An optimal control framework for joint-channel parallel MRI reconstruction without coil sensitivities

•We impose two learnable regularizers: one is the composition of a multi-coil image combination operator and a full-body image regularization using deep neural nets, and another one incorporate k-space prior information of the target coil images to increase the accuracy of the Fourier signals.•Our method advocates an adaptive combination operator that first merges multi-coil images into a full-body image, followed by an effective regularization on this image of more uniform image contrasts compared to the multi-coil ones.•We cast the reconstruction network as a structured discrete-time gradient flow.•We design network training from the view of the Method of Lagrangian Multipliers.•The model conducts a learnable initial reconstruction for the proximal gradient inspired iterative algorithm.