本論文乃探討具時間延遲之網格型類神經網路(Cellular Neural Networks, CNNs)的穩定性和混沌同步問題。網格型類神經網路是由許多陣列排列的神經元所組成，每個神經元內含非線性動態行為，並應用於解決影像處理和偏微分方程式的問題。因為在應用上，網格型類神經網路是否收歛是必須考量的特性，因此穩定性分析是一重要問題；此外，網格型類神經網路在某些參數情況下能表現出混沌行為，並可應用於安全通訊系統，所以網格型類神經網路之同步亦是一重要問題。

　　本論文提出具固定時間延遲之CNNs的全域指數收歛穩定的充分條件，此充分條件不但容易判別且比文獻上的條件更為廣泛。另外，本研究提出具時變時間延遲之CNNs的指數收歛穩定的充分條件，此充分條件和延遲時間相關，乃藉由計算Hamiltonian矩陣(Hamiltonian Matrix)之特徵值是否有根在虛軸上，以判定系統的穩定性，而不需要直接去解複雜的Riccati方程式(Algebraic Riccati Equation)，並且由此條件所推導而得的指數收歛速率比文獻上的方法更容易計算。除了探討上述的具時間延遲之Retard型式(Retard Type)的CNNs外，本研究亦提出了具時間延遲微分項之Neutral型式(Neutral Type) CNNs之全域漸近收歛穩定的充分條件。
　　
　　然而，當CNNs參數無法滿足一些判斷穩定性的充分條件時，我們雖無法判別系統是否穩定，但若適當的選擇CNNs參數(包括系統參數、延遲時間大小等)，在有限時間和空間範圍內，透過計算系統的李亞普諾夫指數(Lyapunov Exponent)有一個正值的特性，能夠區分其是否具混沌或非混沌的動態行為。混沌運動對初始條件具有敏感依賴性，研究其同步特性將有益於進一步了解複雜的動態系統行為，例如，可以做為系統通訊安全方面的開發工具。因此，本研究亦根據Pecora 和 Carroll 的drive-response 概念(Drive-Response Concept)以及李亞普諾夫穩定理論、Hamiltonian矩陣，分別針對具固定時間延遲和時變時間延遲之混沌CNNs設計出與其指數同步的耦合系統。在實際應用上，特別是高階的CNNs，若遠端神經元的狀態無法即時取得做為回饋信號時，本研究亦提出了非集中式控制法 (Decentralized Control Technique).

　　綜合上述，主要研究成果包括：(一)推導出具固定時間延遲和時變時間延遲之網格型類神經網路穩定性的充分條件，此一結果比文獻上的條件更為廣泛。(二) 針對具固定時間延遲和時變時間延遲之混沌CNNs，設計與其指數同步的耦合系統。(三)針對遠端神經元狀態無法即時取得之高階的混沌CNNs，提出了非集中式控制法。這些結果將可應用於解決實際的問題。

　　In this dissertation, the stability and synchronization problems of Cellular Neural Networks (CNNs) with delays are investigated. A new sufficient condition related to the globally exponential stability of CNNs with/without constant delays is proposed. It is shown that the presented condition is easy to check and is less restrictive than some existing sufficient conditions in the literature. Furthermore, we propose the globally exponential stability condition for CNNs with time-varying delays. The condition is delay-dependent and easily applied only by checking the Hamiltonian matrix with no eigenvalues on the imaginary axis instead of directly solving an algebraic Riccati equation. Moreover, the exponential stability degree is more easily assigned than that reported in the literature. In addition, the globally asymptotic stability of a class of delayed neural networks of the neutral type is also investigated.

　　On the other hand, chaotic neural networks have recently received attention due to the potential capability for secure communication. It is worthy of noting that CNNs can exhibit some complicated dynamics and even chaotic behaviors if the network’s parameters and time delays are appropriately chosen. Therefore, the synchronization problems of CNNs with/without constant or time-varying delays are investigated in the dissertation. Using the drive-response concept and Lyapunov stability theory or Hamiltonian matrix, several sufficient delay independent and dependent exponential synchronization conditions are derived to synchronize two identical chaotic neural networks with/without constant and time-varying delays, respectively. For practical implementation, we also derive the synchronization law of coupled chaotic CNNs with time delays by decentralized control technique.

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