Volume 29, 2023
|Number of page(s)||39|
|Published online||24 March 2023|
Optimal control of ensembles of dynamical systems*
School of Computation, Information and Technology, Technical University of Munich,
Garching b. München, Germany
2 Munich Center for Machine Learning (MCML), Munich, Germany
** Corresponding author: firstname.lastname@example.org
Accepted: 15 February 2023
In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a Γ-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the L2-strong topology of the minimizers of the original problem. Using again a Γ-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in ℝ2.
Mathematics Subject Classification: 49J15 / 49K15 / 49M05
Key words: Optimal control / simultaneous control / Γ-convergence / gradient-based minimization / Pontryagin Maximum Principle
A great part of the work presented here was done while the Author was a Ph.D. candidate at Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy. The Author acknowledges partial support from INdAM–GNAMPA. The Author thanks Prof. Andrei Agrachev for encouragement and helpful discussions. Finally, the Author wants to express his gratitude to two anonymous Referees, whose comments helped to improve the quality of the paper. In particular, the results presented in Section 6 were inspired by the observation of a Reviewer.
© The authors. Published by EDP Sciences, SMAI 2023
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