It is well known that optimal control problems with L1-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L2-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method.
Accepté le :
DOI : 10.1051/cocv/2017049
Mots clés : Optimal control, group sparsity, directional sparsity, bang-bang principle, stability analysis, discretization error estimates
@article{COCV_2018__24_2_811_0,
author = {Schneider, Christopher and Wachsmuth, Gerd},
title = {Regularization and discretization error estimates for optimal control of {ODEs} with group sparsity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {811--834},
publisher = {EDP-Sciences},
volume = {24},
number = {2},
year = {2018},
doi = {10.1051/cocv/2017049},
zbl = {1402.49019},
mrnumber = {3816416},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017049/}
}
TY - JOUR AU - Schneider, Christopher AU - Wachsmuth, Gerd TI - Regularization and discretization error estimates for optimal control of ODEs with group sparsity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 811 EP - 834 VL - 24 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017049/ DO - 10.1051/cocv/2017049 LA - en ID - COCV_2018__24_2_811_0 ER -
%0 Journal Article %A Schneider, Christopher %A Wachsmuth, Gerd %T Regularization and discretization error estimates for optimal control of ODEs with group sparsity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 811-834 %V 24 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017049/ %R 10.1051/cocv/2017049 %G en %F COCV_2018__24_2_811_0
Schneider, Christopher; Wachsmuth, Gerd. Regularization and discretization error estimates for optimal control of ODEs with group sparsity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 811-834. doi : 10.1051/cocv/2017049. http://www.numdam.org/articles/10.1051/cocv/2017049/
[1] , , and , Error Bounds for Euler Approximation of Linear-Quadratic Control Problems with Bang-Bang Solutions. Numerical Algebra, Control Optimiz. 2 (2012) 547–570. | DOI | MR | Zbl
[2] and , Linear-quadratic control problems with L1-control cost. Optimal. Control Appl. Methods 36 (2015) 512–534. | DOI | MR | Zbl
[3] , and , Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions. Appl. Math. Comput. 287–288 (2016) 104–124. | MR | Zbl
[4] and , Regularization and Discretization of Linear-Quadratic Control Problems. Control and Cybernetics 40 (2011) 903–920. | MR | Zbl
[5] and , An Implicit Discretization Scheme for Linear-Quadratic Control Problems with Bang–Bang Solutions. Optimization Methods and Software 29 (2014) 535–560. | DOI | MR | Zbl
[6] and , Second Order Optimality Conditions for the L1-Minimization Problem. Appl. Math. Optimiz. 13 (1985) 45–58. | DOI | MR | Zbl
[7] and , Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Series in Statistics. Springer (2011). | DOI | MR | Zbl
[8] and , Analysis of the velocity tracking control problem for the 3d evolutionary navier–stokes equations. SIAM J. Control Optimiz. 54 (2016) 99–128. | DOI | MR | Zbl
[9] , and , Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations. ESAIM: COCV 23 (2017) 263–295. | Numdam | MR | Zbl
[10] and , A note on the approximation of elliptic control problems with bang-bang controls. Comput. Optimiz. Appl. An Inter. J. 51 (2012) 931–939. | DOI | MR | Zbl
[11] and , in Nonlinear Control and Optimization. SIAM J. Control Optimiz. 31 (1993) 569–603. | DOI | MR | Zbl
[12] and , Metric Regularity under Approximations. Control and Cybernetics 38 (2009) 1283–1303. | MR | Zbl
[13] , On stability of bang-bang type controls. SIAM J. Control Optimiz. 41 (2003) 1843–1867. | DOI | MR | Zbl
[14] , Structural Stability Investigations of Bang-Singular-Bang Optimal Controls. J. Optimiz. Theory Appl. 152 (2012) 605–631. | DOI | MR | Zbl
[15] , Discretization of semilinear bang-singular-bang control problems. Computat. Optimiz. Appl. An International J. 64 (2016) 295–326. | DOI | MR | Zbl
[16] , and Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press (2015). | DOI | MR
[17] , and , The Euler Method for Linear Control Systems Revisited. In Proceedings of the 9th International Conference on Large-Scale Scientific Computations, Sozopol (2013) 90–97. | MR
[18] , and , Annular and sectorial sparsity in optimal control of elliptic equations. Computational Optimization and Applications. An International Journal 62 (2015) 157–180. | DOI | MR | Zbl
[19] , and , Directional sparsity in optimal control of partial differential equations. SIAM J. Control and Optimiz. 50 (2012) 943–963. | DOI | MR | Zbl
[20] and , Sensitivity Analysis of Optimal Control Problems with Bang-Bang Controls. In Proceedings of the 42nd IEEE Conference on Decision and Control 4 (2003) 3281–3286.
[21] and Equivalence of Second Order Optimality Conditions for Bang-Bang Control Problems. Part 1: Main Results. Control and Cybernetics 34 (2005). | MR | Zbl
[22] and , Equivalence of Second Order Optimality Conditions for Bang-Bang Control Problems. Part 2: Proofs, Variational Derivatives and Representations. Control and Cybernetics 36 (2007). | MR | Zbl
[23] and , Metric Regularity and Stability of Optimal Control Problems for Linear Systems. SIAM J. Control Optimiz. 51 (2013) 4118–4137. | DOI | MR | Zbl
[24] and , Regularization of Linear-Quadratic Control Problems with L1 -Control Cost. In C. Pötzsche, C. Heuberger, B. Kaltenbacher, and F. Rendl, System Modeling and Optimization, volume 443 of IFIP Advances in Information and Communication Technology, pages 296–305. 26th IFIP TC 7 Conference, CSMO 2013, Springer 2014.
[25] , Elliptic Optimal Control Problems with L1-Control Cost and Applications for the Placement of Control Devices. Comput. Optimiz. Appl. 44 (2009) 159–181. | DOI | MR | Zbl
[26] , Error Analysis of Discrete Approximations to Bang-Bang Optimal Control Problems: The Linear Case. Control and Cybernetics 34 (2005) 967–982. | MR | Zbl
[27] and On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods 27 (2006) 301–321. | DOI | MR
[28] and , Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints. Control and Cybernetics 40 (2011) 1125–1158. | MR | Zbl
[29] and , Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858–886. | Numdam | MR | Zbl
Cité par Sources :
