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Stability radii of positive linear functional differential systems in Banach spaces. (English) Zbl 1127.34044

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.

MSC:

34K20 Stability theory of functional-differential equations
34K30 Functional-differential equations in abstract spaces

Citations:

Zbl 1090.34061
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