Abstract Exergy-based control strategies for ground hybrid electric vehicles (HEVs) enable to pursue unconventional optimization goals that are inaccessible when standard energy-based modeling frameworks based on fuel consumption minimization are used. In this work, we formulate and solve offline and online exergy-based optimization strategies for military HEVs aimed at the minimization of genset exergy destruction and thermal emissions to increase vehicle efficiency and minimize the risk of thermal imaging detection, respectively. We refer to the offline version of these strategies as exergy minimization strategies (ExMSs). Adaptive ExMSs (A-ExMSs) are then formulated for online implementation. Moreover, charge increasing (CI) ExMSs and A-ExMSs are developed to charge the battery as much as possible during a driving mission that is followed by a silent watch phase. To assess the performance of the proposed strategies, the results obtained by the ExMSs and A-ExMSs are compared to the benchmark solutions obtained by Dynamic Programming.