  We introduce a notion of invariance entropy for uncertain control systems,
which is, roughly speaking, the exponential growth rate of "branches" of
"trees" that are formed by controls and are necessary to achieve invariance of
controlled invariant subsets of the state space. This entropy extends the
invariance entropy for deterministic control systems introduced by Colonius and
Kawan (2009). We show that invariance feedback entropy, proposed by Tomar,
Rungger, and Zamani (2020), is bounded from below by our invariance entropy. We
generalize the formula for the calculation of entropy of invariant partitions
obtained by Tomar, Kawan, and Zamani (2020) to quasi-invariant-partitions.
Moreover, we also derive lower and upper bounds for entropy of a
quasi-invariant-partition by spectral radii of its adjacency matrix and
weighted adjacency matrix. With some reasonable assumptions, we obtain explicit
formulas for computing invariance entropy for uncertain control systems and
invariance feedback entropy for finite controlled invariant sets.
