This paper presents a new numerical method for treating the problem of optimal control when there are hard bounds on the control variables (e.g., limit switches on a linear drive, current limits to motor input, etc.) and/or on the state/output variables (e.g., obstacle avoidance). This is accomplished by means of a new approach for discretizing the optimal control problem, while introducing regularization terms to reduce the solution space to smooth functions. Further, by introducing a consistent discretization of the state-space equations with arbitrary boundary conditions, the problem is cast as a problem of quadratic programming, whereby (hard) bounds can be put on any of the state-space variables (i.e., input or output). The method is demonstrated on the example of a pendulum on a cart. Bounded optimal control solutions are computed for two examples: Velocity bounds are placed on the cart in the classic optimal control problem; a variation of trajectory tracking where instead of specifying a single valued path, the bounds of the trajectory of the pendulum bob are specified, and the required input to keep the bob within these bounds during its motion is computed.