A Safety Study for Dynamical Systems on Heisenberg Lie Group of dimension 4

Abstract

The safety property of dynamical systems has typically been studied in Euclidean spaces. In this work, we extend the notion of safety to a non-Euclidean geometry. Motivated by the role of time as a fourth dimension in physical models, we construct the 4-dimensional Heisenberg Lie group H4 and investigate the safety problem of dynamical systems defined on this group. Unlike odd-dimensional Heisenberg Lie groups, which admit a unique structure, even-dimensional cases allow multiple forms; in particular, H4 possesses four distinct forms. Focusing on one such form, we provide a detailed analysis of dynamical systems on H4. Moreover, using a diffeomorphism between the (2n+1)-dimensional Heisenberg Lie group and the Euclidean space of the same dimension, we establish their equivalence, and we extend safety result for H4. Several examples are presented to illustrate the applicability of the theoretical results.