
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-023-01374-8

The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen–Cahn equation. A tracking functional is minimized subject to the Allen–Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin—in time—scheme is considered for the approximation of the control to state and the state to adjoint mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters k , h respectively in terms of the parameter ϵ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon 1 / ϵ . Unlike to previous works for the uncontrolled Allen–Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of ϵ . These estimates and a suitable localization technique, via the second order condition (see Arada et al. in Comput Optim Appl 23(2):201–229, 2002; Casas et al. in Comput Optim Appl 31(2): 193–219, 2005; Casas and Raymond in SIAM J Control Optim 45(5):1586–1611, 2006; Casas et al. in Control Optim 46(3):952–982, 2007), allows to prove error estimates for the difference between local optimal controls and their discrete approximations as well as between the associated state and adjoint state variables and their discrete approximations.