Optimal control for a scalar one-step linear system with additive Cauchy noise

An optimal control scheme is developed for scalar discrete linear dynamic systems driven by Cauchy distributed process and measurement noises. Since the Cauchy density has infinite variance, a cost function is defined for which the unconditional expectation with respect to the Cauchy densities produces a cost criterion that exists. After showing that this cost criterion allows a dynamic programming solution for the multistage problem, an optimal controller is determined for one step time update. Characteristics of the optimal controller is compared with the linear exponential Gaussian (LEG) controller. The dramatic performance difference between the Cauchy and the LEG controllers is studied. Furthermore, through different numerical examples, some interesting properties of the Cauchy controller are examined.