  This paper studies the projected saddle-point dynamics associated to a
convex-concave function, which we term saddle function. The dynamics consists
of gradient descent of the saddle function in variables corresponding to
convexity and (projected) gradient ascent in variables corresponding to
concavity. We examine the role that the local and/or global nature of the
convexity-concavity properties of the saddle function plays in guaranteeing
convergence and robustness of the dynamics. Under the assumption that the
saddle function is twice continuously differentiable, we provide a novel
characterization of the omega-limit set of the trajectories of this dynamics in
terms of the diagonal blocks of the Hessian. Using this characterization, we
establish global asymptotic convergence of the dynamics under local strong
convexity-concavity of the saddle function. When strong convexity-concavity
holds globally, we establish three results. First, we identify a Lyapunov
function (that decreases strictly along the trajectory) for the projected
saddle-point dynamics when the saddle function corresponds to the Lagrangian of
a general constrained convex optimization problem. Second, for the particular
case when the saddle function is the Lagrangian of an equality-constrained
optimization problem, we show input-to-state stability of the saddle-point
dynamics by providing an ISS Lyapunov function. Third, we use the latter result
to design an opportunistic state-triggered implementation of the dynamics.
Various examples illustrate our results.
