### Minimal data rates and entropy for control problems, Part 2

Christoph Kawan

Abstract:
In the second part, I will concentrate on nonlinear systems given by
differential equations. In general, for such systems no formulas for the
invariance entropy are known. However, under additional control-theoretic
assumptions, is is possible to formulate an upper estimate in
terms of Lyapunov exponents that has some similarity with the integral
formula for the topological entropy of smooth maps, proved in
[O.S. Kozlovski. An Integral Formula for Topological Entropy of
C^{\infty}-maps, Ergodic Theory Dyn. Syst. 18 (1998), 405-424.
Without any additional assumptions, a general lower bound of the
entropy is given by a uniform escape rate from the controlled
invariant set. Under the assumption of uniform hyperbolicity, a
skew-product version of the Bowen-Ruelle volume lemma can be used to
relate this escape rate to the unstable volume growth and a quantity
similar to topological entropy. In particular examples, the upper and
lower estimates can be merged into a formula.