切换系统最优控制的热带Kraus映射
Tropical Kraus maps for optimal control of switched systems
摘要：
Kraus映射（完全正的保迹映射）在量子信息中经典地出现，因为它们描述了非对易概率测度的演化。我们介绍热带类似的Kraus地图，通过取代正半定矩阵的一个多值上确界的Löwner秩序。我们表明，热带Kraus映射的非线性特征向量确定切换最优控制问题的值函数的分段二次逼近。这导致了一个新的近似方法，我们说明了两个应用：1）近似的联合谱半径，2）计算的Hamilton-Jacobi PDE的近似解所产生的一类切换线性二次型问题先前研究的McEneaney。我们报告的数值实验，表明与早期的数值方案相比，由于我们的方法的“无LMI”性质的可扩展性方面的重大改进。
英文摘要：
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 Löwner 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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