LAUSR

This brief focuses on the $$\mathscr {H}_-$$H- index problem for nonlinear systems involving state- and input-dependent noises in finite and infinite horizon. A sufficient condition of the $$\mathscr {H}_-$$H- index in finite/infinite case is developed for such systems based on Hamilton–Jacobi equations/inequalities (HJEs/HJIs). Generally, one can hardly solve these HJEs/HJIs. By fuzzy approach, the characterization of $$\mathscr {H}_{-}$$H- index is derived via solving a linear matrix inequality (LMI). Finally, an example verifies the effect of the obtained results.