Characteristics of invariancy for the attainability set of a control system

We study characteristics associated with invariancy or weak invariancy of a given set $\mathfrak M\doteq\bigl\{(t,x)\in [0,+\infty)\times\mathbb R^n: x\in M(t)\bigr\}$ with respect to a control system $\dot x=f(t,x,u)$ on a finite time interval. One of such characteristics is relative frequency ${\rm freq}_{[\tau,\tau+\vartheta]}(D,M)$ of containing the attainability set $D(t,X)$ of this system in the set $\mathfrak M$ on a segment $[\tau,\tau+\vartheta]$. This characteristic is equal to the quotient of the Lebegues measure of those $t$ from $[\tau,\tau+\vartheta]$ at which $D(t,X)\subseteq M(t)$ to the length of the given segment. Other characteristic, ${\rm freq}_{\vartheta}(D,M)\doteq\inf\limits_{\tau\geqslant\,0}\, {\rm freq}_{[\tau,\tau+\vartheta]}(D,M)$ displays uniformity of containing the attainability set $D(t,X)$ in the set $\mathfrak M$ on a segment of the fixed length $\vartheta$. We prove theorems about estimation and calculation of these characteristics for various multivalued functions $M(t)$ and $D(t,X)$. In particular, we receive equalities for ${\rm freq}_{T}(D, M)$ if the function $M(t)$ is periodic with a period $T$ and the function $D(t, X)$ satisfies the inclusion $D(t+T,X)\subseteq D(t,X)$ for all $t\geqslant 0$. We consider examples of calculation and estimations of these characteristics.