The problem considered in this paper is that of controlling downhole pressure during oil and gas well drilling, with a particular focus on handling gas kicks leading to two-phase gas-liquid flow conditions. We identify a first-order approximation to the infinite-dimensional system which captures the dominating mode of the pressure dynamics in the frequency range of interest, while the high-frequency pressure dynamics are represented by a multiplicative uncertainty. This approximation is then modified to accommodate the changes to the dynamics introduced by the two-phase flow. The linearized plant has an open-loop time constant which varies between 2 and 600 s depending on operating point and gas distribution in the well. Robust controller design is then performed using linear matrix inequalities (LMIs) via a polytopic norm-bounded description of both the high-frequency multiplicative, and the low frequency parametric uncertainty. It is shown that, in order to achieve acceptable performance over such a large range of open loop time constants, a time-varying controller gain is required. The main contribution of the paper is to achieve this control objective systematically by formulating the control design problem as an LMI optimization problem. Then optimal solutions of the LMI problem can be obtained in polynomial time by using modern interior point method (IPM) numerical solution algorithms. (C) 2016 Elsevier Ltd. All rights reserved.