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In this study, the authors explore a joint cross-layer optimised routing and dynamic power allocation for intermittently connected deep space backbone layer in deep space information networks via predictable contacts. First, the authors build up the payoff function and state dynamics based on differential game theory via two defined cost function paradigms. Then they establish a general differential game model for dynamic power allocation by the associated payoff function and state dynamics, and further propose the theoretical results of dynamic power allocation in a cooperative or non-cooperative manner. To describe the routing metric, they introduce the concept of hybrid link homeostasis which illustrates the connection between the predictable contact and the transmitted power along the corresponding backbone link. In addition, they propose a polynomial time algorithm of cross-layer optimised routing, which realises joint routing selection, transmitted power allocation and predictable contact schedule simultaneously. The numerical results demonstrate the effectiveness and feasibility of the authors proposed joint cross-layer optimised routing and dynamic power allocation.