On the exponential stability of a class of nonlinear systems including delayed perturbations.
The authors consider the system \[ \dot x= F(x,t)+ G(x, t)u(t), \] whose equilibrium at the origin for \(u(t)\equiv 0\) is exponentially stable, this property being ensured by a \({\mathcal C}^1\) Lyapunov function satisfying \[ \lambda^2_1| x|^2\leq V(x, t)\leq \lambda^2_2| x|^2,\quad {\partial V\over\partial t}+ (\text{grad}_x V)F(x,t)\leq -\lambda_3 V(x,t). \] This system is perturbed by a delayed state dependent disturbance satisfying \(| H(x,t)|\leq \beta| x|\). It is shown that the control law \[ u(t)=- {G^T(x, t)(\text{grad}_x V(x,t))^T \beta^2\chi^2(t)\over |\text{grad}_x V(x,t) G(x,t)|_\beta\chi(t)+ \varepsilon e^{-\alpha t}} \] exponentially stabilizes the system \[ \dot x= F(x(t), t)+ G(x(t), t)[H(x(t- h(t), t)+ u(t)],\quad 0\leq h(t)\leq\overline h. \] Here \(\alpha> 0\), \(\varepsilon> 0\) and \(\chi(t)= \sup_{t-\overline h\leq\theta\leq\overline h}| x(\theta)|\).
scientific article; zbMATH DE number 2006409