Decompositions and bang-bang properties

Gengsheng Wang; Yubiao Zhang

doi:10.3934/mcrf.2017005

Article Contents

2017, Volume 7, Issue 1: 73-170. Doi: 10.3934/mcrf.2017005

This issue Previous Article Control and stabilization of 2 × 2 hyperbolic systems on graphs Next Article

Decompositions and bang-bang properties

Gengsheng Wang^1, and
Yubiao Zhang^2,3, ,

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
* Corresponding author:Yubiao Zhang

Received: March 31, 2016

Revised: June 30, 2016

Published: May 2017

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.

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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.

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Mathematics Subject Classification: Primary: 93B03; Secondary: 93C35.

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Figure 1. The BBP decomposition for $(NP)^{T,y_0}$

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Figure 2. The BBP decomposition for $(TP)^{M,y_0}$

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