  It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems.
Gradients of the objective function are most efficiently computed by
approximate adjoint variables. High accuracy with moderate computing time can
be achieved by such time integration methods that satisfy a sufficiently large
number of adjoint order conditions and supply gradients with higher orders of
consistency. In this paper, we upgrade our former implicit two-step Peer
triplets constructed in [Algorithms, 15:310, 2022] to meet those new
requirements. Since Peer methods use several stages of the same high stage
order, a decisive advantage is their lack of order reduction as for
semi-discretized PDE problems with boundary control. Additional order
conditions for the control and certain positivity requirements now intensify
the demands on the Peer triplet. We discuss the construction of $4$-stage
methods with order pairs $(3,3)$ and $(4,3)$ in detail and provide three Peer
triplets of practical interest. We prove convergence of order $s-1$, at least,
for $s$-stage methods if state, adjoint and control satisfy the corresponding
order conditions. Numerical tests show the expected order of convergence for
the new Peer triplets.
