An optimal predictive controller for linear, vector-state dynamic systems driven by Cauchy measurement and process noises is developed. For the vector-state system, only the characteristic function of the conditional probability density function (pdf), and not the pdf itself, can be expressed analytically in a closed form. Consequently, the conditional performance index formulated for the controller design can also be evaluated only in the spectral domain. In particular, by taking the conditional expectation of an objective function that is a product of functions resembling Cauchy pdfs, the conditional performance index is obtained in closed form by using Parseval's identity and integrating over the spectral vector. This forms a deterministic, non-convex function of the control signal and the measurement history that must be optimized numerically at each time step. A two-state example is used to expose the interesting robustness characteristics of the proposed controller.