Global CLF Stabilization of Nonlinear Systems: A Geometric Point of View

Abstract

AbstractThe aim of this paper is to design regular feedback controls for the global asymptotic stabilization (gas) of systems with compact convex control value sets (cvs) U with 0 ∈ intU, in the framework of Artstein-Sontag's control Lyapunov function (clf) approach. Convex theory allows us to reveal the intrinsic geometry involved in the clf stabilization problem, and to show that it is solvable if there is an optimal control ω¯(x). We study the existence, uniqueness and continuity of ω¯(x) depending on properties of U, and how to attain higher regularity in terms of the geometry (curvature) of U. However, in view that ω¯(x) is singular, we consider a general form of admissible feedbacks for the gas of a system, provided a clf is known. Then, we propose an explicit formula for suboptimal admissible feedback controls. Finally, based on a method to approximate compact convex sets, we design regular feedback controls for the gas of systems with compact convex cvsU with 0 ∈ intU, at expenses of small overflows in the control values.