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Invariance entropy for a class of partially hyperbolic sets. (English) Zbl 1401.93135
Summary: Invariance entropy is a measure for the smallest data rate in a noiseless digital channel above which a controller that only receives state information through this channel is able to render a given subset of the state space invariant. In this paper, we derive a lower bound on the invariance entropy for a class of partially hyperbolic sets. More precisely, we assume that $$Q$$ is a compact controlled invariant set of a control-affine system whose extended tangent bundle decomposes into two invariant subbundles $$E^+$$ and $$E^{0-}$$ with uniform expansion on $$E^+$$ and at most subexponential expansion on $$E^{0-}$$. Under the additional assumptions that $$Q$$ is isolated and that the $$u$$-fibers of $$Q$$ vary lower semicontinuously with the control $$u$$, we derive a lower bound on the invariance entropy of $$Q$$ in terms of relative topological pressure with respect to the unstable determinant. Under the assumption that this bound is tight, our result provides a first quantitative explanation for the fact that the invariance entropy does not only depend on the dynamical complexity on the set of interest.

##### MSC:
 93C62 Digital control/observation systems 93C10 Nonlinear systems in control theory 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 93C15 Control/observation systems governed by ordinary differential equations 93C41 Control/observation systems with incomplete information 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37D30 Partially hyperbolic systems and dominated splittings
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