Stability and robust stability of positive Volterra systems

Journal of Differential Equations, 2008

This paper addresses the local and global stability of n-dimensional Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks. Necessary and sufficient conditions for local stability independent of the choice of the delay functions are given, by imposing a weak nondelayed diagonal dominance which cancels the delayed competition effect. The global asymptotic stability of positive equilibria is established under conditions slightly stronger than the ones required for the linear stability. For the case of monotone interactions, however, sharper conditions are presented. This paper generalizes known results for discrete delays to systems with distributed delays. Several applications illustrate the results.

Systems & Control Letters, 2016

In this paper we propose new explicit tests for positivity and exponential stability of systems with large time-varying delays. Our approach is based on nonoscillation of solutions of the corresponding diagonal scalar delay differential equations. Numerical examples illustrate the efficiency of the results.

Bulletin of the Polish Academy of Sciences Technical Sciences, 2015

By a novel approach, we get explicit robust stability bounds for positive linear time-invariant time delay differential systems subject to time-varying structured perturbations or non-linear time-varying perturbations. Some examples are given to illustrate the obtained results. To the best of our knowledge, the results of this paper are new.

IET Control Theory & Applications, 2018

This work introduces the internally positive representation of linear time-varying delay differential systems, in the general case of multiple time-varying delays. The technique, previously established for the delay-free case and recently extended to various classes of linear delay systems, aims at building a positive representation of systems whose dynamics is, in general, not definite in sign, in order to export results that only hold for positive systems to arbitrary ones. In the special case of constant matrices, this leads to a simple and easy to check condition for the delay-independent stability of differential systems with multiple time-varying delays. The condition is shown to be less conservative than some well-known conditions available in the literature. Numerical examples are proposed to validate the theoretical results.

SIAM Journal on Control and Optimization, 2009

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parameterization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

Bulletin of the Polish Academy of Sciences Mathematics, 2014

Linear Volterra integro-differential equations with infinite delay are studied. Sufficient conditions for exponential asymptotic stability of linear time-varying equations with delay are given. Then several explicit criteria for exponential asymptotic stability of linear timeinvariant equations are presented. Two examples are given to illustrate the obtained results.

International Journal of Robust and Nonlinear Control, 2009

We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron-Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. POSITIVE LINEAR VOLTERRA DIFFERENCE EQUATIONS 553 for many interesting problems in Mathematics, Physics, Economics, Biology, etc. Moreover, in general, obtained results of problems for classes of positive systems are often very interesting, see . In recent times, problems of positive systems have attracted a lot of attention from researchers, see .