A minimum effort optimal control problem for elliptic PDEs
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 911-927

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls.

DOI : https://doi.org/10.1051/m2an/2011074
Classification:  49J52,  49J20,  49K20
Keywords: optimal control, minimum effort, L∞control cost, semi-smooth Newton method
@article{M2AN_2012__46_4_911_0,
author = {Clason, Christian and Ito, Kazufumi and Kunisch, Karl},
title = {A minimum effort optimal control problem for elliptic PDEs},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
pages = {911-927},
doi = {10.1051/m2an/2011074},
zbl = {1270.49023},
mrnumber = {2891474},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_4_911_0}
}

Clason, Christian; Ito, Kazufumi; Kunisch, Karl. A minimum effort optimal control problem for elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 911-927. doi : 10.1051/m2an/2011074. http://www.numdam.org/item/M2AN_2012__46_4_911_0/

[1] J.Z. Ben-Asher, E.M. Cliff and J.A. Burns, Computational methods for the minimum effort problem with applications to spacecraft rotational maneuvers, in IEEE Conf. on Control and Applications (1989) 472-478.

[2] C. Clason, K. Ito and K. Kunisch, Minimal invasion : An optimal L∞ state constraint problem. ESAIM : M2AN 45 (2010) 505-522. | Numdam | MR 2804648 | Zbl 1269.65060

[3] I. Ekeland and R. Témam, Convex analysis and variational problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1999). | MR 1727362 | Zbl 0939.49002

[4] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR 1814364 | Zbl 1042.35002

[5] T. Grund and A. Rösch, Optimal control of a linear elliptic equation with a supremum norm functional. Optim. Methods Softw. 15 (2001) 299-329. | MR 1892589 | Zbl 1005.49013

[6] M. Gugat and G. Leugering, L∞-norm minimal control of the wave equation : On the weakness of the bang-bang principle. ESAIM Control Optim. Calc. Var. 14 (2008) 254-283. | Numdam | MR 2394510 | Zbl 1133.49006

[7] K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 15 (2008). | MR 2441683 | Zbl 1156.49002

[8] K. Ito and K. Kunisch, Minimal effort problems and their treatment by semismooth newton methods. SIAM J. Control Optim. 49 (2011) 2083-2100. | MR 2861426 | Zbl 1234.49017

[9] O.A. Ladyzhenskaya and N.N. Ural'Tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation, edited by L. Ehrenpreis, Academic Press, New York (1968). | MR 244627 | Zbl 0164.13002

[10] L.W. Neustadt, Minimum effort control systems. SIAM J. Control Ser. A 1 (1962) 16-31. | MR 145172 | Zbl 0145.34502

[11] U. Prüfert and A. Schiela, The minimization of a maximum-norm functional subject to an elliptic PDE and state constraints. Z. Angew. Math. Mech. 89 (2009) 536-551. | MR 2553754 | Zbl 1166.49021

[12] A. Schiela, A simplified approach to semismooth Newton methods in function space. SIAM J. Optim. 19 (2008) 1417-1432. | MR 2460749 | Zbl 1169.49032

[13] Z. Sun and J. Zeng, A damped semismooth Newton method for mixed linear complementarity problems. Optim. Methods Softw. 26 (2010) 187-205. | MR 2773654 | Zbl 1251.90369

[14] G.M. Troianiello, Elliptic Differential Equations and Obstacle Problems. The University Series in Mathematics, Plenum Press, New York (1987). | MR 1094820 | Zbl 0655.35002

[15] F. Tröltzsch, Optimal Control of Partial Differential Equations : Theory, Methods and Applications. American Mathematical Society, Providence (2010). Translated from the German by Jürgen Sprekels. | MR 2583281 | Zbl 1195.49001

[16] E. Zuazua, Controllability and observability of partial differential equations : some results and open problems, in Handbook of differential equations : evolutionary equations. Handb. Differ. Equ., Elsevier, North, Holland, Amsterdam III (2007) 527-621. | MR 2549374 | Zbl 1193.35234