Convergence analysis of a full discretization of operator-valued differential Riccati equations
This fills a gap in the theoretical understanding of numerical methods for differential Riccati equations, which are important in optimal control and other applications.
The authors provide the first convergence analysis of a full discretization (finite elements in space, Lie splitting in time) for operator-valued differential Riccati equations, proving order one in time and order two in space (up to logarithmic factors) under weak assumptions, with numerical validation in optimal control.
In recent previous work [E. Hansen, T. Stillfjord and T. Åberg, SIAM J. Numer. Anal., to appear], we analyzed the convergence of operator splitting methods applied to operator-valued differential Riccati equations (DRE). In this paper, we extend these results by analyzing the convergence of a full discretization based on finite elements in space and Lie splitting in time. As far as we are aware, this is the first such analysis for DRE. There are very few analyses of temporal discretizations of DRE overall, and none of them have been combined with spatial discretizations. However, it is clearly vital to know when the full discretization converges, since this is what will be used in practical applications. Our main result is that except for logarithmic factors, the method converges with order one in time and order two in space, under fairly weak assumptions on the problem data. This is illustrated by a numerical experiment based on an application in optimal control.