Pontryagin’s Maximum Principle for a Advection-Diffusion-Reaction Equation


In this paper we investigate optimal control problems governed by a advection-diffusion-reaction equation. We present a method for deriving conditions in the form of Pontryagin’s principle. The main tools used are the Ekeland’s variational principle combined with penalization and spike variation techniques.

Share and Cite:

Y. Xu, C. Xiao and H. Zhu, "Pontryagin’s Maximum Principle for a Advection-Diffusion-Reaction Equation," Applied Mathematics, Vol. 3 No. 12, 2012, pp. 1888-1891. doi: 10.4236/am.2012.312258.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] H. W. Lou and J. M. Yong, “Optimal Controls for Semilinear Elliptic Equations with Leaing Term Containing Controls,” SIAM Journal on Control and Optimization, Vol. 48, No. 4, 2009, pp. 2366-2387. doi:10.1137/080740301
[2] J. T. Oden and J. N. Reddy, “Variational Methods in Theoretical Mechanics,” Springer, Berlin and Heidelberg, 1983. doi:10.1007/978-3-642-68811-9
[3] L. Dede and A. Quarteroni, “Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations,” Mathematical Modelling and Numerical Analysis, Vol. 39, No. 2, 2005, pp. 1019-1040.
[4] N. N. Yan and Z. J. Zhou, “A Priori and a Posteriori Error Analysis of Edge Stabilization Galerkin Method for the Optimal Control Problem Governed by Convection-Dominated Diffusion Equation,” Journal of Computational and Applied Mathematics, Vol. 223, No. 1, 2009, pp. 198-217. doi:10.1016/j.cam.2008.01.006
[5] R. Becker and B. Vexler, “Optimal Control of the Convection-Diffusion Equation Using Stabilized Finite Element Methods,” Numerische Mathematik, Vol. 106, No. 3, 2007, pp. 349-367. doi:10.1007/s00211-007-0067-0
[6] S. Micheletti and S. Perotto, “An Anisotropic Mesh Adaptation Procedure for an Optimal Control Problem of the Advection-Diffusion-Reaction Equation,” MOX-Report No. 15, 2008.
[7] S. S. Collis and M. Heinkenschloss, “Analysis of the Streamline Upwind/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems,” Technical Report 02-01, Department of Computational and Applied Mathematics, Rice University, Houston, 2002.
[8] X. Li and J. Yong, “Optimal Control Theory for InfiniteDimensional Systems,” Birkh?user, Boston, 1995.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.