# Recent progress in an algebraic analysis approach to linear systems

## Abstract

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

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## Notes

1. The assumption that $$D$$ is a left Noetherian domain actually implies (cf. McConnell and Robson 2000, Thm. 2.1.15) that $$D$$ satisfies the left Ore condition, i.e., every pair of non-zero elements of $$D$$ has a non-zero common left multiple, a property shared by all Ore algebras (cf. also Proposition 3.7).

2. As mentioned earlier, every pair of non-zero elements of $$D$$ has a non-zero common left multiple, which implies, e.g., that $${{\mathrm{t}}}(M)$$ is a left $$D$$-module.

3. We thank Burt Totaro for a clarification of this statement and the reference Cohn (2006).

4. In concrete examples we may assume that the matrix $$R$$ is defined over a computable subalgebra of $$D$$.

5. For Janet bases the divisibility relation of terms is actually a restriction of the usual divisibility relation. The concept of Janet division (or, more generally, of an involutive division) determines for each monomial the set of indeterminates which may be multiplied from the left to the monomial when it is used for reduction of other terms. As a consequence, every element of $$D^{1 \times q} \, R$$ has a unique representation as left $$D$$-linear combination of the Janet basis elements taking their so-called multiplicative variables into account. For a survey on the algorithmic development of this efficient alternative to Buchberger’s algorithm we refer to Gerdt (2005).

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Robertz, D. Recent progress in an algebraic analysis approach to linear systems. Multidim Syst Sign Process 26, 349–388 (2015). https://doi.org/10.1007/s11045-014-0280-9

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### Keywords

• Systems of linear functional equations
• Systems theory
• Control theory
• Algebraic analysis
• Homological algebra
• Linear partial differential equations
• Janet bases
• Gröbner bases