Prescribed-time stabilisation of nonlinear time-varying systems exhibiting impulsive behaviour

This article investigates the prescribed-time stabilisation of time-varying impulsive nonlinear systems. This work's principal contributions are twofold. In the first part, in contrast to the traditional approaches to prescribed-time stability, where continuous systems used to be considered globally stable, it is shown here that even the systems with unstable flows can be stabilised globally in prescribed time by properly establishing relations between time-varying gain functions, stabilising impulsive sequences, and the prescribed time. Some adequate results on these aspects are proposed. An analysis of how destabilising impulses affect the stability of time-varying systems is presented in the second part of the article. Some results on local prescribed-time stability under the influence of destabilising impulses are proposed in this respect. Additionally, the suggested findings are used to the construction of prescribed-time controllers for time-varying affine systems that have input disturbances. To further support the veracity of the suggested results, three examples are given.