Multi-bang control of elliptic systems
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, p. 1109-1130
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

Multi-bang control refers to optimal control problems for partial differential equations where a distributed control should only take on values from a discrete set of allowed states. This property can be promoted by a combination of L 2 and L 0 -type control costs. Although the resulting functional is nonconvex and lacks weak lower-semicontinuity, application of Fenchel duality yields a formal primal-dual optimality system that admits a unique solution. This solution is in general only suboptimal, but the optimality gap can be characterized and shown to be zero under appropriate conditions. Furthermore, in certain situations it is possible to derive a generalized multi-bang principle, i.e., to prove that the control almost everywhere takes on allowed values except on sets where the corresponding state reaches the target. A regularized semismooth Newton method allows the numerical computation of (sub)optimal controls. Numerical examples illustrate the effectiveness of the proposed approach as well as the structural properties of multi-bang controls.

DOI : https://doi.org/10.1016/j.anihpc.2013.08.005
Keywords: Optimal control, Elliptic partial differential equations, Nonconvex relaxation, Fenchel duality, Newton methods
@article{AIHPC_2014__31_6_1109_0,
     author = {Clason, Christian and Kunisch, Karl},
     title = {Multi-bang control of elliptic systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     year = {2014},
     pages = {1109-1130},
     doi = {10.1016/j.anihpc.2013.08.005},
     zbl = {1304.49014},
     mrnumber = {3280062},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_6_1109_0}
}
Clason, Christian; Kunisch, Karl. Multi-bang control of elliptic systems. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, pp. 1109-1130. doi : 10.1016/j.anihpc.2013.08.005. http://www.numdam.org/item/AIHPC_2014__31_6_1109_0/

[1] K. Ito, K. Kunisch, L p (Ω)-optimization with p[0,1) , (2012)

[2] G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices, Comput. Optim. Appl. 44 (2009), 159 -181 | MR 2556849 | Zbl 1185.49031

[3] E. Casas, R. Herzog, G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional, SIAM J. Control Optim. 22 (2012), 795 -820 | MR 3023751 | Zbl 1278.49026

[4] C. Clason, K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESAIM Control Optim. Calc. Var. 17 (2011), 243 -266 | Numdam | MR 2775195 | Zbl 1213.49041

[5] F. Tröltzsch, A minimum principle and a generalized bang-bang principle for a distributed optimal control problem with constraints on control and state, Z. Angew. Math. Mech. 59 (1979), 737 -739 | MR 572382 | Zbl 0425.49014

[6] D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lect. Notes Math. vol. 1459 , Springer-Verlag, Berlin (1990) | MR 1090951 | Zbl 0732.49002

[7] M. Bergounioux, D. Tiba, Some examples of optimality conditions for convex control problems with general constraints, Control of Partial Differential Equations and Applications, Laredo, 1994, Lect. Notes Pure Appl. Math. vol. 174 , Dekker, New York (1996), 23 -30 | MR 1364634 | Zbl 0851.49017

[8] M. Bergounioux, F. Tröltzsch, Optimality conditions and generalized bang-bang principle for a state-constrained semilinear parabolic problem, Numer. Funct. Anal. Optim. 17 (1996), 517 -536 | MR 1404833 | Zbl 0858.49021

[9] M. Fu, B.R. Barmish, Adaptive stabilization of linear systems via switching control, IEEE Trans. Autom. Control 31 (1986), 1097 -1103 | MR 864140 | Zbl 0607.93041

[10] D. Liberzon, Switching in Systems and Control, Systems Control Found. Appl. , Birkhäuser Boston Inc., Boston, MA (2003) | MR 1987806 | Zbl 1036.93001

[11] R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems, SIAM Rev. 49 (2007), 545 -592 | MR 2375524 | Zbl 1127.93005

[12] Q. Lü, E. Zuazua, Robust null controllability for heat equations with unknown switching control mode, Discrete Contin. Dyn. Syst., Ser. B (2013) | MR 3195365

[13] E. Zuazua, Switching control, J. Eur. Math. Soc. 13 (2011), 85 -117 | MR 2735077 | Zbl 1203.49011

[14] W. Schirotzek, Nonsmooth Analysis, Universitext , Springer, Berlin (2007) | MR 2330778 | Zbl 1120.49001

[15] I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math. vol. 28 , SIAM, Philadelphia (1999) | MR 1727362 | Zbl 0939.49002

[16] H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books Math./Ouvrages Math. SMC , Springer, New York (2011) | MR 2798533

[17] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York (2010) | MR 2759829

[18] K. Yosida, Functional Analysis, Grundlehren Math. Wiss. vol. 123 , Springer-Verlag, Berlin (1980) | MR 617913 | Zbl 0152.32102

[19] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics Appl. Math. vol. 31 , Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2000) | MR 1786735 | Zbl 0988.49003

[20] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing Co., Amsterdam (1973) | MR 348562 | Zbl 0252.47055

[21] C. Silva, E. Trélat, Smooth regularization of bang-bang optimal control problems, IEEE Trans. Autom. Control 55 (2010), 2488 -2499 | MR 2721891

[22] M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Ser. Optim. vol. 11 , Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011) | MR 2839219 | Zbl 1235.49001

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

[24] A. Logg, G.N. Wells, DOLFIN: Automated finite element computing, ACM Trans. Math. Softw. 37 (2010), 1 -28 | MR 2738227

[25] A. Logg, K.-A. Mardal, G.N. Wells, et al., Automated Solution of Differential Equations by the Finite Element Method, Springer (2012), http://fenicsproject.org | MR 3075806