Parametrization for control of linear PDE systems

Pseudo-differential operators (ψDO) form an extension of combined partial differential equations (PDE) and boundary-value (trace) operators. Applicability of ψDOs lies in the fact that they form an algebra like in the case of ordinary differential operators. Linear pseudo-differential operators are considered. Parametrization, which roughly corresponds to flatness of ordinary differential systems, means that the solution variables on the domain and boundary, the independent controls and output variables can be expressed as functions of some other functions (parameters). In the case of classical PDE systems partial derivations and trace and potential (singular Green) operators are needed to obtain resulting control-solution pairs. Here we formulate a general control system, present two theorems on parametrization, and study the two approaches to parametrization using the same example system. Some ideas to extend these results to nonlinear ψDO systems are outlined, too.