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Zbl 1171.34052
Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions.
(English)
[J] J. Differ. Equations 246, No. 10, 3834-3863 (2009). ISSN 0022-0396

The authors present the existence of integral solutions and extremal integral solutions for the following problem \align y'(t)\in Ay(t)+F(t,y_t), &\quad t\in [0,T],\\ \Delta y|_{t=t_k}\in I_k(y(t_k)), &\quad k=1,\ldots,m, \\ y(t)=\varphi(t),&\quad t\in[-r,0], \endalign where $F: [0,T]\times {\cal D}\to {\cal P}(E)$ are a multivalued maps, ${\cal P}(E)$ is the family of all nonempty subsets of $E$, $A: D(A)\subset E\to E$ is a nondensely defined closed linear operator on $E$, $0<r<\infty$, $0=t_{0}<t_1<\cdots<t_m<t_{m+1}=T$, $$\multline {\cal D}= \{\psi:[-r,0]\to E: \psi\text { is continuous everywhere except for a finite number}\\ \text {of points } \bar{t}\text { at which }\psi(\bar{t}^-)\text { and }\psi(\bar{t}^+)\text { exist and satisfy }\psi(\bar{t}^-)=\psi(\bar{t})\}, \endmultline$$ $\varphi\in {\cal D},$ $I_k\in E\to {\cal P}(E)$ $(k=1,\ldots,m)$, $\Delta y|_{t=t_k}= y(t_k^+)- y(t_k^-)$, $y(t_k^+)= \lim_{h\to 0^+}y(t_k+h)$ and $y(t_k^-)= \lim_{h\to 0^+} y(t_k-h)$ stand for the right and the left limits of $y(t)$ at $t=t_k$, respectively. For any function $y$ defined on $[-r,b]$ and any $t\in J$, $y_t$ refers to the element of ${\cal D}$ such that $$y_t(\theta)=y(t+\theta),\quad \theta\in[-r,0];$$ thus the function $y_t$ represents the history of the state from time $t-r$ up to the present time $t$. Also the controllability of above problem are investigated. An examples is presented.
[Abdelghani Ouahab (Sidi Bel Abbes)]
MSC 2000:
*34K45 Equations with impulses
34K30 Functional-differential equations in abstract spaces
34K35 Functional-differential equations connected with control problems
93B05 Controllability

Keywords: nondensely defined operator; Impulsive semilinear differential inclusions; Fixed point; Integral solutions; Extremal solutions; Controllability

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