摘要
In this paper the Hausdorff measure of sets of integral and fractional dimensions was introduced and applied to control systems. A new concept, namely, pseudo-self-similar set was also introduced. The existence and uniqueness of such sets were then proved, and the formula for calculating the dimension of self-similar sets was extended to the pseudo-self-similar case. Using the previous theorem, it was shown that the reachable set of a control system may have fractional dimensions. It is expected that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.
In this paper the Hausdorff measure of sets of integral and fractional dimensions was introduced and applied to control systems. A new concept, namely, pseudo-self-similar set was also introduced. The existence and uniqueness of such sets were then proved, and the formula for calculating the dimension of self-similar sets was extended to the pseudo-self-similar case. Using the previous theorem, it was shown that the reachable set of a control system may have fractional dimensions. It is expected that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.
基金
This work was supported in part by Laboratory of MADIS
