A theorem on Lyapunov stability for dynamical systems and a conjecture on a property of entropy

For a general dynamical system, it is proved that an equilibrium state belonging to a continuous family of conditionally stable equilibrium states is stable. The result is applied to quantum thermodynamics to clarify in what restricted sense the entropy functional s( ρ)=−k Tr ρ ln ρ can provide a Lyapunov criterion for the stability of thermodynamic equilibrium. A conjecture on a special positive-definiteness property of −k Tr ρ ln ρ remains to be proved.