Metric sub-regularity in optimal control of affine problems with free end state
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 47.

The paper investigates the property of Strong Metric sub-Regularity (SMsR) of the mapping representing the first order optimality system for a Lagrange-type optimal control problem which is affine with respect to the control. The terminal time is fixed, the terminal state is free, and the control values are restricted in a convex compact set U. The SMsR property is associated with a reference solution of the optimality system and ensures that small additive perturbations in the system result in solutions which are at distance to the reference one, at most proportional to the size of the perturbations. A general sufficient condition for SMsR is obtained for appropriate space settings and then specialized in the case of a polyhedral set U and purely bang-bang reference control. Sufficient second-order optimality conditions are obtained as a by-product of the analysis. Finally, the obtained results are utilized for error analysis of the Euler discretization scheme applied to affine problems.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2019046
Classification : 49K40, 49J53, 49J30, 49K15, 47J30
Mots-clés : Optimal control, affine control problems, bang-bang control, metric regularity, Pontryagin’s maximum principle, second-order optimality conditions, Euler discretization 
@article{COCV_2020__26_1_A47_0,
     author = {Osmolovskii, N.P. and Veliov, V.M.},
     title = {Metric sub-regularity in optimal control of affine problems with free end state},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2019046},
     mrnumber = {4144113},
     zbl = {1448.49033},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2019046/}
}
TY  - JOUR
AU  - Osmolovskii, N.P.
AU  - Veliov, V.M.
TI  - Metric sub-regularity in optimal control of affine problems with free end state
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2020
VL  - 26
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2019046/
DO  - 10.1051/cocv/2019046
LA  - en
ID  - COCV_2020__26_1_A47_0
ER  - 
%0 Journal Article
%A Osmolovskii, N.P.
%A Veliov, V.M.
%T Metric sub-regularity in optimal control of affine problems with free end state
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2020
%V 26
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2019046/
%R 10.1051/cocv/2019046
%G en
%F COCV_2020__26_1_A47_0
Osmolovskii, N.P.; Veliov, V.M. Metric sub-regularity in optimal control of affine problems with free end state. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 47. doi : 10.1051/cocv/2019046. http://www.numdam.org/articles/10.1051/cocv/2019046/

[1] W. Alt, U. Felgenhauer and M. Seydenschwanz, Euler discretization for a class of nonlinear optimal control problems with control appearing linearly. Comput. Optim. Appl. 69 (2018) 825–856 | DOI | MR | Zbl

[2] 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 (2016) 104–124 | MR | Zbl

[3] J.-P. Aubin and H. Frankowska, Set-valued Analysis. Birkhäuser, Boston, Basel, Berlin (1990) | MR | Zbl

[4] R. Cibulka, A.L. Dontchev and A.Y. Kruger, Strong metric subregularity of mappings in variational analysis and optimization. J. Math. Anal. Appl. 457 (2018) 1247–1282 | DOI | MR | Zbl

[5] A.L. Dontchev and W.W. Hager, Lipschitzian stability in nonlinear control and optimization. SIAM J. Control Optim. 31 (1993) 569–603 | DOI | MR | Zbl

[6] A.L. Dontchev and R.T. Rockafellar, Implicit Functions and Solution Mappings: A View from Variational Analysis. Second edition. Springer, New York (2014) | DOI | MR | Zbl

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

[8] U. Felgenhauer, Discretization of semilinear bang-singular-bang control problems. Comput. Optim. Appl. 64 (2016) 295–326 | DOI | MR | Zbl

[9] U. Felgenhauer, A Newton-type method and optimality test for problems with bang-singular-bang optimal control. Pure Appl. Funct. Anal. 1 (2016) 197–215 | MR | Zbl

[10] W.W. Hager, Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27 (1990) 1061–1080 | DOI | MR | Zbl

[11] M.R. Hestenes, Calculus of variations and optimal control theory. John Wiley & Sons (1966) | MR | Zbl

[12] A. Pietrus, T. Scarinci and V.M. Veliov, High order discrete approximations to Mayer’s problems for linear systems. SIAM J. Control Optim. 56 (2018) 102–119 | DOI | MR | Zbl

[13] J. Preininger, T. Scarinci and V.M. Veliov, Metric regularity properties in bang-bang type linear-quadratic optimal control problems. Set-Valued Variational Anal. 27 (2019) 381–404 | DOI | MR | Zbl

[14] M. Quincampoix and V. Veliov, Metric regularity and stability of optimal control problems for linear systems. SIAM J. Control Optim. 51 (2013) 4118–4137 | DOI | MR | Zbl

[15] T. Scarinci and V.M. Veliov, Higher-order numerical schemes for linear quadratic problems with bang-bang controls. Comput. Optim. Appl. 69 (2018) 403–422 | DOI | MR | Zbl

[16] M. Seydenschwanz, Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions. Comput. Optim. Appl. 61 (2015) 731–760 | DOI | MR | Zbl

[17] V.M. Veliov, Error analysis of discrete approximation to bang-bang optimal control problems: the linear case. Control Cybern. 34 (2005) 967–982 | MR | Zbl

Cité par Sources :

This research is supported by the Austrian Science Foundation (FWF) under grant No P31400-N32. The second author also acknowledges the partial support of the Erwin Schrödinger International Institute (ESI), Vienna.