平方和理論近來逐漸發展為成熟的技術並且受到廣泛的應用。然而當平方和理論應用於多項式系統並探討其穩定性時，其耦合(coupling)的問題會造成求解上的困難。在此份計畫中，我們會談論到藉由去耦合(decouping)來處理系統矩陣以及Lyapunov P的耦合問題，並藉由引入一寬鬆變數增加求解的便利。為了達成這個目標，我們將略述證明和示範兩個密切相關的模擬實例。而例子中會指出此所提計畫中陳述的概念是可以處理耦合問題的。 Recent theoretical developments show that Sum of Squares (SOS) is an evolving technique that finds its way in many applications. However, when applying SOS to a polynomial systems and checking for stability concerns, there is a coupling issue that hampers solvability when using SOS. In this project, we bring up an idea to tackle the coupling problem by decoupling the system matrices and Lyapunov $P$, and introducing a slack variable. Aiming to this goal, an outline of proof is shown and two examples of relevant simulations are demonstrated to. The examples clearly manifest that this project proposal provides an idea that can tackle the coupling problem. 研究期間：10008 ~ 10107
