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In this article, we consider a class of homing problems with controls only in the diffusion-coefficient of a diffusion process in a given closed interval. The controller seeks to minimize an expected cost that accounts for quadratic control costs, and terminal costs until the controlled process hits the end-points of the given interval. Bounds for the value function and the optimal control in question are established under certain conditions. The present results extend and complement an earlier result proved by Lefebvre (2004). [ABSTRACT FROM AUTHOR]