The problems of interval estimation and stabilization are studied for a class of generalized Persidskii systems. For this class of models with nonlinearities satisfying the incremental passivity property, the nonnegativity conditions are proposed and a nonlinear interval observer is synthesized. A nonlinear feedback is designed that uses the interval estimates. The conditions of stability of estimation and regulation errors are formulated using linear matrix inequalities. The efficiency of the proposed interval observer and controller is demonstrated for a recurrent neural network and a mechanical system with cubic nonlinearity.