Controllability of second-order neutral functional differential inclusions in Banach spaces. (English) Zbl 1025.93006

The authors study the controllability of neutral functional differential inclusions in Banach spaces of the form \[ {d\over dt}\{y'(t)-g(t,y_t)\}\in Ay(t)+Bu(t)+F(t,y_t), \quad t\in J:=[0,T], \]
\[ y_0=\phi, \quad y'(0)=x_0, \] where \(F: J\times C([-r,0],E)\to 2^E\) is a bounded, closed, convex multivalued map, \(g: J\times C([-r,0],E)\to E\) is a given function, \(\phi \in C([-r,0],E),\) \(x_0\in E,\) \(A\) is the infinitesimal generator of a strongly continuous cosine family \(\{C(t): t\in R\}\) in a real Banach space and the control function \(u(\cdot)\) is given in \(L^2([0,T],U),\) a Banach space of admissible control functions with \(U\) as a Banach space. Finally, \(B\) is a bounded linear operator from \(U\) to \(E.\) The results are obtained by using a fixed point theorem for condensing upper semicontinuous mappings due to Martelli.
Reviewer’s remark: In (H7) the constant \(M_0\) depends on \(y,\) so in the conclusion of step 5 \(L\) depends also on \(y.\) To avoid this, condition (H7) must be changed to \(\int_{1}^{+\infty}{ds \over s+\Psi(s)}=+\infty\).


93B05 Controllability
93C23 Control/observation systems governed by functional-differential equations
93C25 Control/observation systems in abstract spaces
34K40 Neutral functional-differential equations
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