Further Constructions of Control-Lyapunov Functions and Stabilizing Feedbacks for Systems Satisfying the Jurdjevic-Quinn Conditions

For a broad class of nonlinear systems, we construct smooth control-Lyapunov functions whose derivatives along the trajectories of the systems can be made negative definite by smooth control laws that are arbitrarily small in norm. We assume our systems satisfy appropriate generalizations of the Jurdjevic-Quinn conditions. We also design state feedbacks of arbitrarily small norm that render our systems integral-input-to-state stable to actuator errors.Comment: 15 pages, 0 figures, accepted for publication in IEEE Transactions on Automatic Control in October 200

Asymptotic controllability and Lyapunov-like functions determined by Lie brackets

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 C^infty 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

Stabilizability in optimal control

We extend the well known concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also associated costs; in particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with W-regulated cost, which roughly means that we require the existence of a stabilizing feedback such that all the corresponding sampling and Euler solutions have finite costs, bounded above by a continuous, state-dependent function W, divided by some positive constant c. We prove that the existence of a special Control Lyapunov Function W, called c-Minimum Restraint function, c-MRF, implies Sample and Euler stabilizability to C with W-regulated cost, so extending [Motta, Rampazzo 2013], [Lai, Motta, Rampazzo, 2016], where the existence of a c-MRF was only shown to yield global asymptotic controllability to C with W-regulated cost

Global Asymptotic Controllability Implies Input to State Stabilization

We study nonlinear systems with observation errors. The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control affine systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed

Global Asymptotic Controllability Implies Input to State Stabilization

The main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable (GAC) control a#ne systems that render the closed loop systems input to state stable with respect to actuator errors. Extensions for fully nonlinear GAC systems with actuator errors are also discussed. Our controllers have the property that they tolerate small observation noise as well