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Universal Regular Control for Generic Semilinear Systems by Bochi, Jairo; Gourmelon, Nicolas is a scholarly article available to read on EtoBox.

What is Universal Regular Control for Generic Semilinear Systems about?

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 $\mathbb{R}P^{d-1}$, inputs $u_t$ are in a manifold $U$ of arbitrary finite dimension, and $A \colon U \to GL(d,\mathbb{R})$ is a differentiable mapping. An input sequence $(u_0,\ldots,u_{N-1})$ is called universally regular if for any initial state $\xi_0 \in \mathbb{R}P^{d-1}$, 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 $(u_0, \dots, u_0)$. 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 $U$. 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 t

Author

Bochi, Jairo; Gourmelon, Nicolas

Published

2012

Language

EN