Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop solution is not required, impractical or impossible. If a trajectory optimization problem can be solved at a rate given by the inverse of the Lipschitz constant, then it can be used iteratively to generate a closed-loop solution in the sense of Caratheodory. If only the first step of the trajectory is executed for an infinite-horizon problem, then this is known as Model Predictive Control (MPC).
Although the idea of trajectory optimization has been around for hundreds of years (calculus of variations, brachystochrone problem), it only became practical for real-world problems with the advent of the computer. Many of the original applications of trajectory optimization were in the aerospace industry, computing rocket and missile launch trajectories. More recently, trajectory optimization has also been used in a wide variety of industrial process and robotics applications.
