Semi-definite programming and functional inequalities for distributed parameter systems

We study one-dimensional integral inequalities on bounded domains, with quadratic integrands. 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.