  In this paper we consider the safety verification and safe controller
synthesis problems for nonlinear control systems. The Control Barrier
Certificates (CBC) approach is proposed as an extension to the Barrier
certificates approach. Our approach can be used to characterize the control
invariance of a given set in terms of safety of a general nonlinear control
system subject to input constraints. From the point of view of controller
design, the proposed method provides an approach to synthesize a safe control
law that guarantees that the trajectories of the system starting from a given
initial set do not enter an unsafe set. Unlike the related control Barrier
functions approach, our formulation only considers the vector field within the
tangent cone of the zero level set defined by the certificates, and is shown to
be less conservative by means of numerical evidence. For polynomial systems
with semi-algebraic initial and safe sets, CBCs and safe control laws can be
synthesized using sum-of-squares decomposition and semi-definite programming.
Examples demonstrate our method.
