Formal elimination for multidimensional systems and applications to control theory

Following Douglas's ideas on the inverse problem of the calculus of variati ons, the purpose of this article is to show that one can use formal integra bility theory to develop a theory of elimination for systems of partial dif ferential equations and apply it to control theory. In particular, we consi der linear systems of partial differential equations with variable coeffici ents and we show that we can organize the integrability conditions on the c oefficients to build an "intrinsic tree". Trees of integrability conditions naturally appear when we test the structural properties of linear multidim ensional control systems with some variable or unknown coefficients (contro llability, observability, invertibility,...) or for generic linearization o f nonlinear systems.