Photochemistry induced by phase-coherent laser light is an intriguing topic. The possibility of weak-field (one-photon) phase-only control of photoisomerization is controversial. Experimental studies on the weak-field coherent control of cis-trans isomerization have led to conflicting results. Here we address this issue by quantum dynamical calculations, focusing on a mechanism where different "phase-shaped" wave packets are quickly stabilized ("dumped") in the trans configuration because of prompt energy dissipation. We systematically investigate different relaxation rates with the system-bath dynamics described within the time-dependent Hartree approximation leading to a friction-type force. We find evidence for phase control of trans-isomer yields (about 5%) in this model with pure energy dissipation given sufficiently strong dampening.