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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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