Département de mathématique

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Sur la résolution des équations de Lyapunov et Riccati
(2024) BEZAI ASSIA
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 B􀀀Y = 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.