Computer algebraic methods for the structural analysis of linear controlsystems

Let D = K[∂1, . . . , ∂n] denote the ring of linear partial differential operators with constant (real or complex) coefficients, and let A = C(R,K). A multidimensional behavioral system is defined as the smooth solution set of a homogeneous system of linear constant-coefficient PDE, that is, B = kerA(R) = {w ∈ A | Rw = 0} for some R ∈ D . The following two properties are fundamental in systems theory: B is autonomous if it has no free variables (inputs), or equivalently, if there exists no 0 = w ∈ B with compact support [5]. B is controllable if it is parametrizable, i.e., it has an image representation B = kerA(R) = imA(M) for some M ∈ D. Equivalently, for all w1, w2 ∈ B and for all open sets U1, U2 ⊂ R n with U1 ∩ U2 = ∅, there exists w ∈ B such that [5]