Abstract: In this paper the Hausdorff measure of sets of integral and fractional dimensions is introduced and applied to control systems.A new concept, namely, pseudo-self-similar set is also introduced.The existence and uniqueness of such sets are then proved, and the formula for calculating the dimension of self-similar sets is extended to the psuedo-self-similar case.Using the previous theorem, we show that the reachable set of a control system may have fractional dimensions.We hope that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.

Key words: fractal set, pseudo-self-similar, fractional dimension, reachable set