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Accueil du site > Résumés des séminaires > Labo > Matrix Riccati Differential Equations, the Cone of Positive Definite Matrices, and the Symplectic Group and Hamiltonian Subsemigroup

Matrix Riccati Differential Equations, the Cone of Positive Definite Matrices, and the Symplectic Group and Hamiltonian Subsemigroup

We consider close connections that exist between the Riccati operator (differential) equation on the cone $P$ of positive definite matrices, which arises in linear control systems, and the symplectic group and its subsemigroup of symplectic Hamiltonian operators. The sympletic group acts by fractional transformations on the cone $P$, and this allows one to "lift" the Riccati equation to the group setting, where the solution evolves in the Hamiltonian subsemigroup. This machinery yields among other things an elementary proof of the existence of a solution for the Riccati equation. Endowing the cone $P$ with the Thompson metric allows one to deduce contractive properties for the fractional transformations of the symplectic group, which in turn give information about convergence and convergence rates for the Riccati equation. An effort will be made to pitch the talk toward a general audience.

CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 23 Fax: +33 1 69 33 46 46