TY - JOUR

T1 - Universal regular control for generic semilinear systems

AU - Bochi, Jairo

AU - Gourmelon, Nicolas

N1 - Funding Information:
J. B. was partially supported by CNPq, Université Bordeaux I, Rede Franco–Brasileira em Matemática, and FAPERJ. N. G. was partially supported by FAPERJ.
Publisher Copyright:
© 2014, Springer-Verlag London.

PY - 2014/12

Y1 - 2014/12

N2 - We consider discrete-time projective semilinear control systems ξt+1=A(ut)·ξt, where the states ξt are in projective space RPd-1, inputs ut are in a manifold U of arbitrary finite dimension, and A: u→GL(d, R) is a differentiable mapping. An input sequence (u0, …, uN-1) is called universally regular if for any initial state ξ0∈RPd-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 (u0, …, u0). 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 C2-open and C∞-dense set of maps A, and N only depends on d and on the dimension of 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.

AB - We consider discrete-time projective semilinear control systems ξt+1=A(ut)·ξt, where the states ξt are in projective space RPd-1, inputs ut are in a manifold U of arbitrary finite dimension, and A: u→GL(d, R) is a differentiable mapping. An input sequence (u0, …, uN-1) is called universally regular if for any initial state ξ0∈RPd-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 (u0, …, u0). 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 C2-open and C∞-dense set of maps A, and N only depends on d and on the dimension of 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.

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U2 - 10.1007/s00498-014-0126-x

DO - 10.1007/s00498-014-0126-x

M3 - Article

AN - SCOPUS:84919877485

SN - 0932-4194

VL - 26

SP - 481

EP - 518

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

IS - 4

ER -