  In this paper, we develop an interior-point method for solving a class of
convex optimization problems with time-varying objective and constraint
functions. Using log-barrier penalty functions, we propose a continuous-time
dynamical system for tracking the (time-varying) optimal solution with an
asymptotically vanishing error. This dynamical system is composed of two terms:
(i) a correction term consisting of a continuous-time version of Newton's
method, and (ii) a prediction term able to track the drift of the optimal
solution by taking into account the time-varying nature of the objective and
constraint functions. Using appropriately chosen time-varying slack and barrier
parameters, we ensure that the solution to this dynamical system globally
asymptotically converges to the optimal solution at an exponential rate. We
illustrate the applicability of the proposed method in two practical
applications: a sparsity promoting least squares problem and a collision-free
robot navigation problem.
