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We present a numerical investigation of the control of few-particlefluctuating clusters with a macroscopic field. Our goal is to reach a giventarget cluster shape is minimum time. This question is formulated as a firstpassage problem in the space of cluster configurations. We find the optimalpolicy to set the macroscopic field as a function of the observed shape usingdynamic programming. Our results show that the optimal policy is non-unique,and its degeneracy is mainly related to symmetries shared by the initial shape,the force and the target shape. The total fraction of shapes for which optimalchoice of the force is non-unique vanishes as the cluster size increases.Furthermore, the optimal policy exhibits a discrete set of transitions when thetemperature is varied. Each transition leads to a discontinuity in thederivative of the time to reach with target with respect to temperature. As thesize of the cluster increases, the change in the policy due to temperaturetransitions grows like the total number of configurations and a continuum limitemerges.

Temperature transitions and degeneracy in the control of small clusters with a macroscopic field