Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau–Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.
DOI: 10.1051/cocv/2015017
Keywords: Optimal control, switching control, partial differential equations, nonsmooth optimization, convexification, semi-smooth Newton method
@article{COCV_2016__22_2_581_0, author = {Clason, Christian and Ito, Kazufumi and Kunisch, Karl}, title = {A convex analysis approach to optimal controls with switching structure for partial differential equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {581--609}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015017}, mrnumber = {3491785}, zbl = {1338.49056}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015017/} }
TY - JOUR AU - Clason, Christian AU - Ito, Kazufumi AU - Kunisch, Karl TI - A convex analysis approach to optimal controls with switching structure for partial differential equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 581 EP - 609 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015017/ DO - 10.1051/cocv/2015017 LA - en ID - COCV_2016__22_2_581_0 ER -
%0 Journal Article %A Clason, Christian %A Ito, Kazufumi %A Kunisch, Karl %T A convex analysis approach to optimal controls with switching structure for partial differential equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 581-609 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015017/ %R 10.1051/cocv/2015017 %G en %F COCV_2016__22_2_581_0
Clason, Christian; Ito, Kazufumi; Kunisch, Karl. A convex analysis approach to optimal controls with switching structure for partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 581-609. doi : 10.1051/cocv/2015017. http://archive.numdam.org/articles/10.1051/cocv/2015017/
H. Attouch and H. Brezis, Duality for the Sum of Convex Functions in General Banach Spaces, in Aspects of Mathematics and its Applications. North-Holland, Amsterdam (1986) 125–133. | MR | Zbl
H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011). | MR | Zbl
Smoothing and first order methods: a unified framework. SIAM J. Optim. 22 (2012) 557–580. | DOI | MR | Zbl
and ,A. Braides, -Convergence for Beginners. Oxford University Press, Oxford (2002). | MR | Zbl
Perturbations of nonlinear maximal monotone sets in Banach space. Commun. Pure Appl. Math. 23 (1970) 123–144. | DOI | MR | Zbl
, and ,Optimal switching for ordinary differential equations. SIAM J. Contr. Optim. 22 (1984) 143–161. | DOI | MR | Zbl
and ,Multi-bang control of elliptic systems. Ann. Inst. Henri Poincaré (C) Anal. Non Lin. 31 (2014) 1109–1130. | DOI | Numdam | MR | Zbl
and ,I. Ekeland and R. Témam, Convex Analysis and Variational Problems, vol. 28 of Classics Appl. Math. SIAM, Philadelphia (1999). | MR | Zbl
Optimal switching boundary control of a string to rest in finite time. ZAMM 88 (2008) 283–305. | DOI | MR | Zbl
,Penalty techniques for state constrained optimal control problems with the wave equation. SIAM J. Control Optim. 48 (2009/2010) 3026–3051. | DOI | MR | Zbl
,Modeling and analysis of modal switching in networked transport systems. Appl. Math. Optim. 59 (2009) 275–292. | DOI | MR | Zbl
, and ,Relaxation methods for mixed-integer optimal control of partial differential equations. Comput. Optim. Appl. 55 (2013) 197–225. | DOI | MR | Zbl
and ,J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis. Springer-Verlag, Berlin (2001). | MR | Zbl
Optimal control of switched distributed parameter systems with spatially scheduled actuators. Automatica J. IFAC 45 (2009) 312–323. | DOI | MR | Zbl
and ,K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia, PA (2008). | MR | Zbl
Optimal control with , , control cost. SIAM J. Control Optim. 52 (2014) 1251–1275. | DOI | MR | Zbl
and ,Robust null controllability for heat equations with unknown switching control mode. Discrete Contin. Dyn. Syst. B 34 (2014) 4183–4210. | DOI | MR | Zbl
and ,Stabilization of the wave equation by on-off and positive-negative feedbacks. ESAIM: COCV 7 (2002) 335–377. | Numdam | MR | Zbl
and ,Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93 (1965) 273–299. | DOI | Numdam | MR | Zbl
,W. Schirotzek, Nonsmooth Analysis. Universitext, Springer, Berlin (2007). | MR | Zbl
Stability criteria for switched and hybrid systems. SIAM Rev. 49 (2007) 545–592. | DOI | MR | Zbl
, , , and ,M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia, PA (2011). | MR | Zbl
Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim. 20 (1989) 223–235. | DOI | MR | Zbl
,Switching control. J. Eur. Math. Soc. 13 (2011) 85–117. | DOI | MR | Zbl
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