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In this paper, a new approach to the existence of time optimal controls of system governed by nonlinear equations on Banach spaces is provided. A sequence of Meyer problems is constructed to approach a class of time optimal control problems. A deep relationship between time optimal control problems and Meyer problems is presented. The method is much different from standard methods.doi:10.14232/ejqtde.2009.1.45 fatcat:k7yslxewxfbgjd7eay5wqgmn4i