  This paper deals with the distributed $\mathcal{H}_2$ optimal control problem
for linear multi-agent systems. In particular, we consider a suboptimal version
of the distributed $\mathcal{H}_2$ optimal control problem. Given a linear
multi-agent system with identical agent dynamics and an associated
$\mathcal{H}_2$ cost functional, our aim is to design a distributed diffusive
static protocol such that the protocol achieves state synchronization for the
controlled network and such that the associated cost is smaller than an a
priori given upper bound. We first analyze the $\mathcal{H}_2$ performance of
linear systems and then apply the results to linear multi-agent systems. Two
design methods are provided to compute such a suboptimal distributed protocol.
For each method, the expression for the local control gain involves a solution
of a single Riccati inequality of dimension equal to the dimension of the
individual agent dynamics, and the smallest nonzero and the largest eigenvalue
of the graph Laplacian.
