量子系統的控制最早在1980年左右被提出來研究，而現在量子狀態的追蹤控制在改善通訊和計算上是很重要的課題。量子系統的動態方程式的描述是來自於 方程式，量子系統的狀態是波函數且含有機率的含意，而量子控制系統可視為是一個雙線性的控制系統。本篇論文旨在依據設計的需求來獲得量子追蹤系統的最佳控制解，並藉由求取偏微分方程式HJE的解來設計量子系統的最佳追蹤控制信號。由於雙線性系統的HJE不能表達成一個簡單完整的解，在此使用張量級數的技術來近似偏微分方程式HJE成一些Ricaati-like方程式，藉由求得那些Ricaati-like方程式的解來獲得HJE的近似解，並藉此來設計量子系統的近似最佳追蹤控制信號。在這篇論文中亦保含考慮量子系統含有隨機係數的擾動，而此擾動在量子控制系統中可視為是一個和狀態有關的雜訊項，而量子隨機系統的最佳追蹤控制的設計也是量子控制系統發展中的一個課題。最後，我們提出氫原子核的自旋 1/2 系統來作為此量子追蹤控制系統的設計示範，分別考慮使用單方向磁場和雙垂直方向的磁場控制，和含有隨機變化的系統控制這三個方面。藉由這三個例子的模擬結果之間的相互比較來討論使用此設計方法是否可行，以及對照不同情況時得到結果之間的差異來比較設計的優劣之處。

In this study, in order to specify quantum state to any desired state we need for the use of communication and computation, the quantum control system is formulated as a bilinear state space tracking system. An optimal tracking control is proposed to achieve state-tracking by solving the Hamilton-Jacobi equation (HJE). In order to avoid the difficulty in solving the HJE with a closed-form solution, the technique of formal tensor power series is employed to treat the partial differential equation HJE to obtain the optimal tracking control in quantum systems from the approximate perspective. If the quantum system suffers from stochastic parameter variations, it could be modeled as state-dependent noise. In this situation, stochastic optimal tracking control design is also developed for quantum systems. Finally, several examples are given to illustrate the design procedure and proposed method.

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