Ngoc, Pham Huu Anh; Minh, Nguyen Van; Naito, Toshiki Stability radii of positive linear functional differential systems in Banach spaces. (English) Zbl 1127.34044 Int. J. Evol. Equ. 2, No. 1, 75-97 (2007). The main purpose of this paper is to extend the results in P. H. A. Ngoc and N. K. Son [SIAM J. Control Optim. 43, No. 6, 2278–2295 (2005; Zbl 1090.34061)], which are based on a finite dimensional space \(\mathbb{R}^n\), to a Banach lattice \(X\). The authors consider a linear retarded system described by the following functional differential equation in a Banach lattice \(X\), \[ \dot{x}(t) = A_0x(t) + L x_t, \quad t\geq 0, \quad x(t) \in X, \]\[ x(\theta) = \phi_0(\theta), \quad \theta \in [-h, 0], \] where \(A_0\) is the generator of a compact \(C_0\) semigroup on \(X\), for every \(t\geq 0, x_t \in C([-h, 0], X)\) is defined by \(x_t(\theta) = x(t+\theta)\), \[ L : C([-h, 0], X) \to X, \quad L\phi = \int_{-h}^0 d[\eta(\theta)]\phi(\theta) , \] is a bounded linear operator, and \(\eta(\cdot)\) denotes a \(L(X)\)-valued bounded variation on \([-h, 0]\), which vanishes at \(-h\) and is continuous from the left on \([-h, 0]\). By means of the assumption of the system being Hurwitz stable, a lower and an upper bound for the complex stability radius with respect to multi-perturbations are given. Then, for the class of positive linear retarded systems, it is shown that the complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by an explicit formula. Reviewer: Ti-Jun Xiao (Hefei) Cited in 2 Documents MSC: 34K20 Stability theory of functional-differential equations 34K30 Functional-differential equations in abstract spaces Keywords:linear functional differential system; Banach lattice; positive system; stability radii Citations:Zbl 1090.34061 PDFBibTeX XMLCite \textit{P. H. A. Ngoc} et al., Int. J. Evol. Equ. 2, No. 1, 75--97 (2007; Zbl 1127.34044)