  This investigation presents novel adaptive control algorithms specifically
designed to address and mitigate thermoacoustic instabilities. Two control
strategies are available to alleviate this issue: active and passive. Active
control strategies have a wider flexibility than passive control strategies
because they can adapt to the operating conditions of the gas turbine. However,
optimizing the control parameters remains a challenge, especially if additional
constraints have to be fulfilled, such as e.g. pollutant emission levels. To
address this issue, we propose three adaptive control strategies based on
Bayesian optimization. The first and foundational algorithm is the safeOpt
algorithm, and the two adaptations that have been made are stageOpt and
shrinkAlgo. The Gaussian Process Regressor (GPR) is employed to approximate
both the objective and constraint functions, with continuous updates occurring
during iterations. The algorithms also enable the transfer of knowledge
obtained from one operating point to another, thereby reducing the number of
iterations needed to reach the optimal point. We demonstrate the effectiveness
of the algorithms both numerically and through two distinct experimental
validations. In the numerical demonstration, we employ a low-order
thermoacoustic network model to simulate a single-stage combustor setup
equipped with loudspeaker actuation and a gain-delay ($n-\tau$) controller for
active stabilization. The first experimental demonstration has the same
structure as the numerical case. For the second experimental validation, we
apply the framework to a sequential combustor configuration utilizing
nanosecond repetitively pulsed discharges (NRPD) as the control actuator. This
demonstrates the framework's adaptability to various control actuation methods
in turbulent combustors where control parameter optimization is required.
