Control effort considerations in the stabilization of uncertain dynamical systems (Q1112997)

The authors consider uncertain nonlinear dynamical systems of the form \[ (1)\quad \dot x(t)=f(x(t),r(t),t)+B(x(t),s(t),t)u(t). \] The system (1) is said to be stabilizable with a Lyapunov function V(\(\cdot)\) and a bounding function \(c_ 0(\cdot)\) if there exists a feedback control law \(u(t)=p(x(t),t)\) such that for any initial value \(x_ 0\in R^ n\), \(t_ 0>0\) and any admissible uncertainties r(\(\cdot)\), s(\(\cdot)\), there is a solution to the closed loop state equation over \([t_ 0,\infty)\). Moreover, the origin \((x=0)\) is a uniformly asymptotically stable equilibrium point with Lyapunov derivative satisfying \[ {\mathcal L}(x,t)\triangleq \frac{\partial V(x,t)}{\partial t}+\nabla V(x,t)'f(x,r(t),t)+\nabla V(x,t)'B(x,s(t),t)p(x,t)\leq -c_ 0(\| x\|) \] for all \((x,t)\in R^ n\times [0,\infty)\). The paper gives necessary and sufficient conditions for (1) to be stabilizable with a given Lyapunov function. The authors focus their attention on the amplitude (norm) of the controller. Their results are new and interesting.