Absorbed and nonabsorbed points of sets of attainability of differential inclusions, and generalized Hamilton–Jacobi equations

The problems considered in this article are in a certain sense alternative to the problem of invariance, namely, they concern conditions under which a particular point $x^*$ belonging at time $t^*$ to the set of attainability $X(t^*)$ of an autonomous differential inclusion $\dot x\in F(x)$ does or does not belong to the sets of attainability $X(t)$ for all $t$ in some sufficiently small interval adjacent to $t^*$ from the right or left. The conditions obtained are used to derive generalized equations of Hamilton–Jacobi type describing the boundaries of the phase and integral funnels of the differential inclusion, as well as the optimal result function and the fast-action time in an optimal control problem.