Abstract
In the abstract hyperbolic-like case, under a mild exact controllability assumption, the Riccati operator is known to be an isomorphism [F. Flandoli, I. Lasiecka, R. Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems, Annali di Matematica Pura e Applicata (iv)CLII (1988) 307–382 (Section 6)]. This property then plays a crucial role in establishing a Dual Algebraic Riccati Theory. Here we strengthen this theory by providing additional results (which we had announced in [V. Barbu, I. Lasiecka, R. Triggiani, Extended algebraic Riccati equations in the abstract hyperbolic case, Non-linear Analysis 40 (2000) 105–129] and [R. Triggiani, The algebraic Riccati equation with unbounded control operator: the abstract hyperbolic case revisited, AMS, Contemporary Mathematics 209 (1997) 315–338]): in particular that P D ( A F ) = D ( A ∗ ) and that P D ( A ) = D ( A F ∗ ) .
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