Abstract and Applied Analysis

The Derivation of Structural Properties of LM-g Splines by Necessary Condition of Optimal Control

Xiongwei Liu, Xinjian Zhang, and Lizhi Cheng

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The structural properties of LM-g splines are investigated by optimization and optimal control theory. The continuity and structure of LM-g splines are derived by using a class of necessary conditions with state constraints of optimal control and the relationship between LM-g interpolating splines and the corresponding L-g interpolating splines. This work provides a new method for further exploration of LM-g interpolating splines and its applications in the optimal control.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 641435, 5 pages.

Dates
First available in Project Euclid: 6 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412606208

Digital Object Identifier
doi:10.1155/2014/641435

Mathematical Reviews number (MathSciNet)
MR3206808

Zentralblatt MATH identifier
07022805

Citation

Liu, Xiongwei; Zhang, Xinjian; Cheng, Lizhi. The Derivation of Structural Properties of LM-g Splines by Necessary Condition of Optimal Control. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 641435, 5 pages. doi:10.1155/2014/641435. https://projecteuclid.org/euclid.aaa/1412606208


Export citation

References

  • Y. Wang, Smoothing Splines: Methods and Applications, CRC Press, NewYork, NY, USA, 2011.
  • M. Egerstedt and C. Martin, Control Theoretic Splines: Optimal Control, Statistics, and Path Planning, Princeton University Press, Princeton, NJ, USA, 2009.
  • X. J. Zhang and S. R. Lu, “A spline method for computing a class of minimum-energy control for multivariable linear systems,” Control Theory & Applications, vol. 19, no. 1, pp. 61–64, 2002.
  • G. S. Sidhu and H. L. Weinert, “Vector-valued $Lg$-splines. I. Interpolating splines,” Journal of Mathematical Analysis and Applications, vol. 70, no. 2, pp. 505–529, 1979.
  • G. S. Sidhu and H. L. Weinert, “Vector-valued $Lg$-splines. II.Smoothing splines,” Journal of Mathematical Analysis and Applications, vol. 101, no. 2, pp. 380–396, 1984.
  • R. J. P. de Figueiredo, “LM-g splines,” Journal of Approximation Theory, vol. 19, no. 4, pp. 332–360, 1977.
  • H. L. Weinert, U. B. Desai, and G. S. Sidhu, “ARMA splines, system inverses, and least-squares estimates,” SIAM Journal on Control and Optimization, vol. 17, no. 4, pp. 525–536, 1979.
  • R. Kohn and C. F. Ansley, “A new algorithm for spline smoothing based on smoothing a stochastic process,” SIAM Journal on Scientific and Statistical Computing, vol. 8, no. 1, pp. 33–48, 1987.
  • X. Zhang and K. Fang, “On a class of minimum energy controls and generalized splines,” Journal of National University ofDefense Technology, vol. 15, no. 4, pp. 84–90, 1993.
  • G. Opfer and H. J. Oberle, “The derivation of cubic splines with obstacles by methods of optimization and optimal control,” Numerische Mathematik, vol. 52, no. 1, pp. 17–31, 1988.
  • S. Fredenhagen, H. J. Oberle, and G. Opfer, “On the construction of optimal monotone cubic spline interpolations,” Journal of Approximation Theory, vol. 96, no. 2, pp. 182–201, 1999.
  • X. J. Zhang, “Generalized interpolating splines with obstacles and optimal control problems with state constraints,” Acta Mathematicae Applicatae Sinica, vol. 23, no. 3, pp. 342–350, 2000.
  • S. Takahashi and C. F. Martin, “Optimal control theoretic splines and its application to mobile robot,” Control Applications, vol. 2, pp. 1729–1732, 2004.
  • M. Alhanaty and M. Bercovier, “Curve and surface fitting and design by optimal control methods,” Computer-Aided Design, vol. 33, no. 2, pp. 167–182, 2001.
  • X. J. Zhang, “Structure and continuity characteristics of operator spline interpolations associated with invertible linear systems,” Mathematica Numerica Sinica, vol. 23, no. 2, pp. 145–154, 2001.
  • X. Zhang and X. Liu, “Derivation of structural characteristics of differential operator interpolating splines by the criteria of optimal control,” Control Theory & Applications, vol. 28, no. 6, pp.851–854, 2011.
  • X. Zhang, “On the inversion of linear systems,” Journal of National University of Defense Technology, vol. 20, no. 2, pp. 109–113, 1998. \endinput