On assigning the derivative of a disturbance attenuation control Lyapunov function (Q1577555)

The paper is motivated by \({\mathcal L}_2\) and \({\mathcal L}_\infty\) disturbance attenuation problems for a nonlinear system of the form \[ \dot x= f(x,d(t))+ g(x)u, \] where \(d(t)\) represents the disturbance. Given a control Lyapunov function candidate \(V(x)\), the authors seek conditions for the existence of a continuity feedback \(u= k(x)\) such that \[ \dot V(x(t))\leq \alpha(x(t), d(t))\quad \text{a.e. }t, \] where \(\alpha\) is a given upper bound and \(x(t)\) is any solution to the closed loop system. The authors give applications to systems where the control is implemented through a perturbed integrator (backstepping), and to linear systems with bounded controls.