Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems ξt`1 “ Aputq ̈ ξt, where the states ξt are in projective space RPd ́1, inputs ut are in a manifold U of arbitrary dimension, and A : U Ñ GLpd,Rq is a differentiable mapping. An input sequence pu0, . . . , uN ́1q is called universally regular if for any initial state ξ0 P 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 pu0, . . . , u0q. Our main result states that for generic such control systems, all constant inputs of sufficient length N are universally regular, except for a discrete set. More precisely, the conclusion holds for a C2-open and C8-dense set of maps A. 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.