Existence and uniqueness of solutions of piecewise nonlinear systems

Wu, Changzhi; Teo, Kok Lay; Rehbock, Volker; Liu, G.

doi:10.1016/j.na.2009.06.001

Cited by 5 publications

(5 citation statements)

References 18 publications

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“…As already remarked in [73], the conditions of the above theorem are difficult to check because of the infinite number of intersections in thed e fi n i t i o no ft h es e t s T i .T h e r e f o r e ,t h ea u t h o r so f[ 7 3 ]a l s oc o n s i d e r e ds p e c i a lc a s es of switched systems, namely systems with single criterion, where the cells χ i ,i∈ 1,mare described by one and only one inequality. The results in [73] extend to nonlinear systems the results in [74] obtained for piecewise linear systems with multiple modes and multiple criteria, described by lexicographic inequalities. In turn, the work in [74] is an extension of papers [49] and [48], where equivalent conditions for the well-posedness of bimodal linear systems with multiple criteria are investigated.…”

Section: Remark 132 It Is Easy To Verify That For Anysupporting

confidence: 56%

“…. Necessary and sufficient conditions for the well-posedness of system (2) in the sense of Definition 13.1 have been studied in [73]. In the sequel we give a correct version of Theorem 3.1 in [73]) that wrongly states necessaryconditions.…”

Section: Remark 132 It Is Easy To Verify That For Anymentioning

confidence: 99%

“…PWL systems have been studied in several papers: see [49] for bimodal systems with single criterion and [49], [48], [74] for multimodal systems with multiple criteria. Recently, the study of the well-posedness developed in the above mentioned works was successfully extended by [73] to the moregeneralclassofnonlinear systems with multiple modes and multiple criteria. Inthissectionweturnback to the general form of discontinuous system (2).…”

Section: Multimodal Systems With Multiple Criteriamentioning

confidence: 99%

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Switching, relay and complementarity systems: A tutorial on their well-posedness and relationships

Georgescu

Brogliato

Acary

2012

Physica D: Nonlinear Phenomena

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International audienceThis paper reviews the relationships between switching systems defined from a partition of the state space into convex cells, and relay or complementarity dynamical systems, which are other classes of discontinuous systems. Well-posedness results (i.e. results on the existence and the uniqueness of solutions) for different classes of relay and complementarity systems which are also switching systems are presented. The reverse issue (when can a switching system be rewritten equivalently as a relay or a complementarity system) is also tackled. Many examples from Mechanics, Circuits, and Biology illustrate the developments throughout the paper. The paper focuses on systems with continuous solutions (i.e. with no state jumps). Convexity is the central property

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Section: Remark 132 It Is Easy To Verify That For Anysupporting

confidence: 56%

Section: Remark 132 It Is Easy To Verify That For Anymentioning

confidence: 99%

Section: Multimodal Systems With Multiple Criteriamentioning

confidence: 99%

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Switching, relay and complementarity systems: A tutorial on their well-posedness and relationships

Georgescu

Brogliato

Acary

2012

Physica D: Nonlinear Phenomena

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“…= 0 (grazing contact). Lexicographical inequalities are ubiquitous in the analysis of systems which evolve on the boundary of some domain of their state space [6,528,532,569]. They are closely related to the so-called semi-observability cones {x ∈ R n |(Cx, CAx, .…”

Section: Maximal Monotone Differential Inclusionsmentioning

confidence: 99%

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Brogliato¹,

Tanwani²

2020

SIAM Rev.

118

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This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations (ODEs) is coupled with a static and time-varying setvalued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems including sweeping processes, differential inclusions with maximal monotone right-hand side, complementarity systems, differential and evolution variational inequalities, projected dynamical systems, some piecewise linear switching systems. Such mathematical models have seen applications in electrical circuits, mechanical systems, hysteresis effects, and many more. When we impose a passivity assumption on the open-loop system, and regard the set-valued operator in the feedback as maximally monotone, we obtain a set-valued Lur'e dynamical system. In this article we review the mathematical formalisms, their relationships, main application fields, well-posedness (existence, uniqueness, continuous dependence of solutions), and stability of equilibria. An exhaustive bibliography is provided.

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“…The set of ( 14) and ( 18) represents a nonlinear system with the two unknown variables, τ and p c , and, in general, closed form expressions of these variables are not available. Thus, we need to use numerical techniques to find them [24].…”

Section: The Stationary Transmission Probabilitymentioning

confidence: 99%

Joint backoff control in time and frequency for multichannel wireless systems and its Markov model for analysis

Lee¹,

Lim²,

Choi³

2011

Discrete &Amp; Continuous Dynamical Systems - B

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A new joint random access scheme is proposed to enable effective uplink access when users are partially aware of channel conditions in multichannel wireless systems. The proposed scheme mitigates packet collisions with joint backoff control in the time and frequency domains, cooperating with a sensing method that exploits the channel conditions. The performance analysis of the joint access scheme is facilitated by a Markov model that provides a closed-form throughput expression. Simulation results show that this channel access scheme, working together with a simple sensing method, offers significant improvements to system throughput.

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Existence and uniqueness of solutions of piecewise nonlinear systems

Abstract: a b s t r a c tIn this paper, we consider the existence and the uniqueness of solutions of piecewise nonlinear systems. We will present some necessary and sufficient conditions for the existence and the uniqueness of solutions of this class of systems.

Cited by 5 publications

References 18 publications

Switching, relay and complementarity systems: A tutorial on their well-posedness and relationships

Switching, relay and complementarity systems: A tutorial on their well-posedness and relationships

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

Joint backoff control in time and frequency for multichannel wireless systems and its Markov model for analysis

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