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

TL;DR

This paper studies linear boundary systems with delays in state, input, and boundary output, establishing well-posedness, spectral properties, and regularity, with applications to population dynamics models involving birth and death processes.

Contribution

It introduces a comprehensive analysis of linear boundary systems with multiple delays, proving well-posedness and regularity, and applies these results to complex population models.

Findings

Proved well-posedness of delayed boundary systems.

Derived spectral properties under regularity conditions.

Established regularity and asymptotic behavior in population models.

Abstract

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 "$B_1(t)=\int_0^\infty\int_{-r}^0\beta_1(\sigma,a)u(t-\tau,a)d\sigma da$" and "$B_2(t)=\int_0^\infty\beta_2(a)u(t-\tau,a)da$", and the well-posedness of population systems with death caused by harvesting.