Reduction theorems for stability of closed sets with application to backstepping control design

We present a solution to the following reduction problem for asymptotic stability of closed sets in nonlinear systems. Given two closed, positively invariant subsets of the state space of a nonlinear system, Γ 1 ⊂ Γ 2 , assuming that Γ 1 is asymptotically stable relative to Γ 2 , find conditions under which Γ 1 is asymptotically stable. We also investigate analogous reduction problems for stability and attractivity. We illustrate the implications of our results on the stability of sets for cascade-connected systems and on a hierarchical control design problem. For upper triangular control systems, we present a reduction-based backstepping technique that does not require the knowledge of a Lyapunov function, and mitigates the problem of controller complexity arising in classical backstepping design.