Stability of solutions for nonlinear singular systems with delay

In this paper, we present new definitions of stability of singular systems with delay based on the works of B.Baji'c and M.Mili'c, and establish theorem on stability. Then the sufficient conditions for uniform stability of a class of special nonlinear singular systems with delay are suggested. Keywords: Nonlinear singular systems, singular systems with delay, stability, uniform stability. 1. INTRODUCTION From a modeling point of view, it is perhaps more realistic to model a phenomenon by a singular such as descriptor systems, semi-state space systems, and differentialalgebraic systems, use of which arise frequently in many fields such as optimal control problems, electrical circuits, neutral network, and some population growth models. Some applicable example and basic results can be found in [1, 2]. Singular systems correspond to normal systems. They have both internal logic and essential difference. But the study of singular systems had mostly referred to the given normal systems theories, which had been generalized and transplanted into singular systems so far. The methods of study for singular system are mostly geometric approach, frequency domain method and state-space techniques. And people still have different views related to the questions of singular systems, thus the research achievement of singular systems appears extremely fragmentary. The stability and asymptomatical stability still haven't have uniform distinct definitions, thus there exists confusions for some concepts as well as inconsistencies between the concept and the theorem. The consistency of the initial conditions still has two different views. On the other hand, there often happen impulses and jumps in the solutions. So, the existence and uniqueness of solutions in singular systems haven't been resolved, because the complexity of singular systems and their stability haven't achieved mutual recognition. Therefore, the research for singular systems not only has a widespread practical significance, but also its theoretical value has broad prospects for development. As one of the major research subjects in nonlinear singular systems, the problem of stability attracts many researchers attention, for example [3-5] discussed the stability of *Address correspondence to this author at the Department of Basic Science, North China Institute of Aerospace Engineering, Langfang, Hebei, 065000, P.R. China; Tel: 15831630452; E-mail: [email protected] singular systems. Since delay often occurs in singular systems such as discussed [6], therefore the research on stability of nonlinear singular systems with delay is given much importance in practice and theory. However, few studies have been done on the stability of nonlinear singular systems with delay. Therefore, in this paper we concentrate on this. The discussion on stability of singular systems, compared with that of nonsingular systems, came up with three main new difficulties: the first is that it isn't easy to satisfy the existence and uniqueness of solutions, since the initial conditions may not be consistent; the second is that it is difficult to calculate the derivatives of Lyapunov functions; the third is that there often happen impulses and jumps in the solutions. In order to overcome these difficulties, this paper presents new definitions of stability of nonlinear singular systems with delay based on the previous studies by [7, 8]. Furthermore, the stability theorem of solutions for the following nonlinear singular systems with delay has been established: A ! x(t) = Bx(t)+Cx(t !" )+ f (t,x(t !" )) x t 0 =# $ % & '& (1) where A is an n n × constant singular matrix, and B , C are n! n constant matrices. 0 τ > , f (t,! )"C([0,+#)$C([%& ,0], R ), R ) , f (t,0) = 0 , for any t ! t 0 ! 0 . x t 0 =! is the initial condition of (1), where ! "C([#$ ,0], Rn ) . 2. PRELIMINARIES In this paper, we assume that the solution of initial value problems for system (1) exists which is called nonperturbation solution, written as 0 ( , , ) u x t t φ , sometimes ( ) u x t for short. At first, we introduce the following notations: 608 The Open Automation and Control Systems Journal, 2015, Volume 7 Jing and Chong T k = [0,t k ) , where 0 < t k ! +" ; T 0k (t 0 ) = [t 0 ,t k ) where t 0 !T k . J is an open interval at R , and T k ! J 0 . D is an open interval at n R , ! C (J " D) is the set of all continuous differentiable functions defined as J ! D , q(t,x)! " C (J # R n , R ) . S I (t)! C(["# ,0], Rn ) , S I (t) is a set of all consistency initial functions of (1.1) in 0 [ ), ) k t t τ − ( through (t,! ) at least. S k (t,t k )! S I (t) , and for any ! "S k (t,t k ) there exists a continuous solution of (1.1) in [t,t k ) through (t,! ) at least. ( , ) { ([ ,0], ); }, 0. ( , ) { ; ( , ) ( , ( )) }, 0. n n u B C R Q t x R q t x q t x t φ δ ψ τ ψ φ δ δ ε ε ε = ∈ − − < > = ∈ − < > ‖ ‖ ‖ ‖ In this paper, we suppose x u (t,t 0 ,! ) exists in [t 0 ,t k ) , !t 0 "T k , implies ! "S k (t 0 ,t k ) . Definition 2.1 If !t 0 "T k , !" > 0 , there always exists ! (t 0 ," ) > 0 , such that for !" #B($ ,% )& S k (t 0 ,t k ) , x(t,t 0 ,! ) satisfies that x(t,t 0 ,! )"Q(t,# ) , !t "[t 0 ,t k ) . Then the solution x u (t,t 0 ,! ) is said to be stable in { ( , ), } k q t x T . If δ is only related to ! and has nothing to do with 0 t , then u (t,t0 ,! ) is said to be uniformly stable in {q(t,x),T k }. Definition 2.2 1) If 0 ( , , ) u x t t φ is stable in { ( , ), } k q t x T , where k t = +∞ , and for 0 k t T ∀ ∈ , there exists a 0 ( ) 0 t Δ > , such that 0 0 ( , ( )) ( , ) k B t S t ψ φ ∀ ∈ Δ ∩ +∞ , 0 0 lim ( , ( , , )) ( , ( , , )) 0 u t q t x t t q t x t t ψ φ →+∞ − = ‖ ‖ , then x u (t,t 0 ,! ) is said to be asymptomatically stable in