  We introduce a new technique to optimize a linear cost function subject to a
one-dimensional affine homogeneous quadratic integral inequality, i.e., the
requirement that a homogeneous quadratic integral functional, affine in the
optimization variables, is non-negative over a space of functions defined by
homogeneous boundary conditions. Such problems arise in stability analysis,
input-to-state/output analysis, and control of many systems governed by partial
differential equations (PDEs), in particular fluid dynamical systems. First, we
derive outer approximations for the feasible set of a homogeneous quadratic
integral inequality in terms of linear matrix inequalities (LMIs), and show
that under mild assumptions a convergent, non-decreasing sequence of lower
bounds for the optimal cost can be computed with a sequence of semidefinite
programs (SDPs). Second, we obtain inner approximations in terms of LMIs and
sum-of-squares constraints, so upper bounds for the optimal cost and strictly
feasible points for the integral inequality can also be computed with SDPs. To
aid the formulation and solution of our SDP relaxations, we implement our
techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our
techniques by solving problems arising from the stability analysis of PDEs.
