Abstract
Bounds are established for integration matrices that arise in the convergence analysis of discrete approximations to optimal control problems based on orthogonal collocation. Weighted Euclidean norm bounds are derived for both Gauss and Radau integration matrices; these weighted norm bounds yield sup-norm bounds in the error analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Axelsson, O.: Global integration of differential equations through Lobatto quadrature. BIT Numer. Math. 4, 69–86 (1964)
Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., Rao, A.V.: Direct trajectory optimization and costate estimation via an orthogonal collocation method. J. Guid. Control Dyn. 29, 1435–1440 (2006)
Dennis, M.E., Hager, W.W., Rao, A.V.: Computational method for optimal guidance and control using adaptive Gaussian quadrature collocation. J. Guid. Control Dyn. (2019). https://doi.org/10.2514/1.G003943
Françolin, C.C., Benson, D.A., Hager, W.W., Rao, A.V.: Costate estimation in optimal control using integral Gaussian quadrature orthogonal collocation methods. Optim. Control Appl. Meth. 36, 381–397 (2015)
Garg, D., Hager, W.W., Rao, A.V.: Pseudospectral methods for solving infinite-horizon optimal control problems. Automatica 47, 829–837 (2011)
Garg, D., Patterson, M.A., Darby, C.L., Françolin, C., Huntington, G.T., Hager, W.W., Rao, A.V.: Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method. Comput. Optim. Appl. 49, 335–358 (2011)
Garg, D., Patterson, M.A., Hager, W.W., Rao, A.V., Benson, D.A., Huntington, G.T.: A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica 46, 1843–1851 (2010)
Hager, W.W., Hou, H., Mohapatra, S., Rao, A.V., Wang, X.-S.: Convergence rate for a Radau hp-collocation method applied to constrained optimal control. Comput. Optim. Appl. (2019). https://doi.org/10.1007/s10589-019-00100-1
Hager, W.W., Hou, H., Rao, A.V.: Convergence rate for a Gauss collocation method applied to unconstrained optimal control. J. Optim. Theory Appl. 169, 801–824 (2016)
Hager, W.W., Hou, H., Rao, A.V.: Lebesgue constants arising in a class of collocation methods. IMA J. Numer. Anal. 37, 1884–1901 (2017)
Hager, W.W., Liu, J., Mohapatra, S., Rao, A.V., Wang, X.-S.: Convergence rate for a Gauss collocation method applied to constrained optimal control. SIAM J. Control Optim. 56, 1386–1411 (2018)
Shen, J., Tang, T., Wang, L.-L.: Spectral Methods. Springer, Berlin (2011)
Szegő, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)
Yang, L., Zhou, H., Chen, W.: Application of linear Gauss pseudospectral in model predictive control. Acta Astronaut. 96, 175–187 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
March 25, 2019, revised April 4, 2019. Support by the National Science Foundation under Grants 1522629 and 1819002, and by the Office of Naval Research under Grants N00014-15-1-2048 and N00014-18-1-2100 is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Chen, W., Du, W., Hager, W.W. et al. Bounds for integration matrices that arise in Gauss and Radau collocation. Comput Optim Appl 74, 259–273 (2019). https://doi.org/10.1007/s10589-019-00099-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-019-00099-5