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Motivated by the regulator theory and adaptive controls, several notions onoutput stability in the framework of input-to-state stability (iss) wereintroduced for finite-dimensional systems. It turned out that these outputstability notions are intrinsically different, reflecting different manners ofhow state variables may affect the transient behavior of output variables. Inthis work, we consider these output stability properties for delay systems. Ourmain objective is to illustrate how the various notions are related for delaysystems and to provide the Razumikhin criteria for the output stabilityproperties. The main results are also critical in developing the converseLyapunov theorems of the output stability properties for delay systems

Remarks on Different Notions on Output Stability for Nonlinear Delay Systems