### On a controllability problem for systems governed by semilinear functional differential inclusions in Banach spaces

DOI: http://dx.doi.org/10.12775/TMNA.2000.011

#### Abstract

For a Banach space $E$, a given pair $(\overline p, \overline x)\in[0,a]\times E$,

and control system governed by a semilinear functional differential includion of the form

$$

x' (t)\in Ax(t) +F(t, x(t), Tx)

$$

the existence of a mild trajectory of $x(t)$ satisfying the condition

$x(\overline p)=\overline x$ is considered. Using topological methods we develop

an unified approach to the cases when a multivalued nonlinearity $F$ is

Carathéodory upper semicontinuous or almost lower semicontinuous

and an abstract extension operator $T$ allows to deal with variable and infinite

delay. For the Carathéodory case, the compactness of the solutions set and,

as a corollary, an optimization result are obtained.

and control system governed by a semilinear functional differential includion of the form

$$

x' (t)\in Ax(t) +F(t, x(t), Tx)

$$

the existence of a mild trajectory of $x(t)$ satisfying the condition

$x(\overline p)=\overline x$ is considered. Using topological methods we develop

an unified approach to the cases when a multivalued nonlinearity $F$ is

Carathéodory upper semicontinuous or almost lower semicontinuous

and an abstract extension operator $T$ allows to deal with variable and infinite

delay. For the Carathéodory case, the compactness of the solutions set and,

as a corollary, an optimization result are obtained.

#### Keywords

Controllability; functional differential inclusion; condensing multimap; fixed point

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