On Stability and Robust Stability of Positive Linear Volterra Equations

On 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.…”

“…In the last few years, problems of positive systems have attracted a lot of attention from researchers, see e.g. [2], [6], [10]- [11], [15]- [16], [36]- [35], [41]- [43].…”

“…[32]- [33], [35]; positive linear Volterra integro-differential system [26], [36] and positive linear Volterra integral system [27]. More precisely, we first introduced various notions of positive system for these classes of systems.…”

$\mu $ -Stability of Nonlinear Positive Systems With Unbounded Time-Varying Delays

“…For positive linear time-delay systems, [8]- [9] show that the presence of constant delays does not affect the stability performance of the system; [10]- [11] (and [12]- [13]) also report this fact for positive Tianping linear systems (and positive systems defined by functional and integro differential equations) of time-varying delays; [14] present some criteria for exponential stability of positive LTI differential systems with distributed delay; [15] gives an extension of the classical Perron-Frobenius theorem to positive quasi-polynomial matrices, then some necessary and sufficient conditions for the exponential stability of positive linear timedelay differential systems are obtained; [16] addresses the asymptotic stability of discrete-time positive systems with bounded time-varying delays and proves that the stability is also determined by the delay-free systems; [17] establishes an equivalent relationship between asymptotical stability and exponential stability for discrete-time positive system for all bounded time-varying delays; [18] studies the asymptotic stability and decay rates with unbounded delays; and for positive linear switched system (PLS), [19]- [20] analyze and prove that for 2-dimensional PLS, the necessary and sufficient condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz, but is not true for 3-dimensional PLS; and [21] also addresses the stability problem of both discrete-time and continuous-time PLSs with arbitrary (even unbounded) time delays.…”

On positivity and stability of linear time-varying Volterra equations

“…The theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius, as references we mention [1] and [4]. In recent time, problems of positive systems have attracted a lot of attention from researchers, see [7]- [10] and the references therein. The characterization of positivity, which we present in Theorem 2.2, generalizes a recent result by [10] where A in (1.1) is constant and the equation is of convolution type.…”