Optimization of anti-windup controller design is achieved for non-linear systems with actuator saturation. The systems are represented as Euler-Lagrange systems, and the proportional and integral (PI) controller, in which the gravity term is cancelled by a non-linear compensation technique, is used. The goal of optimization is to minimize the saturated input, i.e., the difference between the control input and the input limit, using the estimation function with a semi-positive convex function of the saturated input. The condition, including the time derivative of the gravity term, is necessary for the stability and optimization. The proposed controller and the optimality are verified through experimentation and numerical simulations using a two-link robot arm.