  We first introduce the concept of $\mathscr{Y}^{g,\xi}$-submartingale
systems, where the nonlinear operator $\mathscr{Y}^{g,\xi}$ corresponds to the
first component of the solution of a reflected BSDE with generator $g$ and
lower obstacle $\xi$. We first show that, in the case of a left-limited
right-continuous obstacle, any $\mathscr{Y}^{g,\xi}$-submartingale system can
be aggregated by a process which is right-lower semicontinuous. We then prove a
\textit{Mertens decomposition}, by using an original approach which does not
make use of the standard penalization technique. These results are in
particular useful for the treatment of control/stopping game problems and, to
the best of our knowledge, they are completely new in the literature. As an
application, we introduce a new class of \textit{Backward Stochastic
Differential Equations (in short BSDEs) with weak constraints at stopping
times}, which are related to the partial hedging of American options. We study
the wellposedness of such equations and, using the
$\mathscr{Y}^{g,\xi}$-Mertens decomposition, we show that the family of minimal
time-$t$-values $Y_t$, with $(Y,Z)$ a supersolution of the BSDE with weak
constraints, admits a representation in terms of a reflected backward
stochastic differential equation.
