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.

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.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017049
Classification : 49K15, 49J15, 49M15, 49M25, 65K15
Mots clés : Optimal control, group sparsity, directional sparsity, bang-bang principle, stability analysis, discretization error estimates
Schneider, Christopher 1 ; Wachsmuth, Gerd 1

1
@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] W. Alt, R. Baier, M. Gerdts and F. Lempio, 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] W. Alt and C. Schneider, Linear-quadratic control problems with L1-control cost. Optimal. Control Appl. Methods 36 (2015) 512–534. | DOI | MR | Zbl

[3] W. Alt, C. Schneider and M. Seydenschwanz, 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] W. Alt and M. Seydenschwanz, Regularization and Discretization of Linear-Quadratic Control Problems. Control and Cybernetics 40 (2011) 903–920. | MR | Zbl

[5] W. Alt and M. Seydenschwanz, 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] A. Ben-Tal and J. Zowe, Second Order Optimality Conditions for the L1-Minimization Problem. Appl. Math. Optimiz. 13 (1985) 45–58. | DOI | MR | Zbl

[7] P. Bühlmann and S. Van De Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer Series in Statistics. Springer (2011). | DOI | MR | Zbl

[8] E. Casas and K. Chrysafinos, 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] E. Casas, R. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations. ESAIM: COCV 23 (2017) 263–295. | Numdam | MR | Zbl

[10] K. Deckelnick and M. Hinze, 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] A.L. Dontchev and W.W. Hager, Lipschitzian Stability in Nonlinear Control and Optimization. SIAM J. Control Optimiz. 31 (1993) 569–603. | DOI | MR | Zbl

[12] A.L. Dontchev and V.M. Veliov, Metric Regularity under Approximations. Control and Cybernetics 38 (2009) 1283–1303. | MR | Zbl

[13] U. Felgenhauer, On stability of bang-bang type controls. SIAM J. Control Optimiz. 41 (2003) 1843–1867. | DOI | MR | Zbl

[14] U. Felgenhauer, Structural Stability Investigations of Bang-Singular-Bang Optimal Controls. J. Optimiz. Theory Appl. 152 (2012) 605–631. | DOI | MR | Zbl

[15] U. Felgenhauer, Discretization of semilinear bang-singular-bang control problems. Computat. Optimiz. Appl. An International J. 64 (2016) 295–326. | DOI | MR | Zbl

[16] T. Hastie, R. Tibshirani and M. Wainwright. Statistical Learning with Sparsity: The Lasso and Generalizations. CRC Press (2015). | DOI | MR

[17] J.L. Haunschmied, A. Pietrus and V.M. Veliov, 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] R. Herzog, J. Obermeier and G. Wachsmuth, 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] R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations. SIAM J. Control and Optimiz. 50 (2012) 943–963. | DOI | MR | Zbl

[20] J-H.R. Kim and H. Maurer, 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] N.P. Osmolovskii and H. Maurer. Equivalence of Second Order Optimality Conditions for Bang-Bang Control Problems. Part 1: Main Results. Control and Cybernetics 34 (2005). | MR | Zbl

[22] N.P. Osmolovskii and H. Maurer, 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] M. Quincampoix and V.M. Veliov, Metric Regularity and Stability of Optimal Control Problems for Linear Systems. SIAM J. Control Optimiz. 51 (2013) 4118–4137. | DOI | MR | Zbl

[24] C. Schneider and W. Alt, 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] G. Stadler, 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] V.M. Veliov, Error Analysis of Discrete Approximations to Bang-Bang Optimal Control Problems: The Linear Case. Control and Cybernetics 34 (2005) 967–982. | MR | Zbl

[27] G. Vossen and H. Maurer. On L1-minimization in optimal control and applications to robotics. Optimal Control Appl. Methods 27 (2006) 301–321. | DOI | MR

[28] D. Wachsmuth and G. Wachsmuth, Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints. Control and Cybernetics 40 (2011) 1125–1158. | MR | Zbl

[29] G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858–886. | Numdam | MR | Zbl

Cité par Sources :