SECOND ORDER SYSTEMS ON HILBERT SPACES WITH NONLINEAR DAMPING

Shantanu Singh; George Weiss

doi:10.1137/22M154199X

SECOND ORDER SYSTEMS ON HILBERT SPACES WITH NONLINEAR DAMPING

Shantanu Singh, George Weiss

School of Electrical Engineering

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems "out of thin air" introduced by Tucsnak and Weiss in 2003. The modified differential equation contains a new nonlinear damping term that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.

Original language	English
Pages (from-to)	2630-2654
Number of pages	25
Journal	SIAM Journal on Control and Optimization
Volume	61
Issue number	4
DOIs	https://doi.org/10.1137/22M154199X
State	Published - 2023

Funding

Funders	Funder number
Israel Science Foundation	3621/21

Keywords

Crandall-Pazy theorem
Lax-Phillips semigroup
Minty's theorem
Rockafellar's theorem
maximal monotone operator
operator semigroup
scattering passive system
well-posed linear system

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10.1137/22M154199X

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title = "SECOND ORDER SYSTEMS ON HILBERT SPACES WITH NONLINEAR DAMPING",

abstract = "We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems {"}out of thin air{"} introduced by Tucsnak and Weiss in 2003. The modified differential equation contains a new nonlinear damping term that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.",

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AU - Weiss, George

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AB - We investigate a special class of nonlinear infinite dimensional systems. These systems are obtained by modifying the second order differential equation that is part of the description of conservative linear systems "out of thin air" introduced by Tucsnak and Weiss in 2003. The modified differential equation contains a new nonlinear damping term that is maximal monotone and possibly set-valued. We show that this new class of nonlinear infinite dimensional systems is incrementally scattering passive (hence well-posed). Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system. We illustrate our result on the n-dimensional wave equation.

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KW - Rockafellar's theorem

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KW - operator semigroup

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SECOND ORDER SYSTEMS ON HILBERT SPACES WITH NONLINEAR DAMPING

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