Stability Radii of Linear Systems Under Multi-perturbations

Murakami

Integr. equ. oper. theory

et al. 2009

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integrodifferential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. Mathematics Subject Classification (2000). Primary 45J05; Secondary 34K20, 93D09.

Section: Stability Radius Of Positive Linear Volterra-stieltjes Diffementioning

confidence: 99%

mentioning

confidence: 99%

On Positive Linear Volterra-Stieltjes Differential Systems

Murakami

Integr. equ. oper. theory

et al. 2009

“…[4,5,7,8,17,18,[23][24][25][26][27][28][29][30][31][32][33][38][39][40]. In particular, in recent time, we develop theory of positive systems to some new classes of linear systems such as: positive linear functional differential equations, see e.g.…”

Section: Introductionmentioning

confidence: 99%

On stability of a class of positive linear functional difference equations

Math. Control Signals Syst.

Shin

2007

Self Cite

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron-Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae.

“…Moreover, in general, obtained results of problems for class of positive systems are often very interesting, see e.g. [1], [6], [8]- [9], [11], [12]- [14], [18], [19].…”

Section: Introductionmentioning

confidence: 99%

“…For example, it is wellknown that a linear time-delay system of the form _ x xðtÞ ¼ A 0 xðtÞ þ A 1 xðt À hÞ, t b 0, is positive if and only if A 0 is a Metzler matrix and A 1 is a nonnegative matrix and a linear discrete system of the form xðk þ 1Þ ¼ A 0 xðkÞ þ A 1 xðk À hÞ, k A N, k b h is positive if and only if A 0 , A 1 are nonnegative matrices, see e.g. [13], [18], [14]. However, to the best of our knowledge, there is not any criterion for positive linear neutral delay-di¤erential systems of the form d dt ðxðtÞ À Bxðt À hÞÞ ¼ A 0 xðtÞ þ A 1 xðt À hÞ; t b 0; ð1Þ in the literature.…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Positive Linear Functional Differential Equations

Shin

2007

Self Cite

Abstract. In this paper, we first prove that if a linear neutral functional di¤erential equation is positive then it must degrade into a linear functional di¤erential equation of retarded type. Then, we give some explicit criteria for positive linear functional di¤erential equations. Consequently, we obtain a novel criterion for exponential stability of positive linear functional di¤erential equations.