A stochastic control problem similar to the one dimensional linear-quadratic-Gaussian problem but with an asymptotically linear cost for control is studied. The value function is characterized, and it is shown that the optimal control process has both absolutely continuous and singular components. A discussion of the fact that the value function is C 2 is given, and an example of a singular control problem in which the value function is not C 2 is presented.
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Lehoczky et al. (1986) studied this question.
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