Decompositions and bang-bang properties

TL;DR

This paper investigates the conditions under which minimal time and norm control problems exhibit bang-bang properties in systems lacking null controllability or backward uniqueness, considering parameter dependencies and weak controllability assumptions.

Contribution

It provides new insights into the bang-bang property for minimal control problems in systems without null controllability, depending on parameters and weak controllability assumptions.

Findings

Bang-bang property depends on parameters for minimal time and norm problems.

Weak controllability and unique continuation influence bang-bang behavior.

Results apply to systems without null controllability or backward uniqueness.

Abstract

In this paper, minimal time and minimal norm control problems are studied. The target sets considered are the origin of state spaces and controls are point-wisely bounded functions. The system stuided in this paper is assumed to have no the null controllability or the backward uniqueness property. In this study, minimal time and minimal norm control problems depend on two parameters, respectively. Whether these problems hold the bang-bang property also depend on the parameters. We study the bang-bang property for different parameters for minimal time and minimal norm control problems, by assuming some kinds of weak controllability and unique continuation property. These two properties automatically hold for general time-invariant finitely dimensional controlled systems.