  Optimization problems emerging in most of the real-world applications are
dynamic, where either the objective function or the constraints change
continuously over time. This paper proposes projected primal-dual dynamical
system approaches to track the primal and dual optimizer trajectories of an
inequality constrained time-varying (TV) convex optimization problem with a
strongly convex objective function. First, we present a dynamical system that
asymptotically tracks the optimizer trajectory of an inequality constrained TV
optimization problem. Later we modify the proposed dynamics to achieve the
convergence to the optimizer trajectory within a fixed time. The asymptotic and
fixed-time convergence of the proposed dynamical systems to the optimizer
trajectory is shown via Lyapunov based analysis. Finally, we consider the TV
extended Fermat -Torricelli problem (eFTP) of minimizing the sum-of-squared
distances to a finite number of nonempty, closed and convex TV sets, to
illustrate the applicability of the projected dynamical systems proposed in
this paper.
