  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.
