Observability of linear systems on the Heisenberg Lie group

In control theory, understanding the observability property of a system is crucial for effectively managing and controlling dynamical systems. This property empowers us to deduce the internal state of a system from its outputs over time, even when direct measurements are impossible. By harnessing observability, we can accurately estimate the complete state of a system and reconstruct its dynamics using just limited information. In this work, we will find conditions for observability of linear systems in the three dimensional Heisenberg group $\mathcal{H}$. Considering the homomorphisms between the group and its simply connected subgroups, whose kernel is denoted by $K$, we will find sufficient conditions for observability on the system using a quotient space $\mathcal{H}/K$ as the output.