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An optimal control approach is used to solve the problem of routing in sensor networks where the goal is to maximize the network's lifetime. We show that in a fixed topology case there exists an optimal policy consisting of fixed routing probabilities which may be obtained by solving a set of relatively simple Non-Linear Programming (NLP) problems. An alternative problem is also considered where, in addition to routing, we also allocate a total initial energy over the network nodes with the same network lifetime maximization objective. We prove that the solution to this problem is given by a policy that depletes all node energies at the same time and that the corresponding energy allocation and routing probabilities are obtained by solving a single NLP problem. Numerical examples are included to contrast the maximum lifetime we obtain to that resulting from alternative routing policies.