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We study local stabilization of nonlinear control systems under explicit gain constraints on the feedback law. Using a quantitative refinement of Brockett's openness condition, we introduce the notion of a maximal continuous openness rate…

Optimization and Control · Mathematics 2026-02-23 Bryce Christopherson , Farhad Jafari

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…

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

This paper considers small gain theorems for the global asymptotic and exponential input-to-state stability for discrete time time-delay systems using Razumikhin-type Lyapunov function. Among other things, unlike the existing literature, it…

Systems and Control · Electrical Eng. & Systems 2023-10-06 Yuanqiu Mo , Wenwu Yu , Huazhou Hou , Soura Dasgupta

We provide a detectability analysis for nonlinear large-scale distributed systems in the sense of exponential incremental input/output-to-state stability (i-IOSS). In particular, we prove that the overall system is exponentially i-IOSS if…

Systems and Control · Electrical Eng. & Systems 2026-04-09 Christian Gatke , Julian D. Schiller , Matthias A. Müller

We consider a random network of nonlinear maps exhibiting a wide range of local dynamics, with the links having normally distributed interaction strengths. The stability of such a system is examined in terms of the asymptotic fraction of…

Adaptation and Self-Organizing Systems · Physics 2009-11-10 Sitabhra Sinha , Sudeshna Sinha

Some systems of nonlinear wave equations admit global solutions for all sufficiently small initial data, while others do not. The (classical) null condition guarantees that such a result holds, but it is too strong to capture certain…

Analysis of PDEs · Mathematics 2019-06-06 Joseph Keir

In this paper, we establish the sufficient conditions guaranteeing global uniform exponential stability, or at least global asymptotic stability, of all solutions for nonlinear dynamical systems, also known as global incremental stability…

Dynamical Systems · Mathematics 2022-07-06 Robert Vrabel

The stability of solutions to evolution equations with respect to small stochastic perturbations is considered. The stability of a stochastic dynamical system is characterized by the local stability index. The limit of this index with…

Condensed Matter · Physics 2009-11-07 V. I. Yukalov

The increasing share of converter based resources in power systems calls for scalable methods to analyse stability without relying on exhaustive system wide simulations. Decentralized small gain and small-phase criteria have recently been…

Systems and Control · Electrical Eng. & Systems 2026-04-15 Diego Cifelli , Adolfo Anta

This article aims to investigate sufficient conditions for the stability of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic…

Probability · Mathematics 2023-05-22 Taras Lukashiv , Igor V. Malyk , Maryna Chepeleva , Petr V. Nazarov

The small-signal stability is an integral part of the power system security analysis. The introduction of renewable source related uncertainties is making the stability assessment difficult as the equilibrium point is varying rapidly. This…

Optimization and Control · Mathematics 2018-11-12 Parikshit Pareek , Konstantin Turitsyn , Krishnamurthy Dvijotham , Hung D. Nguyen

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of a configuration. For…

Probability · Mathematics 2024-06-13 Jacob Calvert , Dana Randall

For an underactuated (simple) Hamiltonian system with two degrees of freedom and one degree of underactuation, a rather general condition that ensures its stabilizability, by means of the existence of a (simple) Lyapunov function, was found…

Optimization and Control · Mathematics 2019-04-29 Sergio Grillo , Leandro Salomone , Marcela Zuccalli

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

A necessary and sufficient condition, expressed simply as the DC loop gain (ie the loop gain at zero frequency) being less than unity, is given in this paper to guarantee the internal stability of a feedback interconnection of Linear…

Optimization and Control · Mathematics 2014-01-31 A. Lanzon , I. R. Petersen

We give sufficient conditions for asymptotic stabilization of equilibrium points and periodic orbits of a dynamical system when we add a geometric dissipation of gradient type. We also describe the domain of attraction in the case of…

Mathematical Physics · Physics 2016-05-02 Petre Birtea , Dan Comănescu

A networked output feedback loop subject to packetized transmissions of the output signal is considered. Based on the small gain theorem, an easy-to-use stability criterion covering two important cases is presented. In the first case a…

Systems and Control · Electrical Eng. & Systems 2021-08-24 Martin Steinberger , Martin Horn

We study the input-to-state stability (ISS) of boundary control systems allowing for infinitely many boundary couplings. Using semigroup perturbation theory and the theory of positive linear operators on Banach lattices, we derive a…

Optimization and Control · Mathematics 2026-04-14 Yassine El Gantouh , Jun Zheng , Guchuan Zhu , Dingshi Li

As is well known, classical PID control is ubiquitous in industrial processes, yet a rigorous and explicit design theory for nonlinear uncertain MIMO second-order systems remains underdeveloped. In this paper we consider a class of such…

Systems and Control · Electrical Eng. & Systems 2025-12-03 Tianyou Xiang , Cheng Zhao

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