Input-to-State Stability, Integral Input-to-State Stability, and ${\cal L}_{2} $-Gain Properties: Qualitative Equivalences and Interconnected Systems

Input-to-state stability (ISS) and L
₂
-gain are wellknown robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and L
₂
-gain properties, demonstrating several qualitative equivalences between these two properties via coordinate transformations. In particular, we investigate and clearly specify the relationship between ISS and linear L
₂
-gain, and between integral ISS and nonlinear L
₂
-gain. This latter relationship raises an interesting question as it is known that cascade and feedback interconnections of integral ISS systems are not generally well-behaved, whereas we demonstrate that such interconnections for systems with nonlinear L
₂
-gain are well-behaved. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting the derived qualitative equivalences.