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We consider the problem of estimating the common survival function S0 of independent objects. The successive failure times of the objects can be observed during a given time interval [0, T]. In order to get information more quickly, the observer has the freedom to shorten the life duration of the observed objects by applying a predictable stress Z = (Zt)t ∈ [0,T]. The stress intensity is governed by a certain cost constraint. We explore this estimation problem within its natural stochastic control setting.By choosing a nonparametric Bayesian setting, we can formulate simultaneously the problem of estimating S0 as → ∞ and the choice of Z. Under some restrictions, we propose an hybrid procedure Ŝ0(Z) for estimating S0, combining the Nelson–Aalen estimator and a dynamic programming approach in order to choose the stress Z in an optimal way.