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The authors consider `partially` and `fully cooperative` state-dependent relay broadcast channels (RBCs), where partial channel state information (CSI) is available at the nodes causally. First, the authors derive an achievable rate region for general discrete memoryless partially cooperative RBC (PC-RBC) with partial causal CSI, by exploiting superposition coding at the source, decode-and-forward scheme at the relay and Shannon`s strategy at the source and the relay. Then, they establish the capacity region of the discrete memoryless physically degraded PC-RBC with partial causal CSI. They also characterise the capacity region of discrete memoryless PC-RBC with feedback and partial causal CSI, and show that feedback does not affect the capacity region of the physically degraded channel. Moreover, for the fully cooperative RBC (FC-RBC) with partial causal CSI the authors obtain the same results as for the state-dependent PC-RBC. The authors` results subsume the previously known results for the degraded broadcast and relay channels with causal CSI. Finally, they extend their achievable rate regions to the Gaussian cases. Providing some numerical examples for the Gaussian cases, they compare the achievable rate regions derived for different situations.