Extremum seeking for distributed delays

We develop extremum seeking schemes for static maps whose inputs are subject to distributed delays. With distributed delays, even generating a probing signal which, when passed through the distributed delay operator, acts as a sinusoid in the static map, is nontrivial. After constructing such a probing signal, using Artstein's reduction approach, we design a delay-compensating update law for the input into the map and prove the convergence of the estimation error to a neighborhood of the origin with a combination of the averaging theory and a Lyapunov functional. The single-input design is followed by multi-parameter schemes of the gradient and Newton types. The effectiveness of the proposed scheme is confirmed by a numerical simulation. (c) 2023 Elsevier Ltd. All rights reserved.