Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator (- Δ) $$ (0 < s < 1). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.
Keywords: Fractional heat equation, Dirichlet and Robin external optimal control problems, admissible control operator, turnpike property, exponential turnpike property.
@article{COCV_2021__27_1_A3_0,
author = {Warma, Mahamadi and Zamorano, Sebasti\'an},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Exponential {Turnpike} property for fractional parabolic equations with non-zero exterior data},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020076},
mrnumber = {4201973},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020076/}
}
TY - JOUR AU - Warma, Mahamadi AU - Zamorano, Sebastián ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Exponential Turnpike property for fractional parabolic equations with non-zero exterior data JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020076/ DO - 10.1051/cocv/2020076 LA - en ID - COCV_2021__27_1_A3_0 ER -
%0 Journal Article %A Warma, Mahamadi %A Zamorano, Sebastián %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Exponential Turnpike property for fractional parabolic equations with non-zero exterior data %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020076/ %R 10.1051/cocv/2020076 %G en %F COCV_2021__27_1_A3_0
Warma, Mahamadi; Zamorano, Sebastián. Exponential Turnpike property for fractional parabolic equations with non-zero exterior data. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 1. doi: 10.1051/cocv/2020076
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Cité par Sources :
The work of the first author is partially supported by Air Force Office of Scientific Research (AFOSR) under Award NO [FA9550-18-1-0242] and Army Research Office (ARO) under Award NO: W911NF-20-1-0115. The second author is supported by the Conicyt PAI Convocatoria Nacional Subvención a la Instalación en la Academia Convocatoria 2019 PAI77190106.
