Well-posedness of infinite-dimensional linear systems with nonlinear feedback

Hastir, Anthony; Califano, Federico; Zwart, Hans

doi:10.1016/j.sysconle.2019.04.002

Cited by 21 publications

(12 citation statements)

References 23 publications

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“…At the same time, the study of nonlinear systems with boundary controls is a much younger subject. For some recent references, we refer to [212] and [79]. Stabilization of linear port-Hamiltonian systems by means of nonlinear boundary controllers has been studied in [6].…”

Section: Spectral Methodmentioning

confidence: 99%

Input-to-state stability of infinite-dimensional systems: recent results and open questions

Mironchenko¹,

Prieur²

2019

Preprint

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In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows to estimate the impact of inputs and initial conditions on both the intermediate values and the asymptotic bound on the solutions. ISS has unified the input-output and Lyapunov stability theories and is a crucial property in stability theory of control systems as well as for many applications whose dynamics depend on parameters, unknown perturbations, or other inputs. In this paper, starting from classic results for nonlinear ordinary differential equations, we motivate the study of ISS property for distributed parameter systems. Then fundamental properties are given, as an ISS superposition theorem and characterizations of (global and local) ISS in terms of Lyapunov functions. We explain in detail the functional-analytic approach to ISS theory of linear systems with unbounded input operators, with a special attention devoted to ISS theory of boundary control systems. Lyapunov method is shown to be very useful for both linear and nonlinear models, including parabolic and hyperbolic partial differential equations. Next, we show the efficiency of ISS framework to study stability of large-scale networks, coupled either via the boundary or via the interior of the spatial domain. ISS methodology allows reducing the stability analysis of complex networks, by considering the stability properties of its components and the interconnection structure between the subsystems. An extra section is devoted to ISS theory of time-delay systems with the emphasis on techniques, which are particularly suited for this class of systems. Finally, numerous applications are considered in this survey, where ISS properties play a crucial role in their study. This survey contains recent as well as classical results on systems theory and suggests many open problems throughout the paper.

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Section: Spectral Methodmentioning

confidence: 99%

Input-to-state stability of infinite-dimensional systems: recent results and open questions

Mironchenko¹,

Prieur²

2019

Preprint

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“…In Augner (2018) a nonlinear semigroup approach is used to prove exponential stability of dpH systems by means of nonlinear dissipative feedback. In Hastir et al (2019) a different approach is used to study the interconnection of infinite-dimensional linear systems interconnected with a static nonlinearity. The result can be applied in this framework to show well-posedness of a vibrating string in pH form with a nonlinear damper at the boundary.…”

Section: Well-posedness Stabilization and Control Of Dph Systems As Bcsmentioning

confidence: 99%

Twenty years of distributed port-Hamiltonian systems: a literature review

Rashad

Califano

Schaft

2020

IMA Journal of Mathematical Control and Information

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The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups.

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“…Assumption 3 is meant to be in force in the context of nonlinear anti-damping. 3 Theorem 2 (Exponential stability). Under Assumption 3 and with α = 1, A uniformly and exponentially attracts the bounded sets of X .…”

Section: Stability Analysis Of the Closed-loop Systemmentioning

confidence: 99%

“…Assume temporarily that u is a strong solution. For all τ ≥ 0, denoting by 3 Alternatively, if F has a stabilizing effect, e.g. F (0) = 0, F nonincreasing, it can be relaxed to F locally Lipschitz -with appropriate modifications in the proof of well-posedness.…”

Section: Stability Analysis Of the Closed-loop Systemmentioning

confidence: 99%

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Control of a Wave Equation with a Dynamic Boundary Condition

Vanspranghe

Ferrante

Prieur

2020

2020 59th IEEE Conference on Decision and Control (CDC)

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The general problem of this paper is the analysis of wave propagation in a bounded medium where the uncontrolled boundary obeys a coupled differential equation. More precisely, we study a one-dimensional wave equation with a nonlinear second-order dynamic boundary condition and a Neuman-type boundary control acting on the other extremity. A generic class of nonlinear collocated feedback laws is considered. Hadamard well-posedness is established for the closed-loop system, with initial data lying in the natural energy space of the problem. Moreover, we investigate an example of stabilization through a proportional controller. t 0 g(s) ds for a.e. t in (0, T). Such class f is identified with its continuous

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Well-posedness of infinite-dimensional linear systems with nonlinear feedback

Cited by 21 publications

References 23 publications

Input-to-state stability of infinite-dimensional systems: recent results and open questions

Input-to-state stability of infinite-dimensional systems: recent results and open questions

Twenty years of distributed port-Hamiltonian systems: a literature review

Control of a Wave Equation with a Dynamic Boundary Condition

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