  A barrier certificate can separate the state space of a con- sidered hybrid
system (HS) into safe and unsafe parts ac- cording to the safety property to be
verified. Therefore this notion has been widely used in the verification of
HSs. A stronger condition on barrier certificates means that less expressive
barrier certificates can be synthesized. On the other hand, synthesizing more
expressive barrier certificates often means high complexity. In [9], Kong et al
consid- ered how to relax the condition of barrier certificates while still
keeping their convexity so that one can synthesize more expressive barrier
certificates efficiently using semi-definite programming (SDP). In this paper,
we first discuss how to relax the condition of barrier certificates in a
general way, while still keeping their convexity. Particularly, one can then
utilize different weaker conditions flexibly to synthesize dif- ferent kinds of
barrier certificates with more expressiveness efficiently using SDP. These
barriers give more opportuni- ties to verify the considered system. We also
show how to combine two functions together to form a combined barrier
certificate in order to prove a safety property under consid- eration, whereas
neither of them can be used as a barrier certificate separately, even according
to any relaxed condi- tion. Another contribution of this paper is that we
discuss how to discover certificates from the general relaxed condi- tion by
SDP. In particular, we focus on how to avoid the unsoundness because of numeric
error caused by SDP with symbolic checking
