Abstract
In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.
Similar content being viewed by others
References
YOUNG, L. C., Lectures on the Calculus of Variations and Optimal Control Theory, Saunders, Philadelphia, Pennsylvania, 1969.
WARGA, J., Optimal Control of Differential and Functional Equations, Academic Press, New York, NY, 1972.
GAMKRELIDZE, R., Principles of Optimal Control Theory, Plenum Press, London, UK, 1978.
CESARI, L., Optimization Theory and Applications, Springer, New York, NY, 1983.
FATTORINI, H. O., Infinite-Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, UK, 1999.
ROUBICEK, T., Relaxation in Optimization Theory and Variational Calculus, Wester de Gruyter, Berlin, Germany, 1997.
FILIPPOV, A. F., On Certain Questions in the Theory of Optimal Control, SIAM Journal on Control, Vol. 1. pp. 76–84, 1962.
GABASOV, R., and KIRILLOVA, F., Qualitative Theory of Optimal Processes, Nauka, Moscow, Russia, 1971 (in Russian).
MORDUKHOVICH, B. S., Discrete Approximations and Refined Euler-Lagrange Conditions for Nonconvex Differential Inclusions, SIAM Journal on Control and Optimization, Vol. 33, pp. 882–915, 1995.
BUTZEK, S., and SCHMIDT, W. H., Relaxation Gaps in Optimal Control Processes with State Constraints, Variational Calculus, Optimal Control and Applications, Edited by W. H. Schmidt, K. Heier, L. Bittner, and R. Bulirsch, Birkhäuser, Basel, Switzerland, pp. 21–29, 1998.
TICHOMIROV, V. M., Grundprinzipien der Theorie der Extremalaufgaben, Teubner, Leipzig, Germany, 1982.
DONTCHEV, A. L., and LEMPIO, F., Difference Methods for Differential Inclusions: A Survey, SIAM Review, Vol. 34, pp. 263–294, 1992.
MALANOWSKI, K., Finite Difference Approximations to Constrained Optimal Control Problems, Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, Springer, New York, NY, pp. 243–254, 1981.
DONTCHEV, A. L., Discrete Approximations in Optimal Control, Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, Edited by B. S. Mordukhovich and H. J. Sussmann, Springer, New York, NY, Vol. 78, pp. 59–80, 1996.
PYTLAK, R., Numerical Methods for Optimal Control Problems with State Constraints, Springer, Berlin, Germany, 1999.
FEDORENKO, R. P., Approximate Solving Optimal Control Problems, Nauka, Moscow, Russia, 1978 (in Russian).
HAGER, W. W., Rate of Convergence for Discrete Approximations to Unconstrained Control Problems, SIAM Journal on Numerical Analysis, Vol. 13, pp.449–471, 1976.
DONTCHEV, A. L., and HAGER, W. W., Lipschitz Stability in Nonlinear Control and Optimization, SIAM Journal on Control and Optimization, Vol. 31, pp. 569–603, 1993.
HAGER, W. W., and LANCULESCU, G. D., Dual Approximations in Optimal Control, SIAM Journal on Control and Optimization, Vol. 22, pp. 1061–1080, 1990.
AZHMYAKOV, V., and SCHMIDT, W. H., Explicit Approximations of Relaxed Optimal Control Processes, Optimal Control, Edited by B. Kugelmann, G. Sachs, and W.H. Schmidt, Hieronymus Bücherproduktion GmbH, München, Germany, pp. 179–192, 2003.
VELIOV, V. M., Second-Order Discrete Approximations to Linear Differential Inclusions, SIAM Journal on Numerical Analysis, Vol. 29, pp. 439–451, 1992.
CLARKE, F. H., Optimal Solutions to Differential Inclusions, Journal of Optimization Theory and Applications, Vol. 19, pp. 469–478, 1976.
WARGA, J., Controllability, Extremality, and Abnormality in Nonsmooth Optimal Control, Journal of Optimization Theory and Applications, Vol. 41, pp. 239–260, 1983.
POLAK, E., Optimization, Springer, New York, NY, 1997.
HAMEL, G., Über eine mit dem Problem der Rakete zusammenhängende Aufgabe, ZAMM, Vol. 7, pp. 451–552, 1927.
MICLE, A., Extremization of Linear Integrals by Green’s Theorem, Optimization Techniques, Edited by G. Leitmann, Academic Press, New York, NY, pp.69–98, 1962.
MAURER, H., Numerical Solution of Singular Control Problems Using Multiple-Shooting Techniques, Journal of Optimization Theory and Applications, Vol. 18, pp. 235–257, 1976.
STRYK, O., User’s Guide for DIRCOL: A Direct Collocation Method for the Numerical Solution of Optimal Control Problems, Technische Universität, München, Germany, 1999.
Author information
Authors and Affiliations
Additional information
Communicated by H. J. Oberle
The authors thank the referees for helpful comments and suggestions.
Rights and permissions
About this article
Cite this article
Azhmyakov, V., Schmidt, W. Approximations of Relaxed Optimal Control Problems. J Optim Theory Appl 130, 61–78 (2006). https://doi.org/10.1007/s10957-006-9085-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-006-9085-9