Sur la résolution des équations de Lyapunov et Riccati
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Date
2024
Authors
BEZAI ASSIA
Journal Title
Journal ISSN
Volume Title
Publisher
University of Batna 2
Abstract
The aim of this thesis is to study the solvability of the Lyapunov (AX-XB = C),
the Sylvester (AX-Y B = C) and the Riccati (AX-XB + XDX = C ) operator
equations in Hilbert spaces of infinite dimensions using generalized inverses. More
precisely, we give new necessary and sufficient conditions for the solvability to the
operator equations AX-XB = C and AX-Y B = C; where A and B are group
invertible. In addition the general solutions to the equation AX-Y B = C; are
derived in terms of the group inverse of A and B. As a consequence, new necessary
and sufficient conditions for the solvability to the operator equation AY-BY = C;
are derived. Next by application of the generalized Drazin inverse, we give a new
method for solving Riccati and Lyapunov operator equations in Hilbert space. Results
are applied to Riccati and Lyapunov operator differential equations.