The　problems　of　stability　and　controllability　of　the　finite　deformation　are　investigated　for　a　pre-stressed　hyperelastic　cylindrical　tube　subjected　to　the　axial　compression.　The　tube　is　assumed　to　be　composed　of　an　incompressible　neo-Hookean　material.　Based　on　the　theory　of　small　deformation　superposed　on　large　elastic　deformation,　the　mathematical　model　is　formulated　by　a　system　of　incremental　equilibrium　equations　and　incremental　boundary　conditions.　Moreover,　the　general　solutions　describing　the　finite　deformation　of　the　tube　are　obtained　by　the　form　of　Bessel　functions.　Finally,　the　system　of　nonlinear　equations　governing　the　stability　of　the　tube　are　given,　the　criteria　of　the　stability　discussed　in　terms　of　the　corresponding　numerical　simulations.　©　2020,　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.