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This note introduces a new notion of random dynamical system with inputs and outputs, and sketches a small-gain theorem for monotone systems which generalizes a similar theorem known for deterministic systems.

Systems and Control · Computer Science 2013-01-01 Michael Marcondes de Freitas , Eduardo D. Sontag

In this work we consider linear non-autonomous systems of Wazewski type on Hilbert spaces and provide a new approach to study their stability properties by means of a decomposition into subsystems and conditions implied on the…

Dynamical Systems · Mathematics 2025-03-13 Ivan Atamas , Sergey Dashkovskiy , Vitalii Slynko

An asymptotic small parameter expansion of a single Cauchy problem is constructed for a singularly perturbed system of hyperbolic equations describing vibrations of two rigidly connected strings. Equations (such as generalized Korteweg-de…

Analysis of PDEs · Mathematics 2025-10-15 Andrey Nesterov

We consider interconnected nonlinear systems with external inputs, where each of the subsystems is assumed to be input-to-state stable (ISS). Sufficient conditions of small gain type are provided guaranteeing that the interconnection is ISS…

Dynamical Systems · Mathematics 2010-06-14 Sergey Dashkovskiy , Michael Kosmykov , Fabian Wirth

For an ISS system, by analyzing local and non-local properties, it is obtained different input-to-state gains. The interconnection of a system having two input-to-state gains with a system having a single ISS gain is analyzed. By employing…

Optimization and Control · Mathematics 2015-08-12 Humberto Stein Shiromoto , Vincent Andrieu , Christophe Prieur

New trajectory-based small-gain results are obtained for nonlinear feedback systems under relaxed assumptions. Specifically, during a transient period, the solutions of the feedback system may not satisfy some key inequalities that previous…

Optimization and Control · Mathematics 2010-02-25 Iasson Karafyllis , Zhong-Ping Jiang

TheL2-gain characterizes a dynamical system's input-output properties, but can be difficult to determine for nonlinear systems. Previous work designed a nonconvex optimization problem to simultaneously search for a continuous piecewise…

Optimization and Control · Mathematics 2024-10-18 Amy Strong , Reza Lavaei , Leila J. Bridgeman

This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic…

Optimization and Control · Mathematics 2010-09-08 Debasish Chatterjee , Daniel Liberzon

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a…

Quantitative Methods · Quantitative Biology 2007-05-23 David Angeli , Eduardo D. Sontag

Systems that show different characteristics, such as finite-gain and passivity, depending on the nature of the inputs, are said to possess mixed input-output properties. In this paper, we provide a constructive method for characterizing…

Systems and Control · Electrical Eng. & Systems 2024-09-10 Sebastiaan van den Eijnden , Thomas Chaffey

The principle of entropy increase is not only the basis of statistical mechanics, but also closely related to the irreversibility of time, the origin of life, chaos and turbulence. In this paper, we first discuss the dynamic system…

Statistical Mechanics · Physics 2022-10-11 Zou Dan Dan

Convergent, contractive or incremental stability properties of nonlinear systems have attracted interest for control tasks such as observer design, output regulation and synchronization. The convergence property plays a central role in the…

Systems and Control · Electrical Eng. & Systems 2021-05-07 Jiayen Chen , Hendra I. Nurdin

We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which…

Dynamical Systems · Mathematics 2012-12-24 Sergey Dashkovskiy , Andrii Mironchenko

In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory…

Optimization and Control · Mathematics 2007-05-23 David Angeli , Eduardo D. Sontag

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain…

Optimization and Control · Mathematics 2021-07-29 Andrii Mironchenko , Christoph Kawan , Jochen Glück

This paper proposes a new adaptation methodology to find the control inputs for a class of nonlinear systems with time-varying bounded uncertainties. The proposed method does not require any prior knowledge of the uncertainties including…

Optimization and Control · Mathematics 2018-03-16 Yi-Wen Liao , Selina Pan , Francesco Borrelli , J. Karl Hedrick

From the structural perspective, this paper investigates a new formulation of the concept of input-to-state stability (ISS), and based on this formulation, proposes a new stability analysis approach for a class of interconnected system. The…

Systems and Control · Computer Science 2015-05-05 Yong Wang

This paper extends the nonlinear ISS small-gain theorem to a large-scale time delay system composed of three or more subsystems. En route to proving this small-gain theorem for systems of differential equations with delays, a small-gain…

Optimization and Control · Mathematics 2009-11-09 Shanaz Tiwari , Yuan Wang , Zhong-Ping Jiang

In this paper, an asymptotic stability proof for a class of methods for inexact nonlinear model predictive control is presented. General Q-linearly convergent online optimization methods are considered and an asymptotic stability result is…

Optimization and Control · Mathematics 2021-12-01 Andrea Zanelli , Quoc Tran Dinh , Moritz Diehl

Linear parameter-varying (LPV) systems with uncertainty in time-varying delays are subject to performance degradation and instability. In this line, we investigate the stability of such systems invoking an input-output stability approach.…

Systems and Control · Electrical Eng. & Systems 2020-04-10 Shahin Tasoujian , Saeed Salavati , Karolos Grigoriadis , Matthew Franchek