  This paper presents a fundamental relation between Output Asymptotic Gains
(OAG) and Input-to-Output Stability (IOS) gains for linear systems. For any
Input-to-State Stable, strictly causal linear system the minimum OAG is equal
to the minimum IOS-gain. Moreover, both quantities can be computed by solving a
specific optimal control problem and by considering only periodic inputs. The
result is valid for wide classes of linear systems (involving delay systems or
systems described by PDEs). The characterization of the minimum IOS-gain is
important because it allows the non-conservative computation of the IOS-gains,
which can be used in a small-gain analysis. The paper also presents a number of
cases for finite-dimensional linear systems, where exact computation of the
minimum IOS gain can be performed.
