Discrete adjoint implicit Peer methods in optimal control

It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the discrete adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to different formulations and boundary steps. For s-stage methods, we prove convergence of order s for the state variables if the adjoint method satisfies the conditions for order s−1, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of Schröder et al. (2014) and discuss the construction of 3-stage methods for the order pair (3,2) in detail. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to backward differentiation formulas. Three 3-stage methods are constructed, which show the expected orders in numerical tests.