Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases

Abstract

This is the third installment in a series of papers concerning the Bounded Real Lemma for infinite-dimensional discrete-time linear input/state/output systems. In this setting, under appropriate conditions, the lemma characterizes when the transfer function associated with the system has contractive values on the unit circle, expressed in terms of a linear matrix inequality, often referred to as the Kalman– Yakubovich–Popov (KYP) inequality. Whereas the first two installments focussed on causal systems with the transfer functions extending to an analytic function on the disk, in the present paper the system is still causal but the state operator is allowed to have nontrivial dichotomy (the unit circle is not contained in its spectrum), implying that the transfer function is analytic in a neighborhood of zero and on a neighborhood of the unit circle rather than on the unit disk. More generally, we consider bicausal systems, for which the transfer function need not be analytic in a neighborhood of zero. For both types of systems, by a variation on Willems’ storage-function approach, we prove variations on the standard and strict Bounded Real Lemma. We also specialize the results to nonstationary discrete-time systems with a dichotomy, thereby recovering a Bounded Real Lemma due to Ben-Artzi–Gohberg–Kaashoek for such systems.

Dedicated to Rien Kaashoek on the occasion of his 80th birthday.

This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 93039, 90670, and 93406).