  We consider a perimeter defense problem in a planar conical environment in
which a single vehicle, having a finite capture radius, aims to defend a
concentric perimeter from mobile intruders. The intruders are arbitrarily
released at the circumference of the environment and they move radially toward
the perimeter with fixed speed. We present a competitive analysis approach to
this problem by measuring the performance of multiple online algorithms for the
vehicle against arbitrary inputs, relative to an optimal offline algorithm that
has information about entire input sequence in advance. In particular, we
establish two necessary conditions on the parameter space to guarantee (i)
finite competitiveness of any algorithm and (ii) a competitive ratio of at
least 2 for any algorithm. We then design and analyze three online algorithms
and characterize parameter regimes in which they have finite competitive
ratios. Specifically, our first two algorithms are provably 1, and
2-competitive, respectively, whereas our third algorithm exhibits different
competitive ratios in different regimes of problem parameters. Finally, we
provide a numerical plot in the parameter space to reveal additional insights
into the relative performance of our algorithms.
