Convergence analysis for the exponential Lie splitting scheme applied to the abstract differential Riccati equation

We consider differential Riccati equations (DREs).
These equations arise in many areas and are very important within the field of optimal control. In particular, DREs provide the crucial link between the state and the optimal input in the solution of linear quadratic regulator (LQR) problems.
For the approximation of the solutions to DREs we consider the recently introduced splitting methods, with the aim of proving convergence orders in the space of Hilbert--Schmidt operators. The use of this abstract setting yields stronger than usual temporal convergence results, and also implies that these are independent of a subsequent (reasonable) spatial discretization.
The main result is that the exponential Lie splitting is first-order convergent, under no artificial regularity assumptions. As side-effects of the analysis, we also acquire concise proofs of the existence and positivity of the exact solutions to abstract DREs, in a more general setting than previously considered.