Nonsmooth Lyapunov functions and discontinuous Carathéodory systems

Bacciotti, Andrea; Ceragioli, Francesca

doi:10.1016/s1474-6670(17)31330-7

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…is convex and differentiable. By Proposition 2 and using the result in (Bacciotti & Ceragioli, 2004), we can conclude that any saddle point is Lyapunov stable. Recall that f (x) has a locally Lipschitz continuous gradient, hence the uniqueness of the Carathéodory solution to (3) can be guaranteed.…”

Section: Convergence Analysismentioning

confidence: 76%

Projected primal–dual gradient flow of augmented Lagrangian with application to distributed maximization of the algebraic connectivity of a network

Zhang

Wei

et al. 2018

Automatica

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In this paper, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization problems that are not necessarily strictly convex. The optimization variables are restricted by a convex set with computable projection operation on its tangent cone as well as equality constraints. As a supplement of the analysis in , we show that the projected dynamical system converges to one of the saddle points and hence finding an optimal solution. Moreover, the problem of distributedly maximizing the algebraic connectivity of an undirected network by optimizing the port gains of each nodes (base stations) is considered. The original semi-definite programming (SDP) problem is relaxed into a nonlinear programming (NP) problem that will be solved by the aforementioned projected dynamical system. Numerical examples show the convergence of the aforementioned algorithm to one of the optimal solutions. The effect of the relaxation is illustrated empirically with numerical examples. A methodology is presented so that the number of iterations needed to reach the equilibrium is suppressed. Complexity per iteration of the algorithm is illustrated with numerical examples.

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Section: Convergence Analysismentioning

confidence: 76%

Projected primal–dual gradient flow of augmented Lagrangian with application to distributed maximization of the algebraic connectivity of a network

Zhang

Wei

et al. 2018

Automatica

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“…The contradiction follows since (27) implies that δ(t) = 0 for all t ∈ (0, t 1 ) due to δ(0) = 0 by construction.…”

Section: When B Is Lower Semicontinuous and C Is Genericmentioning

confidence: 96%

Necessary and sufficient conditions for the nonincrease of scalar functions along solutions to constrained differential inclusions

2022

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In this paper, we propose necessary and sufficient conditions for a scalar function to be nonincreasing along solutions to general differential inclusions with state constraints. The problem of determining if a function is nonincreasing appears in the study of stability and safety, typically using Lyapunov and barrier functions, respectively. The results in this paper present infinitesimal conditions that do not require any knowledge about the solutions to the system. Results under different regularity properties of the considered scalar function are provided. This includes when the scalar function is lower semicontinuous, locally Lipschitz and regular, or continuously differentiable.

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“…This subsection gives a brief overview of several fundamental concepts from discontinuous systems theory that are relevant to this paper. The reader is referred to [19]- [22] for more detailed information.…”

Section: Review Of Discontinuous Systems Theorymentioning

confidence: 99%

“…where I min (x N ) denotes the indices j such that (e j ) T x N = x Nj = m(x N ). By Definition 8, the set of indices N j such that x Nj = m(x N ) is precisely S m (x N ), which by substitution into (28) yields (22). As a final note, observe that since m(•) is locally Lipschitz by Lemma 2, by the Dilation Rule [19] we can derive ∂(−m(x N )) = ∂((−1)m(x N )) = −(∂m(x N )).…”

Section: Proof Of Lemmamentioning

confidence: 99%

Resilient Finite Time Consensus: A Discontinuous Systems Perspective

Usevitch¹,

Panagou²

2020

Preprint

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Many algorithms have been proposed in prior literature to guarantee resilient multi-agent consensus in the presence of adversarial attacks or faults. The majority of prior work present excellent results that focus on discrete-time or discretized continuous-time systems. Fewer authors have explored applying similar resilient techniques to continuoustime systems without discretization. These prior works typically consider asymptotic convergence and make assumptions such as continuity of adversarial signals, the existence of a dwell time between switching instances for the system dynamics, or the existence of trusted agents that do not misbehave. In this paper, we expand the study of resilient continuous-time systems by removing many of these assumptions and using discontinuous systems theory to provide conditions for normally-behaving agents with nonlinear dynamics to achieve consensus in finite time despite the presence of adversarial agents.• We demonstrate that our analysis holds for the general F -local adversarial model on digraphs, which does not assume the presence of any trusted agents. This paper is organized as follows: Section II introduces the notation and problem formulation, Section III presents our main results, Section IV gives simulations demonstrating our method, and Section V gives a brief conclusion.

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Nonsmooth Lyapunov functions and discontinuous Carathéodory systems

Cited by 7 publications

References 17 publications

Projected primal–dual gradient flow of augmented Lagrangian with application to distributed maximization of the algebraic connectivity of a network

Projected primal–dual gradient flow of augmented Lagrangian with application to distributed maximization of the algebraic connectivity of a network

Necessary and sufficient conditions for the nonincrease of scalar functions along solutions to constrained differential inclusions

Resilient Finite Time Consensus: A Discontinuous Systems Perspective

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