  This paper is concerned with the theory, construction and application of
implicit Peer two-step methods that are super-convergent for variable
stepsizes, i.e., preserve their classical order achieved for uniform stepsizes
when applied to ODE constrained optimal control problems in a
first-discretize-then-optimize setting. We upgrade our former implicit two-step
Peer triplets constructed in [Algorithms, 15:310, 2022] to get ready for
dynamical systems with varying time scales without loosing efficiency. Peer
triplets consist of a standard Peer method for interior time steps supplemented
by matching methods for the starting and end steps. A decisive advantage of
Peer methods is their absence of order reduction since they use stages of the
same high stage order. The consistency analysis of variable-stepsize implicit
Peer methods results in additional order conditions and severe new difficulties
for uniform zero-stability, which intensifies the demands on the Peer triplet.
Further, we discuss the construction of 4-stage methods with order pairs (4,3)
and (3,3) for state and adjoint variables in detail and provide four Peer
triplets of practical interest. We rigorously prove convergence of order $s-1$
for $s$-stage Peer methods applied on grids with bounded or smoothly changing
stepsize ratios. Numerical tests show the expected order of convergence for the
new variable-stepsize Peer triplets.
