We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than the usual one encountered in the hybrid setting, which requires that any optimal solution should cross the boundary of the constraint set transversely. Doing so, we apply the Pontryagin Maximum Principle to a sequence of regular optimal control problems whose integral cost approximates the time of crisis. Optimality conditions are derived by passing to the limit in the Hamiltonian system (without the use of the hybrid maximum principle). This convergence result essentially relies on the boundedness of the sequence of adjoint vectors in L$$. Our main contribution is to relate this property to the boundedness in L1 of a suitable sequence which allows to avoid the use of the transverse hypothesis on optimal paths. An example with non-transverse trajectories for which necessary conditions are derived highlights the use of this new condition.
Accepté le :
Première publication :
Publié le :
Keywords: Time crisis, Necessary optimality conditions, Hybrid Maximum Principle, Regularization, Transverse crossing time
@article{COCV_2021__27_1_A107_0,
author = {Bayen, T\'erence and Boumaza, Kenza and Rapaport, Alain},
title = {Necessary optimality condition for the minimal time crisis relaxing transverse condition \protect\emph{via} regularization},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021102},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021102/}
}
TY - JOUR AU - Bayen, Térence AU - Boumaza, Kenza AU - Rapaport, Alain TI - Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021102/ DO - 10.1051/cocv/2021102 LA - en ID - COCV_2021__27_1_A107_0 ER -
%0 Journal Article %A Bayen, Térence %A Boumaza, Kenza %A Rapaport, Alain %T Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021102/ %R 10.1051/cocv/2021102 %G en %F COCV_2021__27_1_A107_0
Bayen, Térence; Boumaza, Kenza; Rapaport, Alain. Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 105. doi: 10.1051/cocv/2021102
[1] and , Necessary first-order conditions for optimal crossing of a given region. Comput. Math. Model. 18 (2007) 397–419.
[2] , Viability Theory, Systems & Control: Foundations & Applications. Birkhäuser Boston (1991).
[3] , and , Viability Theory New Directions, second editions. Springer, Heidelberg (2011).
[4] , and , Value function for regional problems via dynamic programming and Pontryagin maximum principle. Math. Control Relat. Fields 8 (2018) 509–533.
[5] , and , Penalty function method for the minimal time crisis problem. ESAIM Proc. Surveys 71 (2021) 21–32.
[6] and , Second-order analysis for the time crisis problem. J. Convex Anal. 27 (2020) 139–163.
[7] and , About Moreau-Yosida regularization of the minimal time crisis problem. J. Convex Anal. 23 (2016) 263–290.
[8] and , About the minimal time crisis problem. ESAIM Proc. Surveys 57 (2017) 1–11.
[9] and , Minimal time crisis versus minimum time to reach a viability kernel: a case study in the prey-predator model. Optim. Control Appl. Meth. 40 (2019) 330–350.
[10] , Functional Analysis, Calculus of Variation, Optimal control. Vol. 264 of Graduate Texts in Mathematics. Springer, London (2013).
[11] , , and , Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics. Springer (1998).
[12] , The hybrid maximum principle is a consequence of Pontryagin maximum principle. Syst. Control Lett. 57 (2008) 964–970.
[13] and , Maximum principle for optimal control problems with intermediate constraints. Comput. Math. Model. 22 (2011) 180–215.
[14] and , Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints. Discrete Contin. Dyn. Syst. 29 (2011) 523–545.
[15] and , Scale of viability and minimal time of crisis. Set-Valued Var. Anal. 5 (1997) 227–246.
[16] and , Hybrid necessary principle. SIAM J. Control Optim. 43 (2005) 1867–1887.
[17] and , Convergence results for smooth regularizations of hybrid nonlinear optimal control problems. SIAM J. Control Optim. 49 (2011) 1498–1522.
[18] , , and , The Mathematical Theory of Optimal Processes. The Macmillan Co., New York (1964).
[19] and , Asymptotic approach on conjugate points for minimal time bang–bang controls. Syst. Control Lett. 59 (2010) 720–733.
[20] , Necessary optimality conditions for a class of optimal control problems with discontinuous integrand. Proc. Steklov Inst. Math. 262 (2008) 213–230.
[21] Team Commands, Inria Saclay, BOCOP: an open source toolbox for optimal control. http://bocop.org.
[22] , Optimal Control, Systems and Control: Foundations and Applications. Birkhäuser, Boston (2000).
[23] and , Theory of Chattering Control. Systems & Control: Foundations & Applications, Birkhäuser (1994).
Cité par Sources :
