# American Institute of Mathematical Sciences

March  2017, 7(1): 73-170. doi: 10.3934/mcrf.2017005

## Decompositions and bang-bang properties

 1 School of Mathematics and Statistics, Collaborative Innovation Centre of Mathematics, Wuhan University, Wuhan 430072, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 3 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

* Corresponding author:Yubiao Zhang

Received  April 2016 Revised  July 2016 Published  December 2016

Fund Project: The first author was partially supported by the National Natural Science Foundation of China under grant 11571264. The second author was partially supported by the National Natural Science Foundation of China under grants 11571264 and 11371285.

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.

Citation: Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control & Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005
##### References:

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##### References:
The BBP decomposition for $(NP)^{T,y_0}$
The BBP decomposition for $(TP)^{M,y_0}$
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