Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

TL;DR

This paper proves the existence and convergence of solutions for operator-valued Riccati differential equations using a backward Euler scheme, extending previous algebraic results to infinite-dimensional settings.

Contribution

It generalizes existence results from algebraic Riccati equations to operator-valued differential equations and establishes convergence of numerical approximations.

Findings

Existence of solutions for operator-valued Riccati differential equations.

Weak and strong convergence of backward Euler approximations.

Applicability to a broad class of initial data and right-hand sides.

Abstract

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert--Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.