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Remarks on Different Notions on Output Stability for Nonlinear Delay Systems

Motivated by the regulator theory and adaptive controls, several notions on output stability in the framework of input-to-state stability (iss) were introduced for finite-dimensional systems. It turned out that these output stability notions are intrinsically different, reflecting different manners of how state variables may affect the transient behavior of output variables. In this work, we consider these output stability properties for delay systems. Our main objective is to illustrate how the various notions are related for delay systems and to provide the Razumikhin criteria for the output stability properties. The main results are also critical in developing the converse Lyapunov theorems of the output stability properties for delay systems