We consider a general minimum time problem governed by a mono-input affine control system. Additionally, we suppose that if a singular arc occurs, then it is of order at least two. Such a situation typically arises in the context of Fuller's problem that involves chattering. The objective of this presentation is to show how to explicitly construct a family of control systems approximating the original one and such that the associated minimum time problems do not have a singular arc of order greater or equal than two. Our main result is the construction of such a family of control systems that depends on the original (and arbitrary) control system. For this purpose, we prove that if a singular arc occurs for the approximated problem, the term containing the control in the second-order derivative of the switching function is non-zero
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