Fading memory and necessity of the small gain theorem

The authors consider fading memory for nonlinear time-varying systems and associated problems of robust stability. They establish notation, state some preliminary results, and consider the fading memory property. Uniform and pointwise fading memory are defined. The focus is on the differences between pointwise and uniform fading memory, and conditions are provided for an operator to have pointwise fading memory. Previous results showing that the small-gain theorem is necessary for robust stability of fading memory nonlinear systems are generalized. These results are applied to a robust stabilization problem. Conditions are presented under which a closed-loop system exhibits fading memory