Sur la résolution des équations de Lyapunov et Riccati
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Date
2024
Authors
BEZAI ASSIA
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Abstract
The aim of this thesis is to study the solvability of the Lyapunov (AXXB = 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.
Key words : Hilbert spaces, Inner inverse, Drazin inverse, Group inverse, Generalized
Drazin inverse, Lyapunov equation, Riccati equation, Sylvester equation,
Pseudo-similarity, Pseudo-equivalence.
AMS Classification : Primary 47A62; Secondary 15A09.