Abstract:
A bilinear control system on (d+1)-dimensional Euclidean space induces a
control-affine system on the d-dimensional real projective space P^d. The
well-known theorem of Selgrade about linear flows with chain transitive
base flow yields a description of the chain control sets on P^d. In this
talk, I show that each chain control set has a partially hyperbolic
structure in a skew-product sense. The uniformly hyperbolic ones are
easily characterized via the dimensions of the associated Selgrade
bundles. This kind of uniform hyperbolicity is interesting, because
despite the fact that the system is continuous in time, there is no
central subbundle present (which can only happen in time-dependent
systems). For the invariance entropy of the hyperbolic chain control sets,
a formula is available (arXiv:1408.2416). I will sketch the proof of this
formula, which reveals an interesting analogy between the classical
entropy theory in dynamical systems and the theory of feedback entropy in
control: The entropy of a uniformly hyperbolic set is determined by the
periodic trajectories inside the set. More precisely, in the case of
invariance entropy, there is no cheaper strategy (in the sense of lower
data rate) to keep the system inside the hyperbolic set than by
stabilization at periodic orbits.