  We study the safety verification problem for a class of distributed parameter
systems described by partial differential equations (PDEs), i.e., the problem
of checking whether the solutions of the PDE satisfy a set of constraints at a
particular point in time. The proposed method is based on an extension of
barrier certificates to infinite-dimensional systems. In this respect, we
introduce barrier functionals, which are functionals of the dependent and
independent variables. Given a set of initial conditions and an unsafe set, we
demonstrate that if such a functional exists satisfying two (integral)
inequalities, then the solutions of the system do not enter the unsafe set.
Therefore, the proposed method does not require finite-dimensional
approximations of the distributed parameter system. Furthermore, for PDEs with
polynomial data, we solve the associated integral inequalities using
semi-definite programming (SDP) based on a method that relies on a quadratic
representation of the integrands of integral inequalities. The proposed method
is illustrated through examples.
