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We are concerned with deterministic and stochastic nonstationary discrete-time optimal control problems in infinite horizon. We show, using Gâteaux differentials, that the so-called Euler equation and a transversality condition are necessary conditions for optimality. In particular, the transversality condition is obtained in a more general form and under milder hypotheses than in previous works. Sufficient conditions are also provided. We also find closed-form solutions to several (discounted)doi:10.3934/jdg.2014.1.57 fatcat:stepjj5omfccvdw2jpmtsburm4