*(English)*Zbl 0995.34053

The author considers systems in a Hilbert space governed by impulsive differential inclusions of the form

where $A$ is the infinitesimal generator of a ${C}_{0}$-semigroup, $F$ and the ${G}_{i}$’s are multivalued, $D=\{{t}_{1},{t}_{2},\cdots ,{t}_{n}\}\subseteq (0,T)$ and ${\Delta}x\left({t}_{i}\right)=x({t}_{i}+0)-x\left({t}_{i}\right)$. The ${G}_{i}$’s being set-valued allow for systems in which the jump sizes are uncertain and also problems in which the jump sizes are chosen from a control set. The existence of solutions is proven under the assumptions $F$ is closed convex-valued, satisfies a growth condition and is upper semicontinuous; the ${G}_{i}$’s are closed bounded-valued and map closed and bounded sets into closed and bounded sets; plus several other conditions. Under the additional assumption that the semigroup generated by $A$ is compact, the attainable set is characterized. Special cases of this problem in which $F$ or the ${G}_{i}$’s are singleton-valued are also considered. Finally, the author mentions several open questions in control theory relating to these results.