  In a recent paper, Bouchard, Elie and Reveillac \cite{BER} have studied a new
class of Backward Stochastic Differential Equations with weak terminal
condition, for which the $T$-terminal value $Y_T$ of the solution $(Y,Z)$ is
not fixed as a random variable, but only satisfies a constraint of the form
$E[\Psi(Y_T)] \geq m.$ The aim of this paper is to introduce a more general
class of BSDEs with {\em nonlinear weak terminal condition}. More precisely,
the constraint takes the form $\mathcal{E}^f_{0,T}[\Psi(Y_T)] \geq m,$ where
$\mathcal{E}^f$ represents the $f$-conditional expectation associated to a {\em
nonlinear driver} $f$.
  We carry out a similar analysis as in \cite{BER} of the value function
corresponding to the minimal solution $Y$ of the BSDE with nonlinear weak
terminal condition: we study the regularity, establish the main properties, in
particular continuity and convexity with respect to the parameter $m$, and
finally provide a dual representation and the existence of an optimal control
in the case of concave constraints. From a financial point of view, our study
is closely related to the approximative hedging of an European option under
dynamic risk measures constraints. The nonlinearity $f$ raises subtle
difficulties, highlighted throughout the paper, which cannot be handled by the
arguments used in the case of classical expectations constraints studied in
\cite{BER}.
