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On the exponential stability of a class of nonlinear systems including delayed perturbations. (English) Zbl 1033.93055
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 $$\chi(t)= \sup_{t-\overline h\leq\theta\leq\overline h}| x(\theta)|$$.

##### MSC:
 93D15 Stabilization of systems by feedback 93C10 Nonlinear systems in control theory 93D09 Robust stability 93D21 Adaptive or robust stabilization
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