Solving inverse illumination problems with Liouville's equation

We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can be applied. We show that this approach has great potential by a simple numerical example of an ideal lens