Empirical experiences have shown that simultaneous (rather than conventional sequential) plant and controller design procedure leads to an improvement in performance and saving of plant resources. Such a simultaneous synthesis procedure is called as "co-design". In this letter we study the co-design problem for a class of Lipschitz nonlinear dynamical systems having a quadratic control objective and state-feedback controller. We propose a novel time independent reformulation of the co-design optimization problem whose constraints ensure stability of the system. We also present a gradient-based co-design solution procedure which involves system coordinate transformation and whose output is provably stable solution for the original system. We show the efficacy of the solution procedure through co-design of a single-link robot.