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An analytical method is proposed for the design of a congestion control scheme in the ATM network with available bit rate (ABR) service. The control architecture is rate based with a local feedback controller that is associated with each switching node. It is shown that there exists a set of control gains that results in asymptotic stability of the network model. To use optimal control theory for finding the state feedback gain is the main purposes in this paper. The theoretical results are illustrated by a simulation example.