Stability robustness of almost linear state equations

Sufficient conditions for stability robustness of finite-dimensional autonomous systems are discussed. The system comprises a stable linear nominal part and different types of unstructured norm-bounded perturbations. Quantitative sufficient conditions for stability robustness are introduced for the case in which the perturbations are almost linear, in the sense that both the size and the derivative of the perturbations are bounded. These conditions describe a tradeoff between the size of the perturbations and their distance from linearity. The arbitrary nonlinear case and a special linear case are shown to be limiting cases of the almost linear type. As an illustration, the same quantitative sufficient conditions for stability are applied to a system with slowly varying linear perturbations