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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

x ˙(t)Ax(t)+Fx ( t ),t[0,T]D,x(0)=x 0 ,
Δx(t i )G i x ( t i ),0=t 0 <t 1 <<t n+1 =T,

where A is the infinitesimal generator of a C 0 -semigroup, F and the G i ’s are multivalued, D={t 1 ,t 2 ,,t n }(0,T) and Δx(t i )=x(t i +0)-x(t i ). 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:
34G25Evolution inclusions
34H05ODE in connection with control problems
34A37Differential equations with impulses
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
49N25Impulsive optimal control problems
93B03Attainable sets