  In this paper, we present a methodology for stability analysis of a general
class of systems defined by coupled Partial Differential Equations (PDEs) with
spatially dependent coefficients and a general class of boundary conditions.
This class includes PDEs of the parabolic, elliptic and hyperbolic type as well
as coupled systems without boundary feedback. Our approach uses positive
matrices to parameterize a new class of SOS Lyapunov functionals and combines
these with a parametrization of projection operators which allow us to enforce
positivity and negativity on subspaces of L_2. The result allows us to express
Lyapunov stability conditions as a set of Linear Matrix Inequality (LMI)
constraints which can be constructed using SOSTOOLS and tested using
Semi-Definite Programming (SDP) solvers such as SeDuMi or Mosek. The
methodology is tested using several simple numerical examples and compared with
results obtained from simulation using a standard form of numerical
discretization.
