Complete stabilizability and stability radius of time-varying Sturm-Liouville systems

Abstract

There are many physical systems which can be modeled by a time-varying Sturm-Liouville system i.e. a system is inducted by the negative of a time-varying Sturm-Liouville operator. The system is a time-varying Riesz-spectral system. The goals of this paper are to analyze exact controllability, stabilizability, and stability radius of the time-varying Riesz-spectral systems. The steps of analysis of methods of the paper are to define a time-varying Sturm-Liouville system, to construct sufficient and necessary conditions for exact controllability and stabilizability, and to find stability radius. The analysis uses a strongly continuous quasi-semigroup approach. The results show that sufficient and necessary conditions for exact controllability, exact null controllability, and stabilizability can be constructed. Moreover, in the time-varying Sturm-Liouville systems, the exact null controllability implies the complete stabilizability. This is parallel with infinite dimensional of time-invariant systems. Also, the lower bound for the stability radius of the time-varying Sturm-Liouville systems is a reciprocal of the norm of the input-output operator. An example is provided to confirm the results.