Decompositions and bang-bang properties

Abstract

We study the bang-bang properties of minimal time and minimal norm control problems (where the target set is the origin of the state space and the controlled system is linear and time-invariant) from a new perspective. More precisely, we study how the bang-bang property of each minimal time (or minimal norm) problem depends on a pair of parameters $(M, y_0)$ (or $(T,y_0)$ ), where $M>0$ is a bound of controls and $y_0$ is the initial state (or $T>0$ is an ending time and $y_0$ is the initial state). The controlled system may have neither the $L^∞$ -null controllability nor the backward uniqueness property.