We develop and analyze a model of a minimal synthetic gene circuit, that describes part of the gene expression machinery in Escherichia coli, and enables the control of the growth rate of the cells during the exponential phase. This model is a piecewise non-linear system with two variables (the concentrations of two gene products) and an input (an inducer). We study the qualitative dynamics of the model and the bifurcation diagram with respect to the input. Moreover, an analytic expression of the growth rate during the exponential phase as function of the input is derived. A relevant problem is that of identifiability of the parameters of this expression supposing noisy measurements of exponential growth rate. We present such an identifiability study that we validate in silico with synthetic measurements.