The Symplectic Semigroup and Riccati Differential Equations

In this paper, we study close connections that exist between the Riccati operator (differential) equation that arises in linear control systems and the symplectic group and its subsemigroup of symplectic Hamiltonian operators. A canonical triple factorization is derived for the symplectic Hamiltonian operators, and their closure under multiplication is deduced from this property. This semigroup of Hamiltonian operators, which we call the symplectic semigroup, is studied from the viewpoint of Lie semigroup theory, and resulting consequences for the theory of the Riccati equation are delineated. Among other things, these developments provide an elementary proof for the existence of a solution of the Riccati equation for all t ≥ 0 under rather general hypotheses.