| Publications about 'small-gain' |
| Articles in journal or book chapters |
| A formalism for the study of random dynamical systems with inputs and outputs (RDSIO) is introduced. An axiomatic framework and basic properties of RDSIO are developed, and a theorem is shown that guarantees the stability of interconnected systems. |
| This paper introduces a small-gain result for interconnected orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations correspond to positive feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders. |
| This paper further develops a method, originally introduced in a paper by Angeli and Sontag, for proving global attractivity of steady states in certain classes of dynamical systems. In this aproach, one views the given system as a negative feedback loop of a monotone controlled system. An auxiliary discrete system, whose global attractivity implies that of the original system, plays a key role in the theory, which is presented in a general Banach space setting. Applications are given to delay systems, as well as to systems with multiple inputs and outputs, and the question of expressing a given system in the required negative feedback form is addressed. |
| We provide an almost-global stability result for a particular chemostat model, in which crowding effects are taken into consideration. The model can be rewritten as a negative feedback interconnection of two monotone i/o systems with well-defined characteristics, which allows the use of a small-gain theorem for feedback interconnections of monotone systems. This leads to a sufficient condition for almost-global stability, and we show that coexistence occurs in this model if the crowding effects are large enough. |
| This paper deals with an almost global attractivity result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides sufficient conditions to rule out oscillatory or more complicated behavior which is often observed in predator-prey systems. |
| A small-gain theorem is presented for almost global stability of monotone control systems which are open-loop almost globally stable, when constant inputs are applied. The theorem assumes "negative feedback" interconnections. This typically destroys the monotonicity of the original flow and potentially destabilizes the resulting closed-loop system. |
| The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail. |
| A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems. |
| Conference articles |
| This note introduces a small-gain result for interconnected MIMO orthant-monotone systems for which no matching condition is required between the partial orders in input and output spaces of the considered subsystems. Previous results assumed that the partial orders adopted would be induced by positivity cones in input and output spaces and that such positivity cones should fulfill a compatibility rule: namely either be coincident or be opposite. Those two configurations corresponded to positive-feedback or negative feedback cases. We relax those results by allowing arbitrary orthant orders. |
| We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs. |
| Internal reports |
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