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In this paper, we present Lyapunov-based {\color{black}time varying}controllers for {\color{black}fast} stabilization of a perturbed chain ofintegrators with bounded uncertainties. We refer to such controllers as{\color{black}time varying} higher order sliding mode controllers since theyare designed for nonlinear Single-Input-Single-Output (SISO) systems withbounded uncertainties such that the uncertainty bounds are unknown.%{\color{blue} OLD: Our main result states that, given any neighborhood$\varepsilon$ of the origin, we determine a controller insuring, for everyuncertainty bounds, that every trajectory of the corresponding closed loopsystem enters $\varepsilon$ and eventually remains there. Furthermore, based onthe homogeneity property, a new asymptotic accuracy, which depends on the sizeof $\varepsilon$, is presented.} We provide a time varying control feedback lawinsuring verifying the following: there exists a family $(D(t))_{t\geq 0}$ oftime varying open sets decreasing to the origin as $t$ tends to infinity, suchthat, for any unknown uncertainty bounds and trajectory $z(\cdot)$ of thecorresponding system, there exists a positive positve $t_z$ for which$z(t_z)\in D(t_z)$ and $z(t)\in D(t)$ for $t\geq t_z$. %enters convergence infinite time of all the trajectories to a time varying domain $D(t)$ shrinkingto the origin and their maintenance there. Hence, since the function $\eta(t)$tends to zero, this leads the asymptotic convergence of all the trajectories tozero. The effectiveness of these controllers is illustrated throughsimulations.

A Lyapunov Approach to Barrier-Function Based Time-Varying Gains Higher Order Sliding Mode Controllers