Differential Correspondences and Control Theory

When a differential field K having n commuting derivations is given together with two finitely generated extensions L and M of K, an important problem in algebra to exhibit common extension N order define the new L?M smallest (L,M ) ? containing both M. Such result allows generalize use complex numbers classical algebra. Having now modules over non-commutative ring D = [d1,...,dn] [d] operators coefficients we may similarly look for module L+M. This exactly situation met linear or non-linear OD PD control theory by selecting inputs outputs among variables. However, many recent books papers, have shown that controllability was built-in property system, not depending on choice outputs. The purpose this paper thus revisit showing specific importance previous problems part plaid cases parametrization system. An tool will be study correspondences, modern name what called Bäcklund during last century, namely elimination groups variables systems nonlinear equations. main difficulty homological using noncommutative localization as way symbolic calculus style Heaviside Mikusinski. Finally, when D-module, first time fact system R homK (M,K) D-module Spencer operator acting sections, avoiding behaviours, trajectories signal spaces purely formal way, contrary few works difficult subject.