This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions (LDIs). A recently introduced non-quadratic Lyapunov function, the convex hull of quadratics, will be used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least L-2 gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Numerical examples demonstrate the advantages of using nonlinear feedback control over linear feedback control for LDIs. It is also observed through numerical computation that nonlinear control strategies help to reduce control effort substantially. (C) 2007 Elsevier Ltd. All rights reserved.