A necessary and sufficient condition for local asymptotic stability of a class of nonlinear systems in the critical case

By the theory of linear differential equations, a system described by a chain of integrators with a linear feedback is globally asymptotically stable if and only if the characteristic polynomial is Hurwitz, which implies that all the coefficients in the linear feedback equation are negative. However, negative coefficients may not guarantee the local asymptotic stability of the linear system. In this paper, we reveal that, by monotonizing the powers of the integrators, the strict negativity of the feedback coefficients is not only necessary but also sufficient for the local asymptotic stability of the system. A dual result is also obtained for the dual power integrator systems.