Eventual norm continuity for neutral semigroups on Banach spaces (Q615921)

The paper deals with the infinite-dimensional linear neutral equation \[ \begin{cases}\frac{d}{dt}[x(t)-Kx_t]=A[x(t)-Kx_t]+Lx_t,\,\,\,t\geq 0,\\ \lim_{t\to 0^+}[x(t)-Kx_t]=z\in X,\\ x_0=\varphi,\end{cases} \] where \(X\) is a Banach space, the operator \(A:D(A)\subset X\to X\) generates a strongly continuous semigroup \(T=(T(t))_{t\geq 0}\) on \(X\), \(L,K\in {\mathcal L}(W^{1,p}([-r,0],X),X)\), \(1<p<\infty\) and \(r\in (0,+\infty)\), \(z\in X\), \(\varphi\in L^p([-r,0],X)\) and \(x_t\) is the history function on \([-r,0]\) at \(t\) defined by \(x_t(\theta)=x(t+\theta)\) for \(\theta\in [-r,0]\). The linear operators \(L\) and \(K\) are given by the integrals \[ L\psi= \int_{-r}^0d\mu(s)\psi(s)\text{ and }K\psi= \int_{-r}^0d\eta(s)\psi(s) \] for \(\psi\in W^{1,p}([-r,0],X)\), where \(\mu,\,\eta:[-r,0]\to {\mathcal L}(X)\) are functions of bounded variation and continuous at zero. The authors prove that the solution semigroup of the above problem is eventually norm continuous (eventually compact) whenever the semigroup \(T\) is immediately norm continuous (respectively, immediately compact). The approach is based on a general perturbation theorem obtained from closed-loop systems of infinite-dimensional control systems with unbounded control and observation operators. An example of a partial differential equation of neutral type which illustrates the obtained results is also presented.