In this work, we grant robot agents the capacity to sense and react to their human supervisor.s changing trust state, as a means to maintain the efficiency of their collaboration. We propose the novel formulation of Trust-Aware Conservative Control (TACtiC), in which the agent alters its behaviors momentarily whenever the human loses trust. This trust-seeking robot framework builds upon an online trust inference engine and also incorporates an interactive behavior adaptation technique. We present end-to-end instantiations of trust-seeking robots for distinct task domains of aerial terrain coverage and interactive autonomous driving. Empirical assessments comprise a large-scale controlled interaction study and its extension into field evaluations with an autonomous car. These assessments substantiate the efficiency gains that trust- seeking agents bring to asymmetric human-robot teams.