Teodor Åberg | Searcharxiv

arXiv2026

Convergence analysis of a full discretization of operator-valued differential Riccati equations

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.

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arXiv2025

Convergence analysis of Lie and Strang splitting for operator-valued differential Riccati equations

Differential Riccati equations (DREs) are semilinear matrix- or operator-valued differential equations with quadratic non-linearities. They arise in many different areas, and are particularly important in optimal control of linear quadratic regulators, where they provide the optimal feedback control laws. In the context of control of partial differential equations, these Riccati equations are operator-valued. To approximate their solutions, both spatial and temporal discretizations are needed. While the former have been well analyzed in the literature, there are very few rigorous convergence analyses of time stepping methods applied to DREs, particularly in the infinite-dimensional, operator-valued setting. In view of this, we analyze two numerical time-stepping schemes, the Lie and Strang splitting methods, in such a setting. The analysis relies on the assumption that the uncontrolled system evolves via an operator that generates an analytic semigroup, and that either the initial condition is sufficiently smooth, or the nonlinearity in the DRE is sufficiently smoothing. These assumptions are mild, in the sense that they are not enough to even guarantee continuity in operator-norm of the exact solution to the DRE. However, they imply certain regularity in a pointwise sense, which can be leveraged to prove convergence in operator-norm with the classical orders. The results are illustrated by four numerical experiments, where convergence with the expected order is correlated with the relevant assumptions being fulfilled. The experiments also demonstrate that matrix-valued DREs which arise as spatial discretizations of operator-valued DREs behave similarly, unless the discretization is coarse.

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