We consider a matching system where items arrive one by one at each node of a compatibility network according to Poisson processes and depart from it as soon as they are matched to a compatible item. The matching policy considered is a generalized max-weight policy where decisions can be noisy. Additionally, some of the nodes may have impatience, i.e. leave the system before being matched. Using specific properties of the max-weight policy, we construct several Lyapunov functions, including a simple quadratic one. This allows us to establish stability results, to construct bounds for the stationary mean and variances of the total amount of customers in the system, and to prove exponential convergence speed towards the stationary measure. We finally illustrate some of these results using simulations on toy examples.