Recent evidence suggests that the loudest source of high-speed jet noise can be described as unsteady wavepackets that resemble instabilities. We seek to reduce their acoustic impact by developing a novel control strategy that uses global modes to model their dynamics and structural sensitivity of the linearized compressible Navier-Stokes operator to determine effective linear feedback control. Using co-located actuators and sensors we demonstrate the method on an axisymmetric Mach 1.5 jet issuing from a nozzle. Direct numerical simulations using this control show significant noise reduction, with additional reduction with increase in control gain. Analysis of the mean flow quantities show that the mean flow variation is the same for all the flows for approximately 15 jet radii downstream of the nozzle beyond which the changes become significant. A specific trend is observed in the downstream variation of jet half-width, momentum thickness and centerline axial velocity as the flow gets quieter. The quieter flows are also observed to have a stable shock-cell structure that extends much further downstream.