Thermoacoustic instability is one of the most significant problems faced in the design of some combustion systems. Thermoacoustic oscillations arise due to feedback between acoustic waves and unsteady heat release rate when the fluctuating heat release rate is sufficiently in phase with the unsteady pressure. The primary aim of designers is to design linearly stable thermoacoustic systems in which these dangerous oscillations do not arise. In thermoacoustics, adjoint-based sensitivity analysis has shown promise at predicting the parameters which have the most influence on the linear growth and decay rates as well as oscillation frequency observed during periods of linear growth and decay. Therefore, adjoint-based methods could prove to be a valuable tool for developing optimal passive control solutions. This thesis aims to develop novel experimental sensitivity analysis techniques and provide a first comparison with the predictions of adjoint-based sensitivity analysis. In this thesis experimental sensitivity analysis is performed on (i) a vertical electrically-driven Rijke tube, and (ii) a vertical flame-driven Rijke tube. On the electrically-driven Rijke tube, the feedback sensitivity is studied by investigating the shift in linear growth and decay rates and oscillation frequency observed during periods of linear growth and decay due to the introduction of a variety of passive control devices. On the flame-driven Rijke tube, the base-state sensitivity is studied by investigating how the linear growth and decay rates as well as oscillation frequency during periods of linear growth and decay change as the convective time delay of the flame is modified. Adjoint-based sensitivity analysis gives the shift in linear growth and decay rate and the oscillation frequency when parameters are changed. This thesis provides experimental measurements of the same quantities, for comparison with the numerical sensitivity analysis, opening up new avenues for the development, implementation and validation of optimal passive control strategies for more complex thermoacoustic systems.