Direct optimization using Gaussian quadrature and continuous Runge-Kutta methods: application to an innovation diffusion model

In the present paper the discretization of a particular model arising in the economic field of innovation diffusion is developed. It consists of an optimal control problem governed by an ordinary differential equation.We propose a direct optimization approach characterized by an explicit, fixed step-size continuous Runge-Kutta integration for the state variable approximation. Moreover, high-order Gaussian quadrature rules are used to discretize the objective function. In this way, the optimal control problem is converted into a nonlinear programming one which is solved by means of classical algorithms.