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For N-dimensional (ND) static quadratic map, we present a time-delay approachto gradient-based extremum seeking (ES) both, in the continuous and, for thefirst time, the discrete domains. As in the recently introduced (for 2D maps inthe continuous domain), we transform the system to the time-delay one (neutraltype system in the form of Hale in the continuous case). This system isO($\varepsilon$)-perturbation of the averaged linear ODE system, where$\varepsilon$ is a period of averaging. We further explicitly present theneutral system as the linear ODE, where O($\varepsilon$)-terms are consideredas disturbances with distributed delays of the length of the small parameter$\varepsilon$. Regional input-to-state stability (ISS) analysis is provided byemploying a variation of constants formula that greatly simplifies thepreviously used analysis via Lyapunov-Krasovskii (L-K) method, simplifies theconditions and improves the results. Examples from the literature illustratethe efficiency of the new approach, allowing essentially large uncertainty ofthe Hessian matrix with bounds on $\varepsilon$ that are not too small.

A Robust Time-Delay Approach to Extremum Seeking via ISS Analysis of the Averaged System