# 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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Limebeer, A motorcycle model for stability and control analysis,, Multi-body System Dynamics, 6 (2001), 123. Google Scholar [28] R. Sharp, Optimal preview speed-tracking control for motorcycles,, Multi-body System Dynamics, 18 (2007), 397. 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. Google Scholar [30] N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle,, Proc. IEEE International Conference Robotics and Biometrics, (2006), 1652. 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, (2007), 2610. Google Scholar [32] K. Zhou and J. C. Doyle, Essentials of Robust Control,, NJ: Prentice Hall, (1998). Google Scholar

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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. 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. 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. 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, 1 (2011), 211. doi: 10.3934/naco.2011.1.211. Google Scholar [5] L. Feng, Robust Control Design: An Optimal Control Approach,, Wayne State University, (2007). Google Scholar [6] N. H. Getz, Dynamic Inversion of Nonlinear Maps with Applications to Nonlinear Control and Robotics,, Ph.D. Dissertation, (1995). Google Scholar [7] Y. Harata, Y. Banno and K. Taji, Parametric excitation based bipedal walking: Control method and optimization,, Numerical Algebra, 1 (2011), 171. 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. 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, (2005), 16. Google Scholar [10] R. N. Jazar, Mathematical theory of auto-driver for autonomous vehicles,, J. Vibration and Control, 16 (2010), 253. 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. 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., (1991). Google Scholar [14] M. S. Mahmoud, Robust control of blood gases during extracorporeal circulation,, IET Control Theory and Applications, 5 (2011), 1577. 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. 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. 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. 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, 2 (2012), 413. 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, 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. 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. 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. 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), (2013), 20. Google Scholar [25] M. C. Pai, Sliding mode control of vibration in uncertain time-delay systems,, J. Vibration and Control, 16 (2010), 2131. doi: 10.1177/1077546309350865. Google Scholar [26] H. Schttler and U. Ledzewicz, Perturbation feedback control: A geometric interpretation,, Int. J. Numerical Algebra, 2 (2012), 631. 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. Google Scholar [28] R. Sharp, Optimal preview speed-tracking control for motorcycles,, Multi-body System Dynamics, 18 (2007), 397. 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. Google Scholar [30] N. Umashankar and H. D. Sharma, Adaptive neuro-fuzzy controller for stabilizing autonomous bicycle,, Proc. IEEE International Conference Robotics and Biometrics, (2006), 1652. 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, (2007), 2610. Google Scholar [32] K. Zhou and J. C. Doyle, Essentials of Robust Control,, NJ: Prentice Hall, (1998). Google Scholar
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