External stability of a double integrator with saturated linear control laws

For a double integrator subject to input saturation, it is well-known that linear control laws can achieve global asymptotic stability. But a study of external stability for such a simple system reveals an unexpectedly rich nature. It is shown in this paper that external $L_p$ stability for non-input-additive disturbance only holds for $p < 2,$ but not for $p > 2$ no matter what linear control law is used. However, for input-additive disturbance, $L_p$ stability holds for all $1 < p < \infty.$. As a third result, we show that the double integrator system controlled by a saturating linear feedback is not input-to-state stable (ISS) even when all disturbances have their magnitudes restricted to be arbitrarily small. These results for the first time reveal that external stability of nonlinear systems is essentially different from that of linear systems. A fundamental discovery in this study is that the external stability of nonlinear systems cannot be separated from the internal state behavior.