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Systems governed by impulsive differential inclusions on Hilbert spaces. (English) Zbl 0995.34053

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

$\stackrel{˙}{x}\left(t\right)\in Ax\left(t\right)+F\left(x\left(t\right)\right),\phantom{\rule{4pt}{0ex}}t\in \left[0,T\right]\setminus D,\phantom{\rule{4pt}{0ex}}x\left(0\right)={x}_{0},$
${\Delta }x\left({t}_{i}\right)\in {G}_{i}\left(x\left({t}_{i}\right)\right),\phantom{\rule{4pt}{0ex}}0={t}_{0}<{t}_{1}<\cdots <{t}_{n+1}=T,$

where $A$ is the infinitesimal generator of a ${C}_{0}$-semigroup, $F$ and the ${G}_{i}$’s are multivalued, $D=\left\{{t}_{1},{t}_{2},\cdots ,{t}_{n}\right\}\subseteq \left(0,T\right)$ and ${\Delta }x\left({t}_{i}\right)=x\left({t}_{i}+0\right)-x\left({t}_{i}\right)$. 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.

MSC:
 34G25 Evolution inclusions 34H05 ODE in connection with control problems 34A37 Differential equations with impulses 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 49N25 Impulsive optimal control problems 93B03 Attainable sets