Stabilization of underactuated nonlinear systems to non‐feasible curves

Abstract

This paper focuses on the development of stability conditions for systems of nonlinear non-autonomous ordinary differential equations and their applications to control problems. We present a novel approach for the study of asymptotic stability properties for nonlinear non-autonomous systems based on considering a parameterized family of sets. The proposed approach allows to state asymptotic stability conditions for a family of sets representing the level sets of a time-varying Lyapunov function and to estimate the rate of convergence of solutions to a prescribed neighbourhood of the given curve. The obtained stability results are applied to the trajectory tracking problem for a class of nonholonomic systems. 1 Let us recall that a function W (x) : [0, ρ) → [0, ∞) (ρ > 0) is called to be of class K [0,ρ) if it is continuous, strictly increasing, and W (0) = 0.