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A nonlinear system gives rise to many inherent difficulties when designing a feedback control. Motivated by a fixed-speed, fixed-altitude Unmanned Aerial Vehicle (UAV) that tracks an unpredictable target, we seek to control the turning rate of a planar Dubins vehicle. We introduce stochasticity in the problem by assuming the target performs a random walk, which both aides in the computation of a smooth value function and further accounts for all realizations of target kinematics. A Bellman equation based on an approximating Markov chain that is consistent with the stochastic kinematics is used to compute a control policy that minimizes the expected value of a cost function based on a nominal UAV-target distance. Our results indicate how uncertainty in the target motion affects the control law, and simulations illustrate that the control can further be applied to any continuous, smooth trajectory with no need for further computation.