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In this paper we study the dynamic rate allocation, routing and spectrum sharing for multi-hop cognitive radio networks (CRNs), which opportunistically operate on a set of channels unused by primary user systems (PUs). We formulate this cross layer optimization problem as a sequential decision process which aims to minimize the average total power consumption in each scheduling cycle under the constraint that all cognitive radio (CR) users' traffic demands are guaranteed. We use dynamic programming (DP) to solve the formulated problem and derive an optimal rate allocation, routing and spectrum sharing policy for CRNs. Simulation results are presented to show both the validity of our problem formulation and the performance of optimal policy. Moreover, we extend our problem formulation when imperfect spectrum sensing of CRNs is concerned, an.d adopt the interference constraint to address PUs' protection.