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A powerful approach for dynamic optimization in the presence of uncertainty is to incorporate measurements into the optimization framework so as to track the optimum. For nonsingular control problems, this can be done by tracking active constraints along boundary arcs and using neighboring- extremal (NE) control along interior arcs. Essentially, NE control forces the first-order variation of the necessary conditions of optimality (NCO) to zero. In this paper, an extension of NE control to singular control problems is proposed. The paper focusses on single-input systems, while the extension to multiple-input systems is investigated in the companion paper. The idea is to design NE controllers from successive time differentiations of the first- order variation of the NCO. ApproximateNE-feedbacklaws arealsoproposed, which are both easily implementable and tractable from a real-time optimization perspective. These developments are illustrated by the case study of a semi-batch chemical reactor.