Using the technique of generalized inequalities of the Hamilton--Jacobi--Bellman type, we study here the state estimation problem for a control system which operates under conditions of uncertainty and nonlinearity of a special kind, when the dynamic equations describing the studied system simultaneously contain the different forms of nonlinearity in state velocities. Namely, quadratic functions and uncertain matrices of state elocity coefficients are presented therein. The external ellipsoidal bounds for reachable sets are found, some approaches which may produce internal estimates for such sets are also mentioned. The example is included to illustrate the result
