  A barrier certificate is an inductive invariant function which can be used
for the safety verification of a hybrid system. Safety verification based on
barrier certificate has the benefit of avoiding explicit computation of the
exact reachable set which is usually intractable for nonlinear hybrid systems.
In this paper, we propose a new barrier certificate condition, called
Exponential Condition, for the safety verification of semi-algebraic hybrid
systems. The most important benefit of Exponential Condition is that it has a
lower conservativeness than the existing convex condition and meanwhile it
possesses the property of convexity. On the one hand, a less conservative
barrier certificate forms a tighter over-approximation for the reachable set
and hence is able to verify critical safety properties. On the other hand, the
property of convexity guarantees its solvability by semidefinite programming
method. Some examples are presented to illustrate the effectiveness and
practicality of our method.
