Nonlinear systems with nonlinear ℒ2-gain

Nonlinear ℒ₂-gain is a robust stability property whereby the ℒ₂-norm of the state trajectory is bounded above by a monotone aggregation of nonlinear scalings of the norm of the initial state and the ℒ₂-norm of the input. This property is equivalent, via a change of coordinates, to integral input-to-state stability. In order to better understand the nonlinear ℒ₂-gain property, a variety of simple examples are presented for which it is possible to explicitly calculate the attendant transient and gain bounds. A constructive approach to finding changes of coordinates that yield systems satisfying the linear or nonlinear ℒ₂-gain property is also illustrated by example.