A Class of Linear Boundary Systems with Delays in State, Input and Boundary Output

In this paper, we consider a class of linear boundary systems with delays in state, input and boundary output. We prove the well-posedness and derive some spectral properties of linear system with delayed boundary feedback under some regularity conditions. Moreover, we show the regularity of linear boundary systems with delays in state and boundary output. With the above results, the regularity of linear boundary systems with delays in state, input and boundary output is verified. As applications, we prove the well-posedness and the asymptotic behavior of population systems with bounded and unbounded birth processes “$B_1(t)=\int_0^\infty\int_{-r}^0\beta_1(\sigma,a)u(t-\tau,a)d\sigma da$\mathrm{"}and\mathrm{"}$B_2(t)=\int_0^\infty\beta_2(a)u(t-\tau,a)da$”, and the well-posedness of population systems with death caused by harvesting.