Discrete-time programming for control systems

In general, for optimal control problems that present systems of ordinary differential equations, for numerical resolution of the optimization problems of a cost function in Continuous time (minimization of the integral of J(u)), I use an iterative method with a 4th order Runge kutta scheme and it works very well.

However, I would like to know how I can solve numerically the same problem if we consider a cost function in Discrete time (minimization of the sum of J(u)).