Convergence rates of stochastic homogeneous systems.

This study considers stochastic homogeneous systems and focuses on the relation between their homogeneity and the convergence rates. We give a definition of stochastic homogeneous systems with weighted dilations as an extension of deterministic homogeneous systems. We also introduce the stability properties, such as exponential, rational, and finite-time stability with respect to homogeneous norms. As a main result, we show that their homogeneous degrees imply the convergence rates of asymptotically stable systems. Based on the result, we also deal with the stabilization of stochastic homogeneous control systems. We provide a design method of homogeneous feedback laws with homogeneous control Lyapunov functions, which guarantees the convergence rates.