Augmenting the cost with a logarithmic barrier creates a repulsive potential that keeps extremum-seeking trajectories inside a safe set while still converging to a nearby modified optimum, with averaging-theory proofs of practical safety and local stability.
Extremum seeking (ES) is practically stable whenever model-based ES is stable
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Logarithmic Barrier Functions for Practically Safe Extremum Seeking Control
Augmenting the cost with a logarithmic barrier creates a repulsive potential that keeps extremum-seeking trajectories inside a safe set while still converging to a nearby modified optimum, with averaging-theory proofs of practical safety and local stability.