Sustained friction drag reduction and heat transfer augmentation are simultaneously achieved in a fully developed channel flow where the averaged transport equations and wall boundary conditions for momentum and heat have identical form. Zero-net-mass-flux wall blowing and suction is assumed as a control input and its spatio-temporal distribution is determined based on optimal control theory. When the root-mean-square value of the control input is 5 % of the bulk mean velocity, the friction drag is decreased by 24 % from the uncontrolled value, whereas the heat transfer is more than doubled. Optimizations with different amplitudes of the control input and different Reynolds numbers reveal that the optimal control inputs commonly exhibit the property of a downstream travelling wave, whose wavelength is ∼250 in wall units and phase velocity is ∼30 % of the bulk mean velocity. Detailed analyses of the controlled velocity and thermal fields show that the travelling wave input contributes to dissimilar heat transfer enhancement through two distinct mechanisms, i.e. direct modification of the coherent velocity and thermal fields and an indirect effect on the random fields. The present results show that the divergence-free velocity vector and the conservative scalar are essentially different, and this is a key to achieving dissimilar heat transfer enhancement in turbulent shear flows.