Thèse de Doctorat

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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-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.