# Universal regular control for generic semilinear systems

- 153 Downloads

## Abstract

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\), where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\), inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\), the derivative of the time-\(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\). Our main result states that generically in the space of such systems, for sufficiently large \(N\), all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\)-open and \(C^\infty \)-dense set of maps \(A\), and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\). We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.

## Keywords

Discrete-time systems Semilinear systems Bilinear systems Universal regular control## Notes

### Acknowledgments

We are grateful for the hospitality of Institute Mittag–Leffler, where this work begun to take form. We thank R. Potrie, L. San Martin, S. Tikhomirov, and C. Tomei for valuable discussions. We thank the referees for corrections, references to the literature, and other suggestions that helped to improve the exposition.

## Supplementary material

## References

- 1.Azoff EA (1986) On finite rank operators and preannihilators, vol 64, no 357. Mem. Amer. Math. Soc.Google Scholar
- 2.Benedetti R, Risler J-J (1990) Real algebraic and semi-algebraic sets. Hermann, PariszbMATHGoogle Scholar
- 3.Bochnak J, Coste M, Roy M-F (1998) Real algebraic geometry. Springer, BerlinCrossRefzbMATHGoogle Scholar
- 4.Colonius F, Kliemann W (1993) Linear control semigroups acting on projective space. J Dynam Differ Eqs 5(3):495–528CrossRefzbMATHMathSciNetGoogle Scholar
- 5.Colonius F, Kliemann W (2000) The dynamics of control. Birkhäuser, Boston, MACrossRefGoogle Scholar
- 6.Elliott DL (2009) Bilinear control systems. Springer, DordrechtCrossRefzbMATHGoogle Scholar
- 7.Gibson CG, Wirthmüller K, du Plessis AA, Looijenga EJN (1976) Topological stability of smooth mappings. Lecture Notes in Mathematics, vol. 552. Springer, BerlinGoogle Scholar
- 8.Harris J (1992) Algebraic geometry: a first course. Springer, New YorkCrossRefzbMATHGoogle Scholar
- 9.Horn RA, Johnson CR (1994) Topics in matrix analysis. Corrected reprint of the 1991 original. Cambridge University Press, CambridgeGoogle Scholar
- 10.Levi FW (1942) Ordered groups. Proc Indian Acad Sci 16:256–263zbMATHGoogle Scholar
- 11.Roman S (2008) Advanced linear algebra, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
- 12.Shafarevich IG (1994) Basic algebraic geometry, vol 1, 2nd edn. Springer, BerlinCrossRefGoogle Scholar
- 13.Sontag ED (1992) Universal nonsingular controls. Syst Control Lett 19(3):221–224 Errata: Ibid, 20 (1993), no. 1, 77CrossRefzbMATHMathSciNetGoogle Scholar
- 14.Sontag ED, Wirth FR (1998) Remarks on universal nonsingular controls for discrete-time systems. Syst Control Lett 33(2):81–88CrossRefzbMATHMathSciNetGoogle Scholar
- 15.Wirth F (1998) Dynamics of time-varying discrete-time linear systems: spectral theory and the projected system. SIAM J Control Optim 36(2):447–487CrossRefzbMATHMathSciNetGoogle Scholar