Nonstationary dissipative evolution equations in a Hilbert space

Владимиров, А. А.

doi:10.1016/0362-546x(91)90061-5

Cited by 59 publications

(49 citation statements)

References 4 publications

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“…Let us recall Minty's Theorem in the setting of Hilbert spaces (see [7,8]). Let be given two maximal monotone operators F 1 and F 2 , we recall the definition of pseudo-distance between F 1 and F 2 introduced by Vladimirov [23] as follows…”

Section: Definitionmentioning

confidence: 99%

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

2020

Set-Valued Var. Anal

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Using a new implicit discretization scheme, we study in this paper the existence and uniqueness of strong solutions for a class of Lur'e dynamical systems where the set-valued feedback depends on both time and state. This work is a generalization of [3] where the time-dependent set-valued feedback is considered to acquire only weak solutions. Obviously, strong solutions and implicit discretization scheme are nice properties, especially for numerical simulation. We also provide some conditions such that the solutions are exponentially attractive. The obtained results can be used to study the time-varying Lur'e systems with errors in data. Our result is new even the set-valued feedback depends only on the time.

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Section: Definitionmentioning

confidence: 99%

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

2020

Set-Valued Var. Anal

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“…Let us recall Minty's Theorem in the setting of Hilbert spaces (see [5,6] Let be given two maximal monotone operators F 1 and F 2 , we recall the definition of pseudo-distance between F 1 and F 2 introduced by Vladimirov [32] as follows…”

Section: Mathematical Backgroundsmentioning

confidence: 99%

“…where m(x 0 ) > 0 is defined in (32) depending only on x 0 , L 1 , L 2 , c A , c f and m(·) is a continuous function w.r.t x 0 . Therefore…”

Section: Mathematical Backgroundsmentioning

confidence: 99%

“…The classical maximal differential inclusions when A is a fixed maximal monotone operator have been fruitfully studied in the literature, see for examples [5,6]. There are also various works for the case of time-dependence, i.e., A t,x ≡ A t (see [13,15,17,26,32] and the reference therein). Among important contributions are sweeping processes [3,16,18,22,23,24,25,30,31], Skorohod problem [29], hysteresis operators [14] and recently Lur'e dynamical systems [1,7,8,9,10,19,27,28].…”

Section: Introductionmentioning

confidence: 99%

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Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

2019

Optimization

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In this paper, we introduce for the first time a class of state-dependent maximal monotone differential inclusions. Then the existence and uniqueness of solutions are obtained by using an implicit discretization scheme and a kind of hypo-monotonicity assumption respectively. In addition, a characterization for nonsmooth Lyapunov pairs associated with such systems is provided. Our result can be applied to study state-dependent sweeping processes and Lur'e dynamical systems. It is new even the involved maximal monotone operators depend only on the time.

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“…The derivation of (6) is based on the operator distance introduced by Vladimirov in [9]. Remark 4.3 It is crucial that we are able to estimate the difference of the solutions to the evolution equations with multivalued nonlinear operators by the function, which solves the linear elasticity problem.…”

Section: Resultsmentioning

confidence: 99%

Homogenization in the Theory of Viscoplasticity

Nesenenko

2005

Proc Appl Math and Mech

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We study the homogenization of the quasistatic initial boundary value problem with internal variables which models the deformation behavior of viscoplastic bodies with a periodic microstructure. This problem is represented through a system of linear partial differential equations coupled with a nonlinear system of differential equations or inclusions. Recently it was shown by Alber [2] that the formally derived homogenized initial boundary value problem has a solution. From this solution we construct an asymptotic solution for the original problem and prove that the difference of the exact solution and the asymptotic solution tends to zero if the lengthscale of the microstructure goes to zero. The work is based on monotonicity properties of the differential equations or inclusions.

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Nonstationary dissipative evolution equations in a Hilbert space

Cited by 59 publications

References 4 publications

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

Homogenization in the Theory of Viscoplasticity

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