In　the　paper,　we　study　Regularized　Submodular　Maximization　(RegularizedSM)　problem　over　a　down-closed　family　of　sets　by　applying　the　Lyapunov　method.　The　Regularized　Submodular　Maximization　can　be　viewed　as　a　generalization　of　the　Submodular　Maximization　since　it　adds　an　extra　linear　regular　term　(possibly　negative)　to　the　objective　so　that　it　may　no　longer　be　non-negative.　For　Regularized　Non-monotone　Submodular　Maximization　(RegularizedNSM),　we　systematically　design　an　algorithm　framework　in　two　phases.　With　a　proper　choice　of　coefficients　in　the　framework,　a　(1/e,γ-γ/e-1γ-1)　-approximation　algorithm　is　obtained　in　the　continuous-time　phase,　where　γ∈　[　0,　1　)　∪　(1,　+　∞)　is　a　parameter　reflecting　the　relative　dominance　of　the　positive　and　negative　parts　of　the　linear　optimal　value.　In　the　second　phase,　we　make　the　algorithm　implemented　by　discretization　with　almost　the　same　approximation　guarantee　and　O(n3ϵ)　time　complexity.　Moreover,　the　Lyapunov　method　can　also　be　applied　in　Regularized　Monotone　Submodular　Maximization　(RegularizedMSM)　with　(1　-　1　/　e,　1　)　-approximation　performance,　which　coincides　with　the　state-of-the-art　result　given　by　Feldman　[9].　This　observation　implies　that　the　algorithm　framework　designed　by　the　Lyapunov　method　can　unify　some　of　the　existing　approximation　algorithms.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Switzerland　AG　2024.