Further results on merging control Lyapunov functions for linear differential inclusions

For the robust stabilization of constrained linear differential inclusions, we consider non-homogeneous, smooth, composite control Lyapunov functions (CLFs) which belong to the class of “merging” CLFs. Previous work showed the equivalence between the possibility to always merge two given CLFs and the fact that these two share a common control law. In presence of state and control constraints, this latter property may hold only in a small domain of the state space. For such cases, we provide a novel constructive procedure to merge a CLF having a large controlled invariant set and a CLF with locally-optimal performance but with a smaller controlled invariant set. Our merging allows the explicit derivation of a Lyapunov-based, robustly stabilizing, continuous control law with the large controlled invariant set of the former CLF and the locally-optimal performance of the latter. The theoretical results are illustrated via a numerical example.