In an acoustic cavity with a heat source, such as a flame in a gas turbine, the thermal energy of the heat source can be converted into acoustic energy, which may generate loud oscillations. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure.

An important objective of manufacturers is to design systems that are linearly stable, i.e. in which small perturbations decay. However, thermoacoustic systems are nonlinear, which means that they can be linearly stable, but nonlinearly unstable. Additionally, numerical and experimental studies in the literature show that nonlinear thermoacoustic oscillations can be chaotic. In this thesis, we develop strategies to analyse and minimise chaotic acoustic oscillations, for which traditional stability and sensitivity methods fail. We investigate a prototypical thermoacoustic system in which chaotic oscillations arise from the interactions between the acoustics, the heat source, and the turbulent hydrodynamics.

In the first part of the thesis, the focus is on stability, sensitivity and gradient-based optimisation of the chaotic thermoacoustic system. First, we propose covariant Lyapunov vector analysis as a tool to calculate the stability of chaotic acoustics, making connections with eigenvalue and Floquet analyses. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. We find that thermoacoustic systems can display both hyperbolic and non-hyperbolic chaos, as well as discontinuities in the time-averaged acoustic energy. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. These sensitivities, which are based on shadowing methods, are validated against costly finite-difference computations. Fourth, we embed the sensitivities in a gradient-based optimisation routine to identify bifurcations to chaos, and to suppress existing chaotic acoustic oscillations by optimal design of the heat source.

Our findings suggest that gradient-based optimisation of chaotic acoustics is challenging because: (i) tangent solutions are required; (ii) the statistics of the solution, or the sensitivity, may have slow convergence; (iii) the time-averaged acoustic energy may be physically discontinuous; and (iv) thermoacoustic chaos is not necessarily hyperbolic. Therefore, in the second part of the thesis, we develop a non-intrusive and gradient-free optimisation method with a model-informed data-driven technique to overcome the limitations of gradient-based optimisation of chaos. The method is based on reservoir computing, in particular, echo state networks. We analyse the predictive capabilities of echo state networks both in the short- and long-time predictions of the dynamics. We find that both fully data-driven and model-informed architectures are able to predict the chaotic acoustic dynamics time-accurately, as well as statistically. Informing the training with a physical reduced-order model markedly improves the accuracy of the echo state networks, whilst keeping the computational cost low. By coupling echo state networks with Bayesian optimisation to explore the design space, we find the set of heat-source parameters that minimises the time-averaged acoustic energy. The optimal parameters are found with the same accuracy as brute-force grid search, but with a convergence rate that is more than one order of magnitude faster.

The proposed analysis and methods enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is versatile, these techniques can be used in other chaotic systems.