Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-11T01:18:25.116Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing optimal control With a hyperbolic partial differential equation

Published online by Cambridge University Press:  17 February 2009

V. Rehbock ,
S. Wang  and
K.L. Teo
V. Rehbock
Affiliation:
School of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, 6845, Australia
S. Wang
Affiliation:
School of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, 6845, Australia
K.L. Teo
Affiliation:
School of Mathematics and Statistics, Curtin University of Technology, GPO Box U1987, Perth, 6845, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present a method for solving a class of optimal control problems involving hyperbolic partial differential equations. A numerical integration method for the solution of a general linear second-order hyperbolic partial differential equation representing the type of dynamics under consideration is given. The method, based on the piecewise bilinear finite element approximation on a rectangular mesh, is explicit. The optimal control problem is thus discretized and reduced to an ordinary optimization problem. Fast automatic differentiation is applied to calculate the exact gradient of the discretized problem so that existing optimization algorithms may be applied. Various types of constraints may be imposed on the problem. A practical application arising from the process of gas absorption is solved using the proposed method.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 2 , October 1998 , pp. 266 - 287
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Ahmed, N. U., Optimization and identification of systems governed by evolution equations on Banach space (Longman Scientific and Technical, Essex, England, 1988).Google Scholar
[2]Ahmed, N. U., Semigroup theory with applications to systems and control (Longman Scientific and Technical, Essex, England, 1991).Google Scholar
[3]Ahmed, N. U. and Teo, K. L., Optimal control of distributed parameter systems (Elsevier North Holland, New York, 1981).Google Scholar
[4]Banks, H. T. (ed.), Control and estimation in distributed parameter systems, Frontiers in Applied Mathematics (SIAM, Philadelphia, 1992).Google Scholar
[5]Banks, H. T. and Kunisch, K., Estimation techniques for distributed parameter systems (Birkhäuser, Boston, 1989).CrossRefGoogle Scholar
[6]Bensoussan, A., da Prato, G., Delfour, M. C. and Mitter, S. K., Representation and control of infinite dimensional systems, Volume 1 (Birkhäuser, Boston, 1992).Google Scholar
[7]Bensoussan, A., da Prato, G., Delfour, M. C. and Mitter, S. K., Representation and control of infinite dimensional systems, Volume 2 (Birkhäuser, Boston, 1992).Google Scholar
[8]Brenner, C. and Scott, L. R., The mathematical theory of finite element methods (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar
[9]Butkovsky, A. G., Distributed control systems (American Elsevier, New York, 1969).Google Scholar
[10]Craven, B. D., Control and Optimization (Chapman & Hall, London, 1995).CrossRefGoogle Scholar
[11]Curtain, R. F. and Pritchard, A. J., Infinite dimensional linear systems theory, Lecture Notes in Control and Information Sciences 8 (Springer-Verlag, Berlin, 1978).CrossRefGoogle Scholar
[12]Curtain, R. F. and Zwart, H. J., An introduction to infinite dimensional linear systems theory (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
[13]Evtushenko, Y. G., “Fast automatic differentiation and related topics”, submitted.Google Scholar
[14]Griewank, A. and Corliss, G. F., Automatic differentiation of algorithms: theory, implementation and applications (SIAM, Philadelphia, 1991).Google Scholar
[15]Idczak, D., Kibalczyc, K. and Walczak, S., “On an oprimization problem with cost of rapid variation of control”, J. Austral. Math. Hoc. Ser. B 36 (1994) 117–131.CrossRefGoogle Scholar
[16]Li, X. J. and Yong, J. M., Optimal control theory for infinite dimensional systems (Birkhäuser, Boston, 1995).Google Scholar
[17]Lions, J. L., Optimal control of systems described by partial differential equations (Springer-Verlag, Berlin, 1971).Google Scholar
[18]Rehbock, V., Teo, K. L. and Jennings, L. S., “An exact penalty function approach to all-time-step constrained discrete-time optimal control problems”, Applied Mathematics and Computation 49 (1992) 215–230.Google Scholar
[19]Teo, K. L., Goh, C. J. and Wong, K. H., A unified computational approach to optimal control problems (Longman Scientific and Technical, Essex, England, 1991).Google Scholar
[20]Teo, K. L. and Wu, Z. S., Computational methods for optimizing distributed systems (Academic Press, Orlando, 1984).Google Scholar
[21]Tikhonov, A. N. and Samarski, A. A., Equations of mathematical physics (Moscow, 1978).Google Scholar
[22]Tröltzsch, F., Optimality conditions for parabolic control problems and applications (Teubner-Texte zur Mathematik, Band 62, Leipzig, 1982).Google Scholar
[23]Tzafestas, S. G. (ed.), Distributed parameter systems: theory and applications (Pergamon Press, New York, 1982).Google Scholar
[24]Vichnevetsky, R. and Bowles, J. B., Fourier analysis of numerical approximations of hyperbolic equations (SIAM, Philadelphia, 1982).CrossRefGoogle Scholar
[25]Wang, P. K. C., “Control of distributed parameter systems”, in Advances in control systems, theory and applications (ed. Leondes, C. T.), Volume 1, (Academic Press, New York, 1964) 75–172.Google Scholar
[26]Wu, Z. S., “On the optimal control of a class of hyperbolic systems”Phd, University of New South Wales, 1981.Google Scholar