Controllability of integrodifferential systems in Banach spaces. (Q5931696)

The authors study the controllability of integrodifferential systems in Banach spaces of the form NEWLINE\[NEWLINEx'(t)= Ax(t)+(Bu)(t)+f\left(t,x(t),\int_ {0}^ {t}g(t,s,x(s))\,ds\right), \quad t\in J:=[0,b],NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=x_ 0,NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a strongly continuous semigroup \(T(t), t\geq 0\) in a Banach space \(X,\) \(f: J\times X\times X\to X,\) \(g: \Delta\times X\to X,\) are given functions, \(\Delta=\{(t,s): 0\leq s\leq t\leq b\},\) and the control function \(u(\cdot)\) is given in \(L^ 2(J,U),\) a Banach space of admissible control functions, with \(U\) as a Banach space. Finally, \(B\) is a bounded linear operator from \(U\) to \(X.\) The results are obtained by using Schaefer's fixed point theorem.