Exact Null Controllability, Stabilizability, and Detectability of Linear Nonautonomous Control Systems: A Quasisemigroup Approach

Abstract

In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated $C_{\mathrm{0}}$-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.