Semi-definite programming and functional inequalities for Distributed   Parameter Systems
  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. 
G. Valmorbida and M. Ahmadi and A. Papachristodoulou
math.OC