Dynamical systems with a distributed yet interconnected structure, like multi-rigid-body robots or large-scale multi-agent systems, introduce valuable sparsity into the system dynamics that can be exploited in an optimal control setting for speeding up computation and improving numerical conditioning. Conventional approaches for solving the Optimal Control Problem (OCP) rarely capitalize on such structural sparsity, and hence suffer from a cubic computational complexity growth as the dimensionality of the system scales. In this paper, we present an OCP formulation that relies on graphical models to capture the sparsely-interconnected nature of the system dynamics. Such a representational choice allows the use of contemporary graphical inference algorithms that enable our solver to achieve a linear time complexity in the state and control dimensions as well as the time horizon. We demonstrate the numerical and computational advantages of our approach on a canonical dynamical system in simulation.
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