具有任意代价的$n$能级量子系统极小元的共振
Resonance of Minimizers for $n$-level Quantum Systems with an Arbitrary Cost
摘要：
研究了一个描述$n$能级量子系统中激光诱导布居转移的最优控制问题。对于一个凸成本只依赖于模的控制（即激光强度），我们证明了总是存在一个极小的共振。这允许证明在实验物理学中使用的一些策略。这也是相当重要的，因为它允许显着降低问题的复杂性（并扩展我们以前的一些结果$n=2$和$n=3$）：而不是寻找极小的领域$S^{2n-1}\subset\C^n$之一是减少到寻找极小的领域$S^{n-1}\subset \R^n$。此外，对于约化问题，我们还研究了严格反常极小的存在性问题。
英文摘要：
We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\subset\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\subset \R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.
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