Before revision:  Using a simple energy argument , we show an exponential convergence of the ODE solution to its steady state . The PAGD method may be viewed as an explicit-type time-discretization scheme of the ODE system, which requires a natural time step restriction for energy stability. Assuming this restriction, an exponential rate of convergence of the PAGD sequence is demonstrated, by mimicking the convergence of the solution to the ODE via energy methods.
After revision:  Using a simple energy argument , we show an exponential convergence of the ODE solution to its steady state using a simple energy argument. At the discrete level, assuming the aforementioned step size restriction, the existence of an invariant set is proved and a matching exponential rate of convergence of the PAGD sequence is demonstrated, by mimicking the convergence of the solution to the ODE via energy methods.