A high-order Discontinuous Galerkin Method with mesh refinement for optimal control. (November 2017)

• Record Type: Journal Article
• Authors: Henriques, João C.C.; Lemos, João M.; Eça, Luís; Gato, Luís M.C.; Falcão, António F.O.
• Is Part Of: Automatica. Volume 85(2017)
• Page Start: 70
• Page End: 82
• Publication Date: 2017-11
• Subjects: Optimal control -- Finite elements -- Numerical methods -- Bang–bang control -- Energy control; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2017.07.029
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: A high-order Discontinuous Galerkin (DG) finite element time-stepping method is applied for the numerical solution of optimal control problems within the framework of Pontryagin's Maximum Principle. The method constitutes an efficient and versatile alternative to the well-known Pseudospectral (PS) methods. The two main advantages of DG in comparison with the PS methods are: the local nature of the piecewise polynomial solution and the straightforward implementation of element-wise mesh and polynomial refinement if required. Two types of non-linear optimal control problems were analysed: continuous and bang–bang time-solutions. In the case of bang–bang optimal control problems, an h -refinement strategy was developed to achieve agreement between the observed and the formal order of accuracy. The paper also deals with sub-optimal control problems where: (i) time-step is fixed and non-infinitesimal; (ii) the control has two modes (on/off); (iii) the control command is only applied at the beginning of each time-step; and iv) the number of switching instants is large and not known a priori .