A subalgebra of the Hardy algebra relevant in control theory and its algebraic-analytic properties

We denote by A_0+AP_+ the Banach algebra of all complex-valued functions f defined in the closed right half plane, such that f is the sum of a holomorphic function vanishing at infinity and a ``causal'' almost periodic function. We give a complete description of the maximum ideal space M(A_0+AP_+) of A_0+AP_+. Using this description, we also establish the following results: (1) The corona theorem for A_0+AP_+. (2) M(A_0+AP_+) is contractible (which implies that A_0+AP_+ is a projective free ring). (3) A_0+AP_+ is not a GCD domain. (4) A_0+AP_+ is not a pre-Bezout domain. (5) A_0+AP_+ is not a coherent ring. The study of the above algebraic-anlaytic properties is motivated by applications in the frequency domain approach to linear control theory, where they play an important role in the stabilization problem.Comment: 17 page

Inertia theorems for operator Lyapunov inequalities

We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded, but are admissible. We prove an inertia result: under mild conditions, we show that the number of unstable eigenvalues of the generator is less than or equal to the number of negative eigenvalues of the self-adjoint solution of the operator Lyapunov inequality. (C) 2001 Elsevier Science B.V. All rights reserved

Optimal Hankel norm approximation for the Pritchard-Salamon class of infinite-dimensional systems

The optimal Hankel norm approximation problem is solved under the assumptions that the system Sigma (A, B, C) is an exponentially stable, regular Pritchard-Salamon infinite-dimensional system. An explicit parameterization of all solutions is obtained in terms of the system parameters A, B, C