本研究計畫之目的在求解2 點廣義頻譜Nevanlinna-Pick 插值問題，此類差值問題在求從單位開圓盤對應到單位頻譜球內之解析函數，並使函數值與導數值滿足特定插值條件，其中函數值與導數值都是 2×2 複數矩陣。此一問題為結合頻譜Nevanlinna-Pick 以及頻譜Caratheodory-Fejer 兩種插值問題而得，並為建立 μ 控制器合成之數學理論的關鍵，雖經國際學術界研究達20 年，但尚未有確切的結果。此計畫係97 年研究計畫之延續，結合研究2 點頻譜Nevanlinna-Pick 以及Caratheodory-Fejer 插值問題的結論，延伸處理兩點2×2 廣義頻譜Nevanlinna-Pick 插值問題；第一年研究的重點在於將兩點問題的解之存在性與解的求法；第二年則以第一年的研究成果為基礎，討論如何建立該插值問題解的實現問題，並討論延伸到多插值點的相關情形；第三年則專注在解三點SNP 及其對應GSNP 插值問題。 The two-point generalized spectral Nevanlinna-Pick interpolation problem will be investigated in this project which is a consecutive project of previous year. The aim is to find an analytic function from the open unit disc into 2×2 open spectral unit ball such that this function satisfies certain interpolation conditions on its values and derivatives. It is obvious that this problem is to combine the spectral Nevanlinna-Pick and spectral Caratheodory-Fejer problems, and has been studies almost 20 years; which is considered as the key to build up a definitive mathematical theory for μ-synthesis. This research is the continuation of our previous project, which want to study the two-point generalized spectral Nevanlinna-Pick interpolation problem with interpolating points based on the result of our previous study on spectral Nevanlinna-Pick and spectral Caratheodory-Fejer problems. The solvability condition of the generalized spectral Nevanlinna-Pick problem is analyzed in the first year of this project. Furthermore, the construction and realization of the interpolation functions will be considered in the consecutive year. In the meanwhile, the extension to multiple interpolating points will also be investigated. In the third year, the research effort will focus on solving 3-point SNP interpolation problems and its corresponding GSNP problems.
