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We develop an analytical device model for graphene bilayer field-effect transistors (GBL-FETs) with the back and top gates. The model is based on the Boltzmann equation for the electron transport and the Poisson equation in the weak nonlocality approximation for the potential in the GBL-FET channel. The potential distributions in the GBL-FET channel are found analytically. The source-drain current in GBL-FETs and their transconductance are expressed in terms of the geometrical parameters and applied voltages by analytical formulas in the most important limiting cases. These formulas explicitly account for the short-gate effect and the effect of drain-induced barrier lowering. The parameters characterizing the strength of these effects are derived. It is shown that the GBL-FET transconductance exhibits a pronounced maximum as a function of the top-gate voltage swing. The interplay of the short-gate effect and the electron collisions results in a nonmonotonic dependence of the transconductance on the top-gate length.