Recent evidence suggests that the loudest source of high-speed jet noise appears to involve unsteady wavepackets that resemble instabilities. We seek to reduce their acoustic impact with 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 fitted with a nozzle. Direct numerical simulations using this control show significant noise reduction, with additional reduction with increase in control gain. Eigenanalysis of the controlled mean flows reveal fundamental changes in the spectrum at frequencies lower than that used by the control with the quieter flows having unstable eigenvalues that correspond to eigenfunctions without significant support in the acoustic field.