A chebyshev finite difference method for solving a class of optimal control problems

International Journal of Computer Mathematics

Volumn 80, Issue 7, 2003, Pages 883-895

  El Kady, M ^a,b  

^a ASWAN UNIVERSITY   (Egypt)

^b Tabuk Teachers College   (Saudi Arabia)

Author keywords

Chebyshev approximations; Optimal control problems

Indexed keywords

ALGORITHMS; CHEBYSHEV APPROXIMATION; INTEGRATION; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING;

CHEBYSHEV FINITE DIFFERENCE METHOD; OPTIMAL CONTROL PROBLEMS; SYSTEM DYNAMICS;

FINITE DIFFERENCE METHOD;

EID: 28244487189     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/0020716031000070625     Document Type: Article

References (24)

1. Clenshaw, C. W. and Curtis, A. R. (1960). A method of numerical integration on an automatic computer. Numer. Math. (2), 197-205.
2. Dontchev, A. L. (1993). Discrete approximations in optimal control. In: Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control. Minneapolis, MN, IMA Vol. Math. Appl., 78, Springer, New York pp. 59-81. MR 97h:49043 (1996).
3. Dontchev, A. L., Hager, W. W. and Veliov, V. M. (2000). Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Numer. Anal., 38(1), 202-226.
4. Elbarbary, E. M. E. and El-Kady, M. Chebyshev finite difference approximation for the boundary value problems. Applied Math. and Computations (accepted for publication).
5. Elbarbary, E. M. E. Chebyshev finite difference method for the solution of boundary layers equations. J. Computational and Applied Math. (sent for publication).
6. El-Gendi, S. E. (1969). Chebyshev solutions of differential, integral and integro-differential equations. The Computer Journal, 12, 282-287.
7. El-Gindy, T. M., El-Hawary, H. M., Salim, M. S. and El-Kady, M. (1995). A Chebyshev approximation for solving optimal control problems. Computers Math. Appllc., 29(6), 35-45.
8. El-Gindy, T. M., El-Hawary, H. M., Salim, M. S. and El-Kady, M. (1995). The computational approach for optimal control problems of parabolic systems. J. Egypt Math. Scio., 3, 17-23.
9. El-Kady, M. (2002). Numerical solution for large-scale systems with general multiple linear-quadratic structure. Intern. J. Computer Math., 79(5), 615-624.
10. Gill, P. W. M., Saunders, M. and Wright, M. (1998). Users Guide for NPSOL 5.0: A Fortran package for nonlinear programming, Systems optimization laboratory. Stanford University, Stanford, CA 94305.
11. Gonzalez, S. and Rodriguez, S. (1986). Modified Quasilinearization algorithm for optimal control problems with nondifferential constraints and general boundary conditions. J. Optimization Theory and Applications, 50(1), 109-128.
12. Hager, W. W. (1976). Rales of convergence for discrete approximations to unconstrained control problems. SAIM J. Numer. Anal., 13(4) 449-172.
13. Hager, W. W. and Lanculescu, G. D. (1984). Dual approximations in optimal control. SIAM J. Control Optim., 22, 423-465.
14. Hargraves, C. R. and Paris, S. W. (1996). Direct trajectory optimization using nonlinear programming and collocation. J. Guidance, Control and Dynamics, 19(3) 592-599.
15. Milam, M. B., Mushambi K. and Murray, R. M. (2000). A new computational approach to real-time trajectory generation for constrained mechanical system. IEEE Conference on Decision and Control.
16. Polak, E. (1973). A historical survey of computations methods in optimal control. SIAM Review, 15, 553-548. MR 53: 1956.
17. Polak, E. (1997). Optimization: Algorithms and Consistent Approximation. Springer, New York, MR 98g: 49001.
18. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F. (1962). The Mathematical Theory of Optimal Processes. John Wiley & Sons, New York (Translated from Russian by K. N. Trirogoff).
19. Schwartz, A. (1996). Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems. PhD thesis, U.C., Berkeley.
20. Schwartz, A. L. and Polak, E. (1996). Consistent approximation for optimal control problems based on Runge-Kutta integration. SIAM J. Control Optim., 34, 1235-1269. MR 97h: 49045.
21. Shen, J. (1995). Efficient spectral-galerkin method II. Direct solver of second and fourth-order equations using Chebyshev polynomials. SIAM J. Sci. Comput., 16, 74-87.
22. Stryk, O. and Bulirsch, R. (1992). Direct and indirect methods for trajectory optimization. Annals of Operations Research, 37, 357-373.
23. Van Dooren, R. and Vlassenbroeck, J. (1988). A Chebyshev technique for solving nonlinear optimal control problems. IEEE Trans. Automat. Contr., 33(4), 333-339.
24. Wright, S. E. (1995). Consistency of primal-dual approximations for convex optimal control problems. SIAM J. Control Optim., 33, 1489-1509. MR 96h: 49057.