Volume 26, 2020
|Number of page(s)||38|
|Published online||25 June 2020|
Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints
Denis Poisson Institute, UMR CNRS 7013, University of Orléans, France.
2 XLIM Research Institute, UMR CNRS 7252, University of Limoges, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 5 April 2019
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.
Mathematics Subject Classification: 34K35 / 26A33 / 34A08 / 49J15 / 49K40 / 93C15
Key words: 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
© The authors. Published by EDP Sciences, SMAI 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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