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, Γ1⊂Γ2Γ1⊂Γ2, assuming that Γ1Γ1 is asymptotically stable relative to Γ2Γ2, find conditions under which Γ1Γ1 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.