  This paper studies the asymptotic convergence properties of the primal-dual
dynamics designed for solving constrained concave optimization problems using
classical notions from stability analysis. We motivate the need for this study
by providing an example that rules out the possibility of employing the
invariance principle for hybrid automata to study asymptotic convergence. We
understand the solutions of the primal-dual dynamics in the Caratheodory sense
and characterize their existence, uniqueness, and continuity with respect to
the initial condition. We use the invariance principle for discontinuous
Caratheodory systems to establish that the primal-dual optimizers are globally
asymptotically stable under the primal-dual dynamics and that each solution of
the dynamics converges to an optimizer.
