  This paper studies a stochastic extremum seeking method to steer a
nonholonomic vehicle to the unknown source of a static spatially distributed
filed in a plane. The key challenge lies in the lack of vehicle's position
information and the distribution of the scalar field. Different from the
existing stochastic strategy that keeps the forward velocity constant and
controls only the angular velocity, we design a stochastic extremum seeking
controller to regulate both forward and angular velocities simultaneously in
this work. Thus, the vehicle decelerates near the source and stays within a
small area as if it comes to a full stop, which solves the overshoot problem in
the constant forward velocity case. We use the stochastic averaging theory to
prove the local exponential convergence, both almost surely and in probability,
to a small neighborhood near the source for elliptical level sets. Finally,
simulations are included to illustrate the theoretical results.
