A unified criterion for stability is dealt for a class of chemostat model which includes a generic growth rate function representing substrate, biomass, and product inhibitions. The stability criterion is unified at sense that the growth kinetic rate takes a general form into the vector field and includes both the maintenance coefficient and the death rate parameters. Sufficient conditions are provided for multiplicity of the steady states. In addition, from a local analysis, an expression is proposed as a unified criterion for the local stability and for local bifurcations existence. Hence, a control design relies the unified criterion to account local stability and bifurcation with independence of the specific growth rate governing a chemostat. The criterion is based on partial derivatives which allows us to reduce the calculation needed to determine the equilibrium point stability and the bifurcation points existence. Analytical features of the chemostat were investigated when the maximum of productivity was set as the control objective. Results are illustrated through bifurcation diagrams for the specific growth rate models and parameter values reported in the scientific literature, and with examples for which the maximum of productivity is structurally stable. In this sense, the unified criterion is a tool for the formulation of the extremum seeking problem as well. Finally, some examples for applicability of the results are shown for chemostat that might be governed by different growth rates. (C) 2018 Elsevier Ltd. All rights reserved.