  We study one-dimensional integral inequalities, with quadratic integrands, on
bounded domains. Conditions for these inequalities to hold are formulated in
terms of function matrix inequalities which must hold in the domain of
integration. For the case of polynomial function matrices, sufficient
conditions for positivity of the matrix inequality and, therefore, for the
integral inequalities are cast as semi-definite programs. The inequalities are
used to study stability of linear partial differential equations.
