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

Monika Eisenmann; Etienne Emmrich; Volker Mehrmann

doi:10.3934/eect.2019017

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2019, Volume 8, Issue 2: 315-342. Doi: 10.3934/eect.2019017

This issue Previous Article Generation of semigroups for the thermoelastic plate equation with free boundary conditions Next Article Stability of the anisotropic Maxwell equations with a conductivity term

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
^* Corresponding author

Received: May 2018

Revised: August 2018

Early access: March 2019

Published: June 2019

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).

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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.

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Mathematics Subject Classification: Primary: 47J35, 35K58, 65J15, 65M12; Secondary: 34H05, 49J99.

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