BIBO Stability of Linear Control Systems on Lie Group Examples

We develop a collection of nontrivial examples that illustrate and test recent stability results for linear control systems (LCS) on Lie groups. We treat the main structural classes: Abelian (Rn), nilpotent (Heisenberg), solvable non-nilpotent (rigid motions of the plane SE(2)), compact semisimple (SO(3)), noncompact semisimple (SL(2,R) via Iwasawa decomposition) and mixed/Levi-type groups. The examples are designed to (i) show the sharpness of geometric boundedness criteria, (ii) exhibit typical failure modes (exponential escape, polynomial central drift, noncompact neutrals), and (iii) demonstrate how the canonical quotient and suitable outputs recover BIBO stability. The executive framework (ICS existence/uniqueness, canonical quotient G/Γ, BIBO characterization, robustness and ISS-type bounds) is briefly recalled; the main part of the paper consists of detailed worked examples implementing the practical checklist for applying these theorems.