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Tropical Kraus maps for optimal control of switched systems

Stéphane Gaubert 1, 2 Nikolas Stott 2, 1
1 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.
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https://hal.inria.fr/hal-01674496
Contributor : Stephane Gaubert <>
Submitted on : Wednesday, January 3, 2018 - 12:17:52 AM
Last modification on : Friday, April 30, 2021 - 10:00:13 AM

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  • HAL Id : hal-01674496, version 1
  • ARXIV : 1706.04471

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Stéphane Gaubert, Nikolas Stott. Tropical Kraus maps for optimal control of switched systems. CDC 2017 - 57th IEEE Annual Conference on Decision and Control, Dec 2017, Melbourne, Australia. pp.1-15. ⟨hal-01674496⟩

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