  By computing a feedback control via the linear quadratic regulator (LQR)
approach and simulating a non-linear non-autonomous closed-loop system using
this feedback, we combine two numerically challenging tasks. For the first
task, the computation of the feedback control, we use the non-autonomous
generalized differential Riccati equation (DRE), whose solution determines the
time-varying feedback gain matrix. Regarding the second task, we want to be
able to simulate non-linear closed-loop systems for which it is known that the
regulator is only valid for sufficiently small perturbations. Thus, one easily
runs into numerical issues in the integrators when the closed-loop control
varies greatly. For these systems, e.g., the A-stable implicit Euler methods
fails.\newline On the one hand, we implement non-autonomous versions of
splitting schemes and BDF methods for the solution of our non-autonomous DREs.
These are well-established DRE solvers in the autonomous case. On the other
hand, to tackle the numerical issues in the simulation of the non-linear
closed-loop system, we apply a fractional-step-theta scheme with
time-adaptivity tuned specifically to this kind of challenge. That is, we
additionally base the time-adaptivity on the activity of the control. We
compare this approach to the more classical error-based
time-adaptivity.\newline We describe techniques to make these two tasks
computable in a reasonable amount of time and are able to simulate closed-loop
systems with strongly varying controls, while avoiding numerical issues. Our
time-adaptivity approach requires fewer time steps than the error-based
alternative and is more reliable.
