In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 < α ≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥ α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.
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DOI : 10.1051/cocv/2019021
Mots-clés : Optimal control, fractional calculus, Riemann-Liouville and Caputo operators, Filippov’s existence theorem, Pontryagin maximum principle, needle-like variations, Ekeland’s variational principle, adjoint vector, Hamiltonian system, Hamiltonian maximization condition, transversality conditions
@article{COCV_2020__26_1_A35_0,
author = {Bergounioux, Ma{\"\i}tine and Bourdin, Lo{\"\i}c},
title = {Pontryagin maximum principle for general {Caputo} fractional optimal control problems with {Bolza} cost and terminal constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019021},
mrnumber = {4116679},
zbl = {1447.49035},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019021/}
}
TY - JOUR AU - Bergounioux, Maïtine AU - Bourdin, Loïc TI - Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019021/ DO - 10.1051/cocv/2019021 LA - en ID - COCV_2020__26_1_A35_0 ER -
%0 Journal Article %A Bergounioux, Maïtine %A Bourdin, Loïc %T Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019021/ %R 10.1051/cocv/2019021 %G en %F COCV_2020__26_1_A35_0
Bergounioux, Maïtine; Bourdin, Loïc. Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 35. doi : 10.1051/cocv/2019021. http://www.numdam.org/articles/10.1051/cocv/2019021/
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