On the Exponential Stability of the Singular Distributed Parameter Systems

Abstract

Exponential stability for the singular distributed parameter systems is discussed in the light of the theory of GE0-semigroup in Hilbert space. The necessary and sufficient conditions concerning the exponential stability are given.

Mixed PD-Type Iterative Learning Control Algorithm for a Class of Parabolic Singular Distributed Parameter Systems

“…differential-algebraic equations was solved based on the Kronecker-Weierstrass form of the matrix pencil. Meanwhile, there are still also other research attentions on SDPSs, such as decoupling [11], controllability and observability [12], passivity criteria [13], exponential stability [14], solvability [15], and so forth.…”

L2 norm-based finite-time stability and boundedness of singular distributed parameter systems with parabolic type

“…Jiang et al 1 resolved the local and global stability problem for SDPSs and the sufficient conditions of global stability were examined in the boyciana-fish ecological system. Based on the semi-group theory in Hilbert space, the exponential stability of the first-order SDPSs was concerned, and the sufficient conditions were derived in the work by Ge et al 18 Most of the existing studies have been done on the system Lyapunov asymptotic stability, which describes the asymptotic behavior of the state trajectory as time approaches infinity. 3,10,12 However, in many practical applications, the primary attention may be paid to the transient response of the systems within a specified time interval.…”

“…Jiang et al 1 resolved the local and global stability problem for SDPSs and the sufficient conditions of global stability were examined in the boyciana–fish ecological system. Based on the semi-group theory in Hilbert space, the exponential stability of the first-order SDPSs was concerned, and the sufficient conditions were derived in the work by Ge et al 18…”

“…First, up to now, most of the current references that tackle the stability of partial differential equations PDEs-based systems are mainly focused on the asymptotic stability, which is defined on an infinite time interval. 3,10,12,14,18 Second, many industrial and engineering processes can be governed by coupled partial differential equations with singular matrix coefficients, and the SDPSs have been of interest from a practical viewpoint. 1–5 Third, to the best of the authors’ knowledge, results on the control problem of FT stability and FT boundedness for SDPSs with parabolic type have not been investigated in the literature yet.…”

Some results for exponential stability of singular distributed parameter systems

“…In 2006 and 2007, Ge, Zhu and Feng [14,15] studied the stabilization of the second order singular distributed parameter control systems by using the theory of operator semigroup on certain conditions in Hilbert space. In 2008, Ge, Zhu and Feng [16] studied exponential stability of the first order singular distributed parameter systems via degenerate semi-group methods in Hilbert space. In 2009, Ge [18] discussed the exponential stability of a class for the singular distributed parameter system by using the theory of GE 0 -semigroup (i.e., strongly continuously generalized operator semigroup).…”