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We first introduce the concept of $\mathscr{Y}^{g,\xi}$-submartingalesystems, where the nonlinear operator $\mathscr{Y}^{g,\xi}$ corresponds to thefirst component of the solution of a reflected BSDE with generator $g$ andlower obstacle $\xi$. We first show that, in the case of a left-limitedright-continuous obstacle, any $\mathscr{Y}^{g,\xi}$-submartingale system canbe aggregated by a process which is right-lower semicontinuous. We then prove a\textit{Mertens decomposition}, by using an original approach which does notmake use of the standard penalization technique. These results are inparticular useful for the treatment of control/stopping game problems and, tothe best of our knowledge, they are completely new in the literature. As anapplication, we introduce a new class of \textit{Backward StochasticDifferential Equations (in short BSDEs) with weak constraints at stoppingtimes}, which are related to the partial hedging of American options. We studythe wellposedness of such equations and, using the$\mathscr{Y}^{g,\xi}$-Mertens decomposition, we show that the family of minimaltime-$t$-values $Y_t$, with $(Y,Z)$ a supersolution of the BSDE with weakconstraints, admits a representation in terms of a reflected backwardstochastic differential equation.

A new Mertens decomposition of $\mathscr{Y}^{g,\xi}$-submartingale systems. Application to BSDEs with weak constraints at stopping times