On the L-2-gain of a double integrator with feedback loop saturation nonlinearity

We consider the open problem of whether there exists finite L-2 gain in eac h channel of a double integrator in feedback interconnection with a saturat ion nonlinearity [1], [4]. It has been shown by Liu, Chitour, and Sontag [3 ] that the L-2 gain from input to the output of the first integrator is inf inite. In this note, by constructing a special excitation, we show that the L-2 gain from input to the output of the secund integrator is also infinit e.