Lyapunov-like functions involving Lie brackets

Abstract

For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set valued) Lie brackets of locally Lipschitz vector fields are considered as well.

References in corpus (3)

• Global Asymptotic Controllability Implies Input to State Stabilization
• A Geometric Approach to Feedback Stabilization of Nonlinear Systems with Drift
• Integral representation for bracket-generating multi-flows

Cited by in corpus (5)

• A Hamiltonian approach to small time local attainability of manifolds for nonlinear control systems
• Some results on second order controllability conditions
• HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation
• The Aronsson equation, Lyapunov functions and local Lipschitz regularity of the minimum time function
• A degenerate elliptic equation for second order controllability of nonlinear systems