Monge Parametrizations and Integration of Rectangular Linear Differential Systems

Most algorithms for computing local or global solutions of linear differential systems handle only the case of square systems. However, in many applications such as control theory, systems that appear are generally non-square. In this paper, we use a constructive algebraic analysis approach to reduce the integration of a rectangular linear ordinary differential system to that of a square system. In this way, algorithms for computing a given type of solutions of square systems can also be used to provide analogous solutions of rectangular systems. This method has already been sketched in [4] for computing regular formal solutions. An implementation is provided. 1 Algebraic analysis approach to linear systems theory A linear functional system (e.g., linear system of ODEs, PDEs, OD time-delay equations, difference equations) can always be written as Rη = 0 where R ∈ Dq×p is a q×p matrix with coefficients in a noncommutative polynomial ring D of functional operators (e.g., OD or PD operators, time-delay operators, shift operators, difference operators) and η is a vector of unknown functions. More precisely, if F denotes a left D-module, namely, ∀ d1, d2 ∈ D, ∀ η1, η2 ∈ F : d1 η1 + d2 η2 ∈ F , then we can consider the linear functional system kerF (R.) = {η ∈ F | Rη = 0}, i.e., the abelian group formed by the solutions in F of the linear system Rη = 0. The algebraic analysis approach to mathematical systems theory (see [5, 10, 13] and references therein) is based on the fact that the linear functional system kerF (R.) can be studied by means of the left D-module M = D1×p/(D1×q R) given by the following finite presentation: D1×q .R −→ D1×p π −→ M −→ 0 λ = (λ1 . . . λq) 7−→ λR.