Controllability of neutral functional differential equations with infinite delay (Q656175)

The paper deals with the partial neutral functional differential equation with infinite delay \[ \left\{\begin{aligned} &\dfrac{\partial} {\partial t}{\mathcal D}x_t=A{\mathcal D}x_t+Cu (t)+F (t,x_t),\;t\geq 0,\\ &x_0=\phi\in {\mathcal B}, \end{aligned}\right.\tag{1} \] where the state variable \(x (\cdot)\) takes values in a Banach space \( (E,|\cdot|)\) and the control \(u (\cdot)\) is given in \(L^2 ([0,T],U)\), \(T>0\), the Banach space of admissible control functions, with \(U\) a Banach space. Here, \(C:U\to E\) is a bounded linear operator; \(A:D (A)\subseteq E\to E\) is a linear operator on \(E\), non-densely defined, which satisfies the resolvent estimates of the Hille-Yosida theorem; \({\mathcal B}\) is the phase space of functions mapping \( (-\infty,0]\) into \(E\); \({\mathcal D}:{\mathcal B}\to E\) is a bounded linear operator defined by \({\mathcal D}\varphi=\varphi (0)-{\mathcal D}_0\varphi\), \(\varphi\in {\mathcal B}\), where \({\mathcal D}_0:{\mathcal B}\to E\) is a bounded linear operator; \(F:\mathbb{R}_+\times {\mathcal B}\to E\) is a nonlinear continuous mapping; and \(x_t (\theta)=x (t+\theta)\) for \(t\in [0,T]\), \(\theta\in (-\infty,0]\). By using the integrated semigroups theory and the Banach fixed point theorem, the authors give sufficient conditions for controllability of problem(1). An application which illustrates the obtained result is finally presented.