The derivation of structural properties of LM-g splines by necessary condition of optimal control. (English) Zbl 1474.41016

Liu, Xiongwei; Zhang, Xinjian; Cheng, Lizhi

Abstr. Appl. Anal. 2014, Article ID 641435, 5 p. (2014).

Summary: 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.

MSC:

41A15

Spline approximation

References:

[1]	Wang, Y., Smoothing Splines: Methods and Applications, xxiv+370 (2011), NewYork, NY, USA: CRC Press, NewYork, NY, USA · Zbl 1223.65011
[2]	Egerstedt, M.; Martin, C., Control Theoretic Splines: Optimal Control, Statistics, and Path Planning, x+217 (2009), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 1213.41005
[3]	Zhang, X. J.; Lu, S. R., A spline method for computing a class of minimum-energy control for multivariable linear systems, Control Theory & Applications, 19, 1, 61-64 (2002) · Zbl 1041.49502
[4]	Sidhu, G. S.; Weinert, H. L., Vector-valued \(L g\)-splines. I. Interpolating splines, Journal of Mathematical Analysis and Applications, 70, 2, 505-529 (1979) · Zbl 0435.65007
[5]	Sidhu, G. S.; Weinert, H. L., Vector-valued \(L g\)-splines. II. Smoothing splines, Journal of Mathematical Analysis and Applications, 101, 2, 380-396 (1984) · Zbl 0574.65006
[6]	de Figueiredo, R. J. P., LM-g splines, Journal of Approximation Theory, 19, 4, 332-360 (1977) · Zbl 0357.41005
[7]	Weinert, H. L.; Desai, U. B.; Sidhu, G. S., ARMA splines, system inverses, and least-squares estimates, SIAM Journal on Control and Optimization, 17, 4, 525-536 (1979) · Zbl 0418.93087
[8]	Kohn, R.; Ansley, C. F., A new algorithm for spline smoothing based on smoothing a stochastic process, SIAM Journal on Scientific and Statistical Computing, 8, 1, 33-48 (1987) · Zbl 0627.65010
[9]	Zhang, X.; Fang, K., On a class of minimum energy controls and generalized splines, Journal of National University of Defense Technology, 15, 4, 84-90 (1993)
[10]	Opfer, G.; Oberle, H. J., The derivation of cubic splines with obstacles by methods of optimization and optimal control, Numerische Mathematik, 52, 1, 17-31 (1988) · Zbl 0628.41012
[11]	Fredenhagen, S.; Oberle, H. J.; Opfer, G., On the construction of optimal monotone cubic spline interpolations, Journal of Approximation Theory, 96, 2, 182-201 (1999) · Zbl 0934.41009
[12]	Zhang, X. J., Generalized interpolating splines with obstacles and optimal control problems with state constraints, Acta Mathematicae Applicatae Sinica, 23, 3, 342-350 (2000) · Zbl 0957.49020
[13]	Takahashi, S.; Martin, C. F., Optimal control theoretic splines and its application to mobile robot, Control Applications, 2, 1729-1732 (2004)
[14]	Alhanaty, M.; Bercovier, M., Curve and surface fitting and design by optimal control methods, Computer-Aided Design, 33, 2, 167-182 (2001) · Zbl 1206.65043
[15]	Zhang, X. J., Structure and continuity characteristics of operator spline interpolations associated with invertible linear systems, Mathematica Numerica Sinica, 23, 2, 145-154 (2001) · Zbl 1495.65018
[16]	Zhang, X.; Liu, X., Derivation of structural characteristics of differential operator interpolating splines by the criteria of optimal control, Control Theory & Applications, 28, 6, 851-854 (2011)
[17]	Zhang, X., On the inversion of linear systems, Journal of National University of Defense Technology, 20, 2, 109-113 (1998)