Error estimates for the approximation of multibang control problems

This work is concerned with optimal control problems where the objective functional consists of a tracking-type functional and an additional “multibang” regularization functional that promotes optimal control taking values from a given discrete set pointwise almost everywhere. Under a regularity condition on the set where these discrete values are attained, error estimates for the Moreau–Yosida approximation (which allows its solution by a semismooth Newton method) and the discretization of the problem are derived. Numerical results support the theoretical findings.

[1]  Gerd Wachsmuth,et al.  Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints , 2011 .

[2]  Daniel Wachsmuth,et al.  Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations , 2017, 1705.01427.

[3]  Christian Clason,et al.  Convex regularization of discrete-valued inverse problems , 2017, 1707.01041.

[4]  Michael Hinze,et al.  A note on variational discretization of elliptic Neumann boundary control , 2009 .

[5]  Gerd Wachsmuth,et al.  Convergence and regularization results for optimal control problems with sparsity functional , 2011 .

[6]  Anders Logg,et al.  DOLFIN: Automated finite element computing , 2010, TOMS.

[7]  Georg Stadler,et al.  Elliptic optimal control problems with L1-control cost and applications for the placement of control devices , 2009, Comput. Optim. Appl..

[8]  Kazufumi Ito,et al.  Lagrange multiplier approach to variational problems and applications , 2008, Advances in design and control.

[9]  K. Kunisch,et al.  Total variation regularization of multi-material topology optimization , 2017, 1708.06165.

[10]  Anders Logg,et al.  DOLFIN: a C++/Python Finite Element Library , 2012 .

[11]  Michael Hintermüller,et al.  Moreau-Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment , 2009, SIAM J. Numer. Anal..

[12]  Anders Logg,et al.  The FEniCS Project Version 1.5 , 2015 .

[13]  Christian Clason,et al.  Vector-Valued Multibang Control of Differential Equations , 2016, SIAM J. Control. Optim..

[14]  Gerd Wachsmuth,et al.  Necessary Conditions for Convergence Rates of Regularizations of Optimal Control Problems , 2011, System Modelling and Optimization.

[15]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[16]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[17]  D. Wachsmuth,et al.  An iterative Bregman regularization method for optimal control problems with inequality constraints , 2016, 1603.05792.

[18]  Michael Hinze,et al.  A note on the approximation of elliptic control problems with bang-bang controls , 2010, Computational Optimization and Applications.

[19]  Michael Ulbrich,et al.  Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces , 2011, MOS-SIAM Series on Optimization.

[20]  Michael Hintermüller,et al.  The Length of the Primal-Dual Path in Moreau-Yosida-Based Path-Following Methods for State Constrained Optimal Control , 2014, SIAM J. Optim..

[21]  K. Kunisch,et al.  A convex analysis approach to multi-material topology optimization. , 2016, 1702.07525.

[22]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[23]  Karl Kunisch,et al.  Multi-bang control of elliptic systems , 2014 .

[24]  K. Kunisch,et al.  A convex analysis approach to optimal controls with switching structure for partial differential equations , 2016, 1702.07540.

[25]  V. Barbu,et al.  Convexity and optimization in banach spaces , 1972 .