A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as usual, a particular case of the Mangassarian-Fromovitz type condition case. Illustrative examples are presented.
Keywords: Minimax optimal control problems, mixed constrained, maximum principle, nonsmooth analysis
@article{COCV_2021__27_1_A74_0,
author = {Patzi Aquino, Paola Geovanna and de Pinho, M. D. R. and Silva, Geraldo Nunes},
title = {Necessary optimality conditions for minimax optimal control problems with mixed constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021069},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021069/}
}
TY - JOUR AU - Patzi Aquino, Paola Geovanna AU - de Pinho, M. D. R. AU - Silva, Geraldo Nunes TI - Necessary optimality conditions for minimax optimal control problems with mixed constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021069/ DO - 10.1051/cocv/2021069 LA - en ID - COCV_2021__27_1_A74_0 ER -
%0 Journal Article %A Patzi Aquino, Paola Geovanna %A de Pinho, M. D. R. %A Silva, Geraldo Nunes %T Necessary optimality conditions for minimax optimal control problems with mixed constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021069/ %R 10.1051/cocv/2021069 %G en %F COCV_2021__27_1_A74_0
Patzi Aquino, Paola Geovanna; de Pinho, M. D. R.; Silva, Geraldo Nunes. Necessary optimality conditions for minimax optimal control problems with mixed constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 72. doi: 10.1051/cocv/2021069
[1] , , and , A weak maximum principle for optimal control problems with mixed constraints under a constant rank condition. IMA J. Math. Control Inf . 37 (2020) 1021–1047.
[2] , Mixed constrained control problems. J. Math. Anal. Appl. 278 (2003) 293–307.
[3] and , Weak maximum principle for optimal control problems with mixed constrained. Nonlinear Anal. 48 (2002) 1179–1196.
[4] and , Necessary conditions for constrained problems under Mangasarian-Fromowitz conditions. SIAM J. Control Optim. 47 (2008) 535–552.
[5] and , Necessary conditions for optimal control problems involving nonlinear differential algebraic equations. J. Math. Anal. Appl. 212 (1997) 493–516.
[6] , and , A weak maximum principle for optimal control problems with nonsmooth mixed constraints. Set Valued Variat. Anal. 17 (2009) 203–221.
[7] ; and , A maximum principle for optimal control problems with mixed constraints. IMA J. Math. Control Inf . 18 (2001) 189–205.
[8] , Real Analysis-Modern Techniques and Their Applications. John Wiley and Sons (1999).
[9] , Calculus of Variations and Optimal Control Theory. Willey, New York (1966).
[10] , , and , Minimax optimal control problem with state constraints. Eur. J. Control 32 (2016) 24–31.
[11] and , Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints. Set-Valued Var. Anal. 24 (2016) 449–470.
[12] and , Necessary optimality conditions for implicit control systems with applications to control of differential algebraic equations. Set-Valued Var. Anal. 26 (2018) 179–203.
[13] , Variational Analysis and Applications. Springer (2018).
[14] , Optimization, A Theory of Necessary Conditions. Princeton University Press, Princeton, New Jersey (1976).
[15] , Second order conditions for a weak local minimum in an optimal control problem (necessity and sufficiency. Soviet Math. Dokl. 16 (1975) 1480–1484.
[16] and , First and second order necessary conditions for control problems with constraints. Trans. Am. Math. Soc. 346 (1994) 421–453.
[17] and , Optimality conditions for constrained control problems. SIAM J. Control Optim. 34 (1996) 635–659.
[18] , Optimal Control, Birkhauser Basel, Boston (2000).
[19] , Minimax optimal control. SIAM J. Control Optim. 44 (2005) 939–967.
[20] and , A maximum principle for nonsmooth-control problems with state constraints. J. Math. Anal. Appl. 89 (1982) 212–232.
[21] and , Necessary conditions for optimal control problems with state constraints. Trans. Am. Math. Soc. 350 (1998) 1181–1204.
[22] , Necessary conditions for bilevel dynamic optimization problems. SIAM J. Control Optim. 33 (1995) 1208–1223.
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