We consider the optimal problem $\sup_{\tau\in\mathcal{T}_{\eps,T}}\mathbb{E}\left[\sum_{i=1}^n \phi_{(\tau-\eps^i)^ }^i\right]$, where $T0$ is a fixed time horizon, $(\phi_t^i)_{0\leq t\leq T}$ is progressively measurable with respect to the Brownian filtration, $\eps^i\in[0,T]$ is a constant,...