# A differential algebraic approach of systems theory

Sette Diop

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Sette Diop. A differential algebraic approach of systems theory. The 28th International Conference on Applications of Computer Algebra (ACA'2023), July 17 – 21, 2023, Warsaw, Poland, The Applications of Computer Algebra Working Group, Jul 2023, Warsaw (POLAND), Poland. hal-04327570

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Submitted on 8 Dec 2023

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# A differential algebraic approach of systems theory

Sette Diop

Université Paris-Saclay, CNRS, CentraleSupélec,
Laboratoire des signaux et systèmes,
Gif sur Yvette, France

Differential algebraic geometry (and differential algebra) [1,2] was earlier recognized as particularly adapted as a language for the description of some of the systems theory problems. See the pioneering works by Jean-François Pommaret [3] and Michel Fliess [4]. In the latter paper the notion of invertibility, which was long studied in the control literature, was given a better clarification. And in the [5] differential algebraic elimination theory was invoked as, not only a better description of questions in the control literature, but a constructive answer, too. One of the fundamental notions of systems theory, that of observability, was also given a differential algebraic geometry description which clarified many aspects of that questions [6]. This contribution is a tentative comprehensive expose of differential algebraic geometry known answers to some of the systems theory questions. The latter include previously mentioned ones, and notions of invariants, structural issues, and constructivity.

## References

[1] J. F. RITT, *Differential Algebra*. American Mathematical Society, Providence, RI, 1950.

[2] E. R. KOLCHIN, *Differential Algebra and Algebraic Groups*. Academic Press, New York, 1973.

[3] J. F. POMMARET, *Differential Galois Theory*. Gordon and Breach, New York, 1983.

[4] M. FLIESS, A note on the invertibility of nonlinear input-output differential systems. *Systems Control Lett.* **4**, 17–32 (1986).

[5] S. DIOP, Elimination in control theory. *Math. Control Signals Systems* **8**(2), 147–151 (1991).

[6] S. DIOP, From the geometry to the algebra of nonlinear observability. In *Contemporary Trends in Nonlinear Geometric Control Theory and its Applications*, A. Anzaldo-Meneses, B. Bonnard, J. P. Gauthier and F. Monroy-Perez (eds.), 305–345. World Scientific Publishing Co., Singapore, 2002.