Stability of Linear Continuous Singular and Discrete Descriptor Systems over Infinite and Finite Time Interval

1.1 Classes of systems to be considered It should be noticed that in some systems we must consider their character of dynamic and static state at the same time. Singular systems (also referred to as degenerate, descriptor, generalized, differential-algebraic systems or semi-state) are those, the dynamics of which are governed by a mixture of algebraic and differential (difference) equations. Recently many scholars have paid much attention to singular systems and have obtained many good consequences. The complex nature of singular systems causes many difficulties in the analytical and numerical treatment of such systems, particularly when there is a real need for their control. It is well-known that singular systems have been one of the major research fields of control theory. During the past three decades, singular systems have attracted much attention due to the comprehensive applications in economics as the Leontief dynamic model (Silva & Lima 2003), in electrical (Campbell 1980) and mechanical models (Muller 1997), etc. Discussion of singular systems originated in 1974 with the fundamental paper of (Campbell et al. 1974) and latter on the anthological paper of (Luenberger 1977). The research activities of the authors in the field of singular systems stability have provided many interesting results, the part of which were documented in the recent references. Still there are many problems in this field to be considered. This chapter gives insight into a detailed preview of the stability problems for particular classes of linear continuous and discrete time delayed systems. Here, we present a number of new results concerning stability properties of this class of systems in the sense of Lyapunov and non-Lyapunov and analyze the relationship between them.