  In this paper, we consider the problem of tuning the edge weights of a
networked system described by linear time-invariant dynamics. We assume that
the topology of the underlying network is fixed and that the set of feasible
edge weights is a given polytope. In this setting, we first consider a
feasibility problem consisting of tuning the edge weights such that certain
controllability properties are satisfied. The particular controllability
properties under consideration are (i) a lower bound on the smallest eigenvalue
of the controllability Gramian, which is related to the worst-case energy
needed to control the system, and (ii) an upper bound on the trace of the
Gramian inverse, which is related to the average control energy. In both cases,
the edge-tuning problem can be stated as a feasibility problem involving
bilinear matrix equalities, which we approach using a sequence of convex
relaxations. Furthermore, we also address a design problem consisting of
finding edge weights able to satisfy the aforementioned controllability
constraints while seeking to minimize a cost function of the edge weights,
which we assume to be convex. In particular, we consider a sparsity-promoting
cost function aiming to penalize the number of edges whose weights are
modified. Finally, we verify our results with numerical simulations over many
random network realizations as well as with an IEEE 14-bus power system
topology.
