Global CLF stabilization of systems with respect to a hyperbox, allowing the null-control input in its boundary (positive controls)

Our main aim in this work is to study how to render an affine control system globally asymptotically stable (GAS), when the control value set (CVS) is given by an m-hyperbox B_r^m (∞) := [-r₁^-, r₁⁺] × ... × [-r_m^-, r_m⁺] with 0 ∈ B_r^m (∞). Hence we allow the null-control input in its boundary, 0 ∈ ∂B_r^m (∞), i.e. positive/signed control input components. Working along the line of Artstein and Sontag´s control Lyapunov function (CLF) approach, we study the conditions that feedback controls of the decentralized form u(x) = (ρ₁(x) ω̅₁(x), ..., ρ_m(x) ω̅_m(x))^T, should satisfy in order to be admissible (regular and valued in B_r^m (∞)) and render a system GAS. Here, ω̅(x) is an optimal control w.r.t. a CLF and ρ_j(x) are rescaling functions. Finally, we design of an explicit control formula valued in B_r^m (∞) (with signed/positive input components) that renders a system GAS, given a known CLF.