  Reach-avoid analysis combines the construction of safety and specific
progress guarantees, and is able to formalize many important engineering
problems. In this paper we study the reach-avoid verification problem of
systems modelled by ordinary differential equations using Lyapunov densities.
Firstly, the weak reach-avoid verification is considered. Given an initial set,
a safe set and a target set, the weak reach-avoid verification is to verify
whether the reach-avoid property (i.e., the system will enter the target set
eventually while staying inside the safe set before the first target hitting
time) holds for almost all states in the initial set. We propose two novel
sufficient conditions using Lyapunov densities for the weak reach-avoid
verification. These two sufficient conditions are shown to be weaker than
existing ones, providing more possibilities of verifying weak reach-avoid
properties successfully. Then, we generalize these conditions to the
verification of reach-avoid properties for all states in the initial set.
Finally, an example demonstrates theoretical developments of proposed
conditions.
