Universal regular control for generic semilinear systems

TL;DR

This paper proves that for generic discrete-time projective semilinear systems, most constant input sequences are universally regular when the input length is sufficiently large, with only a finite set of exceptions.

Contribution

It establishes the generic universal regularity of constant inputs in semilinear control systems and characterizes the exceptional cases using algebraic geometry and matrix analysis.

Findings

Most constant inputs are universally regular for large N.

The set of non-regular inputs is finite and well-characterized.

The result holds for a dense and open set of system maps.

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 $\mathbb{R}P^{d-1}$, 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 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.