# American Institute of Mathematical Sciences

2014, 4(2): 93-101. doi: 10.3934/naco.2014.4.93

## A sufficient condition of Euclidean rings given by polynomial optimization over a box

 1 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China

Received  June 2013 Revised  December 2013 Published  May 2014

A sufficient condition of Euclidean rings is given by polynomial optimization. Then, through computation, we give all norm-Euclidean square number fields, four examples of norm-Euclidean cubic number fields and two examples of norm-Euclidean cyclotomic fields, with the absolute of a norm less than 1 over the corresponding box, respectively.
Citation: Shenggui Zhang. A sufficient condition of Euclidean rings given by polynomial optimization over a box. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 93-101. doi: 10.3934/naco.2014.4.93
##### References:
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##### References:
 [1] K. J. Astrom, R. E. Klein and A. Lennartsson, Bicycle dynamics and control, IEEE Control Systems Magazine, 25 (2005), 26-47. doi: 10.1109/MCS.2005.1499389.  Google Scholar [2] C. K. Chen and T. K. Dao, Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic, Vehicle System Dynamics, 48 (2010), 133-147. Google Scholar [3] C. Cornejo and L. Alvarez-Icaza, Passivity based control of under-actuated mechanical systems with nonlinear dynamic friction, J. Vibration and Control, 18 (2012), 1025-1042. doi: 10.1177/1077546311408469.  Google Scholar [4] M. L. Fair and S. L. Campbell, Active incipient fault detection in continuous time systems with multiple simultaneous faults, Numerical Algebra, Control and Optimization, 1 (2011), 211-224. doi: 10.3934/naco.2011.1.211.  Google Scholar [5] L. Feng, Robust Control Design: An Optimal Control Approach, Wayne State University, USA and Tongji University, China, John Wiley and Sons Ltd, 2007. Google Scholar [6] N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics, Ph.D. Dissertation, University of California, 1995. Google Scholar [7] Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization, Numerical Algebra, Control and Optimization, 1 (2011), 171-190. doi: 10.3934/naco.2011.1.171.  Google Scholar [8] C. L. Hwang, H. M. Wu and C. L. Shih, Fuzzy sliding-mode underactuated control for autonomous dynamic balance of an electrical bicycle, IEEE Trans. Control Systems Technology, 17 (2009), 658-670. Google Scholar [9] N. H. K. Iuchi, H. Niki and T. Murakami, Attitude control of bicycle motion by steering angle and variable COG control, Proc. 31st Annual Conference of IEEE Industrial Electronics Society, IECON, (2005), 16-21. Google Scholar [10] R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles, J. Vibration and Control, 16 (2010), 253-279. doi: 10.1177/1077546309104467.  Google Scholar [11] R. Khaled and N. G. Chalhoub, A dynamic model and a robust controller for a fully-actuated marine surface vessel, J. Vibration and Control, 17 (2011), 801-812. Google Scholar [12] L. Lujng, System Identification Theory for User,, Linkopping University, ().   Google Scholar [13] M. S. Mahmoud, Computer-Operated Systems Control, Marcel Dekker Inc., New York, 1991. Google Scholar [14] M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation, IET Control Theory and Applications, 5 (2011), 1577-1585. doi: 10.1049/iet-cta.2010.0665.  Google Scholar [15] M. S. Mahmoud, Resilient $\begin{eqation*}\frac{L_2}{L_\infty} \end{equation*}$ filtering of polytopic systems with state delays, IET Control Theory And Applications, 1 (2007), 141-154. doi: 10.1049/iet-cta:20045281.  Google Scholar [16] M. S. Mahmoud and A. Y. Al-Rayyah., Efficient parameterisation to stability and feedback synthesis of linear time-delay systems, IET control theory and applications, 3 (2009), 1107-1118. doi: 10.1049/iet-cta.2008.0152.  Google Scholar [17] M. S. Mahmoud and Yuanqing Xia, Robust filter design for piecewise discrete-time systems with time-varying delays, International Journal of Robust and Nonlinear Control, 20 (2010), 544-560. doi: 10.1002/rnc.1447.  Google Scholar [18] M. S. Mahmoud and M. M. Hussain, Design of linear systems with saturating actuators: A survey, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 413-435. doi: 10.3934/naco.2012.2.413.  Google Scholar [19] J. Meijaard, J. Papadopoulos, A. Ruina and A. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review, Proc. the Royal Society A: Mathematical, Physical and Engineering Science, 463 (2007). doi: 10.1098/rspa.2007.1857.  Google Scholar [20] K. Mendrok and Tadeusz Uhl, Load identification using a modified modal filter technique, J. Vibration and Control, 16 (2010), 89-105. doi: 10.1177/1077546309103274.  Google Scholar [21] G. T. Michaltsos, Bouncing of a vehicle on an irregularity: A mathematical model, J. Vibration and Control, 16 (2010), 181-206. doi: 10.1177/1077546309104878.  Google Scholar [22] H. Moradi, M. R. Movahhedy, and G. Vossoughi, Sliding mode control of machining chatter in the presence of tool wear and parametric uncertainties, J. Vibration and Control, 16 (2010), 231-251. Google Scholar [23] U. Nenner, R. Linker and P. Gutman, Robust feedback stabilization of an unmanned motorcycle, Control Engineering Practice, 2010. Google Scholar [24] Omar S. Al-Buraiki and El Ferik, Sami, Adaptive control of autonomous bicycle kinematics, Proc. 13th Automation and Systems (ICCAS), Gwangju, Korea, Oct. (2013), 20-23. Google Scholar [25] M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems, J. Vibration and Control, 16 (2010),2131-2145. doi: 10.1177/1077546309350865.  Google Scholar [26] H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation, Int. J. Numerical Algebra, Control and Optimization, 2 (2012), 631-654. doi: 10.3934/naco.2012.2.631.  Google Scholar [27] R. Sharp and D. Limebeer, A motorcycle model for stability and control analysis, Multi-body System Dynamics, 6 (2001), 123-142. Google Scholar [28] R. Sharp, Optimal preview speed-tracking control for motorcycles, Multi-body System Dynamics, 18 (2007), 397-411. Google Scholar [29] S. Sivrioglu, H ∞ control for suppressing acoustic modes of a distributed structure using cluster sensing and actuation, J. Vibration and Control, 16 (2010), 439-453. Google Scholar [30] N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle, Proc. IEEE International Conference Robotics and Biometrics, ROBIO06, (2006), 1652-1657. Google Scholar [31] T. Yamaguchi, T. Shibata and T. Murakami, Self-sustaining approach of electric bicycle by acceleration control based backstepping, Proc. 33rd Annual Conference of the IEEE Industrial Electronics Society, IECON, (2007), 2610-2614. Google Scholar [32] K. Zhou and J. C. Doyle, Essentials of Robust Control, NJ: Prentice Hall, 1998. Google Scholar
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