New approach to fixed-time stability of a nonlinear system

The problem of optimal fixed-time controllability to zero for an ordinary differential equation is investigated. We were inspired by the papers Lopez-Ramirez et al. (2018) and Polyakov (2012). We assume that our system is finite time stable. The initial conditions of trajectories we treat as control. We want to maximize the time when the trajectory starting at initial condition terminates at zero. We continue the control approach to stability from Polyakov (2012) to construct sufficient optimality conditions for the fixed-time stability in terms of the optimal control theory and find closed-loop control (feedback control) to steer the trajectory to zero.