# American Institute of Mathematical Sciences

June  2019, 8(2): 315-342. doi: 10.3934/eect.2019017

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

 Technische Universität Berlin, Institut für Mathematik, Straẞe des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author

Received  May 2018 Revised  August 2018 Published  June 2019 Early access  March 2019

Fund Project: The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 901 "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" (projects A2, A8).

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.

Citation: Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations & Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
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