Controllability and Observability of Nonautonomous Riesz-Spectral Systems

Abstract

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a 0 -quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.

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

“…The exact null controllability is special case of exact controllability. A part of the results of exact controllability in the linear non-autonomous control systems can be studied in our previous works [14].…”

“…where denotes an element of the dual space ; see [14,18]. The complete stabilizability is an extension of the concept of stabilizability in Definition 16.…”

“…As described in our recent work [14], a quasisemigroup is an alternative approach that can be implemented to investigate the non-autonomous systems (1). This approach was first introduced by Leiva and Barcenas [15].…”

Transfer Function, Stabilizability, and Detectability of Non-autonomous Riesz-spectral Systems

“…Symbol (A(t), B(t), −) and (A(t), −, C(t)) denote the state linear system (1) if C(t) = 0 and B(t) = 0, respectively. In particular, this paper shall investigate stability of the state linear system (1) where each A(t) is a generalized Riesz-spectral operator [1] using transfer function, stabilizability, and detectability . The following explanations give some reasons why these investigations are important for the systems.…”

“…Even Barcenas et al [22] have characterized the controllability of the non-autonomous control systems using the quasi semigroup approach, although it is still limited to the autonomous controls. In particular, Sutrima et al [1] characterized the controllability and observability of non-autonomous Riesz-spectral systems. The references explain that the transfer function, stabilizability, and detectability of the control systems are important indicators for the stability.…”