Sub-fixed-time stability of a class of nonlinear system

In this paper, a new concept "sub-fixed-time stability"(SFTS) is proposed and studied, which means the states can converge to a region of equilibrium points in a fixed time for any initial states' values. Then, a sufficient condition for it is given and proven. Though SFTS is similar to "practical fixed-time stability"(PFTS), they are not the same, and the sufficient condition for SFTS is much clearer and simpler than PFTS. Next, a sub-fixed-time controller is proposed for a class of second order system and the stability analyses are given. At last, to illustrate the robustness of the proposed sub-fixed-time controller to different initial conditions, one hundred of numerical simulations are conducted for one hundred of initial states' values.