  The primary objective of this paper is to demonstrate that problems related
to stability and robust control in the harmonic context can be effectively
addressed by formulating them as semidefinite optimization problems, invoking
the concept of infinite-dimensional Toeplitz Block LMIs (TBLMIs). One of the
central challenges tackled in this study pertains to the efficient resolution
of these infinite-dimensional TBLMIs. Exploiting the structured nature of such
problems, we introduce a consistent truncation method that effectively reduces
the problem to a finite-dimensional convex optimization problem. By consistent
we mean that the solution to this finite-dimensional problem allows to closely
approximate the infinite-dimensional solution with arbitrary precision.
Furthermore, we establish a link between the harmonic framework and the time
domain setting, emphasizing the advantages over Periodic Differential LMIs
(PDLMIs). We illustrate that our proposed framework is not only theoretically
sound but also practically applicable to solving H 2 and H$\infty$ harmonic
control design problems. To enable this, we extend the definitions of H 2 and
H$\infty$ norms into the harmonic space, leveraging the concepts of the
harmonic transfer function and the average trace operator for Toeplitz Block
operators. Throughout this paper, we support our theoretical contributions with
a range of illustrative examples that demonstrate the effectiveness of our
approach.
