A high-order Discontinuous Galerkin Method with mesh refinement for optimal control

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, anh-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.