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Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

Department of Mathematics and Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania
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Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2019, 11(7), 893; https://doi.org/10.3390/sym11070893
Received: 7 June 2019 / Revised: 30 June 2019 / Accepted: 4 July 2019 / Published: 8 July 2019
(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)
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Abstract

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs. View Full-Text
Keywords: maximum principle; optimal control; Einstein manifold; evolution dynamics; cost functional; submanifold integral maximum principle; optimal control; Einstein manifold; evolution dynamics; cost functional; submanifold integral
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Bejenaru, A.; Udriste, C. Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals. Symmetry 2019, 11, 893.

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