Axiomatic system theory and optical images

Several practical problems which arise in optics are related to achieving a desired three-dimensional signal distribution inside a bounded spatial domain. The author deals with harmonic time dependence he finds an example in integrated circuit microfabrication; if time dependence is arbitrary, pulse compression is thought to be in a dispersive media. To all of these problems there is a unifying approach based on axiomatic system theory. This theory is well-known to rely on the state space formulation. The way in which the input acts on the state is quantified by the `controllability' concept. Similarly `observability' relates output data to the state. Strictly related to this approach is the `optimal control problem', where the task is to find an input which minimizes a functional consisting of two addenda: a physical term comparing the obtained output with the desired one by some quadratic criterion, and an economical term related to the cost of a given input. These concepts are widely used in signal processing, control theory, etc. Their application to optical problems requires them to be extended to distributed parameter systems. For the cases discussed in the text controllability results will be given and optimal control problems will be stated.