Prescribed-Time Mean-Nonovershooting Control under Finite-Time Vanishing Noise

We develop a prescribed-time mean-nonovershooting stabilizing feedback law for stochastic nonlinear systems with noise that vanishes in finite time. The prescribed time of stabilization must be strictly after the noise vanishes, but it may occur as early as the user desires after the noise vanishes. In fact, more generally, the feedback is stabilizing and mean-overshoot free for any noise that, given the prescribed time of stabilization, satisfies certain decay rate properties which require, in particular, that no noise component vanish slower than linearly in the “time to go” until the prescribed time. In contrast to the existing stochastic prescribed-time designs where only multiplicative noise is allowed, our design can deal with multiplicative and additive noise simultaneously. A new controller is designed to guarantee that the mean of the system output prescribed time tracks a given trajectory without overshooting, that the fourth moment of the tracking error between states and derivatives of the reference trajectory converges to zero in prescribed time, and that the controller and all of the states are mean-square bounded. Finally, a simulation example is given to illustrate the prescribed-time mean-nonovershooting design.