This paper investigates the use of the method of inequalities (MoI) to design output-feedback compensators for the problem of the control of instabilities in a laminar plane Poiseuille flow. In common with many flows, the dynamics of streamwise vortices in plane Poiseuille flow are very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger nonlinearities and lead to turbulence even though such perturbations would, in a linear flow model, eventually decay. Such a system can be described as a conditionally linear system. The sensitivity is measured using the maximum transient energy growth, which is widely used in the fluid dynamics community. The paper considers two approaches. In the first approach, the MoI is used to design low-order proportional and proportional-integral (PI) controllers. In the second one, the MoI is combined with McFarlane and Glover s H∞ loop-shaping design procedure in a mixed-optimization approach. In common with many flows, the dynamics of streamwise vortices in plane Poiseuille flow are very non-normal. Due, small perturbations grow rapidly with a large transient that may trigger nonlinearities and lead to turbulence even though such perturbations would, in a linear flow model, eventually decay. Such a system can be described as a conditionally linear system. In the first approach, the MoI is used to design low-order proportional and proportional-integral (PI) controllers . In the second one, the MoI is combined with McFarlane and Glover s H∞ loop-shaping design procedure in a mixed-optimization approach.