Previous works have analyzed finite/fixed-time tracking control for nonlinear systems. In these works, achieving the accurate time convergence of errors must be under the premise of known initial values and careful design of control parameters. Then, how to break through the constraints of initial values and design parameters for this issue is an unsolved problem. Motivated by this, we successfully studied prescribed-time tracking control for single-input single-output nonlinear systems with uncertainties. Specifically, we designed a state feedback controller on $[0,T_p)$, based on the backstepping method, to make the tracking error (TE) tend to zero at $T_p$, in which $T_p$ is the arbitrarily selected prescribed-time. Furthermore, on $[T_p,\mathrm{\infty }),$ another controller, similarly to that on $[0,T_p)$, was designed to keep TE within a precision after $T_p$, while TE may not stay at zero. Therefore, on $[T_p,\mathrm{\infty })$, another new controller, based on sliding mode control, was built to ensure that TE stays at zero after $T_p.$

We address the prescribed-time stability of a class of nonlinear system with uncertainty/disturbance. With the help of the parametric Lyapunov equation (PLE), we designed a state feedback control to regulate the full-state of a controlled system within prescribed time, independent of initial conditions. The result illustrated that the controlled state converges to zero as $t$ approaches the settling time and remains zero thereafter. It was further proved that the controller is bounded by a constant that depends on the system state. A numerical example is presented to verify the validity of the theoretical results.

For a hybrid stochastic system, most existing feedback controllers need to observe modes at continuous times, which is feasible when the system's mode is observable and does not incur any cost. However, in most cases, the mode is not readily apparent, and identifying it always incurs a certain expense. Therefore, in order to reduce control costs, when designing a feedback controller, both the state and the mode should be observed at discrete moments. This paper introduces an intermittent feedback controller for stabilizing an unstable hybrid stochastic system through discrete delayed observations of state and mode. By utilizing M-matrix theory, intermittent control approach, and the comparison principle, we propose sufficient conditions for the stabilization theory of hybrid stochastic systems. An illustrative example is taken to validate the proposed theory.

Krishnamurthy et al. investigated the adaptive output feedback control for prescribed-time stability (PTS) of nonlinear uncertain systems on $[0,T)$, where $T>0$. However, there are special constraints on system structure, and the PTS issue is considered on a finite interval $[0,T)$. For systems without required constraints, the existing adaptive output feedback control seems not to be applicable; the PTS issue on $[0,\mathrm{\infty })$ is more practical than that on $[0,T)$. Motivated by the above, an improved observer and controller were proposed to achieve PTS for nonlinearly uncertain systems with lower-triangular linear growth condition on uncertainties. Compared with the existing work, we emphasized the subsequent contributions: 1) Relax the original structure constraint of objective system; 2) achieve PTS on $[T,\mathrm{\infty })$; and 3) keep the proposed controller bounded. The effectiveness of the controller was verified by numerical simulations across varying initial conditions and prescribed times.