A universal class of non-homogeneous control Lyapunov functions for linear differential inclusions

The constrained stabilization of Linear Differential Inclusions (LDIs) via non-homogeneous control Lyapunov functions (CLFs) is addressed in this paper. We consider the class of “merging” CLFs, which are composite functions whose gradient is a positive combination of the gradients of two given parents CLFs. In particular, we consider the constructive merging procedure based on recently-introduced composition via R-functions, which represents a parametrized trade-off between the two given CLFs. We show that this novel class of non-homogeneous Lyapunov functions is “universal” for the stabilization of LDIs, besides some equivalence results between the control-sharing property under constraints, i.e. the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given CLFs, and the existence of merging CLFs. We also provide an explicit stabilizing control law based on the proposed merging CLF. The theoretical results are finally applied to a perturbed constrained double integrator system.