We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation’s dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.
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Keywords: Nonlinear optimal control, sensitivity analysis, Turnpike property, model predictive control
@article{COCV_2021__27_1_A58_0,
author = {Gr\"une, Lars and Schaller, Manuel and Schiela, Anton},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic {PDEs}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021030},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021030/}
}
TY - JOUR AU - Grüne, Lars AU - Schaller, Manuel AU - Schiela, Anton ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021030/ DO - 10.1051/cocv/2021030 LA - en ID - COCV_2021__27_1_A58_0 ER -
%0 Journal Article %A Grüne, Lars %A Schaller, Manuel %A Schiela, Anton %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021030/ %R 10.1051/cocv/2021030 %G en %F COCV_2021__27_1_A58_0
Grüne, Lars; Schaller, Manuel; Schiela, Anton. Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 56. doi: 10.1051/cocv/2021030
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This work was supported by the DFG Grants GR 1569/17-1 and SCHI 1379/5-1.
