  In some recent papers, the so called $(H,\rho)$-induced dynamics of a system
$\mathcal{S}$ whose time evolution is deduced adopting an operatorial approach,
has been introduced. According to the formal mathematical apparatus of quantum
mechanics, $H$ denotes the Hamiltonian for $\mathcal{S}$, while $\rho$ is a
certain rule applied periodically on $\mathcal{S}$. In this approach the rule
acts at specific times $k\tau$, with $k$ integer and $\tau$ fixed, by modifying
some of the parameters entering $H$ according to the state variation of the
system. As a result, a dynamics admitting an asymptotic equilibrium state can
be obtained. Here, we consider the limit for $\tau\rightarrow 0$, so that we
introduce a generalized model leading to asymptotic equilibria. Moreover, in
the case of a two-mode fermionic model, we are able to derive a relation
linking the parameters involved in the Hamiltonian to the asymptotic
equilibrium states.
