Abstract
In this chapter, we start to investigate nonlinear systems. Applying similar techniques as in the linear case, we can derive at least rough bounds for the invariance entropy. In particular, we can show that under appropriate compactness assumptions the outer invariance entropy of an admissible pair (K, Q) of a smooth system is finite, and an upper bound can be expressed in terms of a Lipschitz constant of the transition map and the upper capacitive dimension of K.
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Notes
- 1.
This corollary asserts in particular that for every point x on a Riemannian manifold (M, g) there exists a neighborhood U of x and \(\varepsilon> 0\) such that for all y ∈ U the map exp y is defined on \(B(0_{y},\varepsilon ) \subset T_{y}M\).
- 2.
Note that the eigenvalues depend continuously on the operator and continuity of \(\nabla \tilde{F}_{v}(z)\) follows from the assumption that the derivative of F with respect to the first argument exists and is continuous as a function of (x, u).
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Kawan, C. (2013). General Estimates. In: Invariance Entropy for Deterministic Control Systems. Lecture Notes in Mathematics, vol 2089. Springer, Cham. https://doi.org/10.1007/978-3-319-01288-9_4
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