  This paper considers a two-dimensional persistent monitoring problem by
controlling movements of second-order agents to minimize some uncertainty
metric associated with targets in a dynamic environment. In contrast to common
sensing models depending only on the distance from a target, we introduce an
active sensing model which considers the distance between an agent and a
target, as well as the agent's velocity. We propose an objective function which
can achieve a collision-free agent trajectory by penalizing all possible
collisions. Applying structural properties of the optimal control derived from
the Hamiltonian analysis, we limit agent trajectories to a simpler parametric
form under a family of 2D curves depending on the problem setting, e.g.
ellipses and Fourier trajectories. Our collision-free trajectories are
optimized through an event-driven Infinitesimal Perturbation Analysis (IPA) and
gradient descent method. Although the solution is generally locally optimal,
this method is computationally efficient and offers an alternative to other
traditional time-driven methods. Finally, simulation examples are provided to
demonstrate our proposed results.
